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ompound Chemical element chemical energy chemical potential chemical reaction chemical vapor deposition chemistry Chen Ning Yang Christopher Stasheff church bells circle circular detection probability CKM matrix clairvoyance classical electrodynamics classical ory Claude Cohen-Tannoudji Claude E. Shannon Claus Jönsson Clay Math#The Millennium Prize problems Clebsch-Gordan coefficients cients coherence (physics) coherent Coherent state colatitude Cold cathode cold emission collapse of the wavefunction collision Colloid ute compact compact Lie group compact space compactification (physics) complementarity complementarity (physics) complementary mplex projective line Complex_numbers complexity Composite field composite particle Compton effect Compton suppression Compton raction configuration space confinement conical intersection conjugate quantities conjugate transpose connected correlation function bability consistent histories conspiracy theory constants of motion constraint Constructive quantum field theory context contextualism tradiction Control theory convergent series convex hull convex set Cooper pair coordinate coordinate system Coordinates (mathematics) mit correspondence principle cosmic inflation cosmic radiation cosmic ray cosmological constant cotangent bundle coulomb Coulomb riant derivative CP violation CP-symmetry CPT invariance CPT symmetry CPT violation CPT-symmetry creation and annihilation operators cal atom Curie curl Current (electricity) curvature form curvilinear Cyclotron#Mathematics_of_the_cyclotron cymbals d'Alembert operator Politzer David Thouless David Wineland Davisson-Germer experiment de Broglie de Broglie hypothesis de Broglie wavelength de Rham n degeneracy degeneracy (disambiguation) degeneracy pressure degenerate degenerate dwarf Degenerate energy level degenerate gases onal theory Density matrix density of states density operator Density state Depleted uranium derivation derivative DESY detailed balance tion diffraction pattern Diffusion diffusion equation Digital Devil Saga Digital object identifier dimension dimensional analysis dimensional rac sea Dirac string Dirac's constant direct product direct sum direct sum#Direct sum of Hilbert spaces Dirichlet problem Dirk Gently's Displacement operator Dissociation (chemistry) distance distant anticipation Distribution (mathematics) divergence theorem divergent experiment Douglas Adams down quark drag (physics) Driven harmonic motion drop dual space Duality (mathematics) Duncan MacInnes don Edward Teller Edward Witten effective action effective field theory effective theory Effects of nuclear explosions Ehrenfest theorem sambiguation) Einstein's summation convention Einstein-Podolsky-Rosen paradox elastic elastic collision electric Electric charge electric g electrical network electrical potential electrical resistance electrical resonance electricity electrochemistry electrode electrodynamics tic waves electromagnetically induced transparency electromagnetism Electron Electron affinity Electron capture electron charge electron ctron spin electron subshell electron volt electron-degenerate matter electron-neutrino Electronic band structure electronic configuration interaction electroweak theory Elementary charge Elementary particle elementary particles Elitzur-Vaidman bomb-testing problem ellipse rgy spectrum energy states Energy-dispersive_X-ray_spectroscopy Englert Englert-Greenberger duality Enriched uranium Enrico Fermi ean epistemology eponym EPR paradox equation equation of state equations of state equilibrium equipartition theorem equivalence class statistics Erwin Schroedinger Erwin Schrödinger Estermann Euclidean quantum gravity Euclidean space Eugene Wigner Euler-Lagrange citons exemplar existence of God exotic (hadron) Exotic atom exotic baryon exotic meson expectation value expected value experiment olator Fast breeder reactor Fast neutron reactor faster-than-light fate fault tolerant feminist science fiction Fermi energy Fermi gas Fermi gnet Fertile material Feshbach resonance Feshbach-Fano partioning Feynman Feynman diagram Feynman path integral Feynman slash nite potential well first-order phase transition flash memory Flatland Flatterland flavour (particle physics) Fluorescence flutter flux Flux Fourier series Fourier transform Fowler-Nordheim equation fractal fractional statistics frames of reference Franck-Hertz experiment Frank e free states Freeman Dyson freezing frequencies frequency frequency spectrum Frequency-resolved optical gating friction Fritz London vative functional derivative operator functional integral functional integration functional integration (neurobiology) functional integration amental particles fundamental representation Fusion power Futurama Future energy development G-parity G. Johnstone Stoney Galaxy box Gas in a harmonic trap gauge anomaly gauge boson gauge field gauge fixing gauge group gauge invariance gauge invariant gauge Marsden experiment Geissler tube General relativity General Semantics General Theory of Relativity generalized coordinate Generalized geometrical optics geophysical George Alec Effinger George Chapline George Eugene Uhlenbeck George Gamow George Mackey George r_Schwerionenforschung GeV GHZ Gibbs paradox Gibbs state gigabyte Gilbert N. Lewis Glauber P representation global minimum global tion theory gravastar gravitation gravitational coupling constant gravitational field gravitational singularities gravitational wave gravitino ation group velocity Group_representation GURPS Gustav Ludwig Hertz Gyroscopic_precession#Torque-induced_precession Göttingen all effect Hamel basis Hamilton-Jacobi equation Hamilton-Jacobi equations Hamiltonian hamiltonian (quantum mechanics) Hamiltonian c oscillator harmonic series (music) harmonics Hartle-Hawking state Hartree product Hartree theory Hartree-Fock Hartree-Fock method Principle Heisenberg uncertainty relations Heisenberg's microscope Heisenberg's uncertainty principle Heisenbug helium Hellsing HEMT e Hermitian matrix hermitian operator Hermitian operator#Spectral theorem hertz heterostructure hidden variable hidden variable theories ature superconductivity high-energy physics Hilary Putnam Hilbert space Hilbert-Schmidt hill potential Hindu History of nuclear weapons lonomy Homodyne detection Homogeneity_(physics) homogeneous space homotopy group Hopf algebra Horst Ludwig Störmer Howard ogen atom hydrogen molecule Hydrogen-2 hydrogen-like atom hypercube hypercylinder hyperfine structure Hyperion Cantos#Endymion ntity function Igor Tamm image image (mathematics) imaginary number imaginary unit Immanuel Kant impact ionization Impedance Index divisibility infinite potential well Infinitesimal influence information information entropy information retrieval Information theory infrared titute for Theoretical Physics (Frankfurt) instrumentalism insulator integer integrable integral Integral Fast Reactor integrate Integration nal conversion (chemistry) internal conversion coefficient internal symmetry International Phonetic Alphabet International Space Station nt invariant (mathematics) invariant theory Inverse scattering inverse scattering problem Inverse scattering transform inverse-square law y iron irreducible representation irreducible representations Is logic empirical? Isaac Newton Isidor Isaac Rabi Isidor Rabi Islam Islamic er Jacobi identity Jahn-Teller effect Jain James Chadwick James Clerk Maxwell James Edward Zimmerman James Franck James Jeans n Cramer John Dalton John F. Allen John Hasbrouck van Vleck John Searle John Stewart Bell John Strutt, 3rd Baron Rayleigh John von h Willard Gibbs joule joule second Julian Schwinger June 15 June 5 K. K. Darrow Kanada kaon Karl K. Darrow Karl Popper Karl Pribram a formula Klystrode Klystron knowledge Kochen-Specker theorem Korteweg-de Vries equation Kristofer Straub Kronecker delta Ladder flow Landau pole Lande interval rule Landé g-factor Laplace Laplace operator Laplacian Large Hadron Collider Larmor_precession laser ity lens (optics) Lenz Leo Kadanoff Leon Lederman Leon Rosenfeld Leonid Mandelshtam Lepton leptons leptoquark Lester Germer Lester t quantum light wave lightbulb limit (mathematics) limits to computing Lindblad equation line broadening line bundle Line integral linear ransformation Linewidth Linus Pauling Liouville equation Liouville's theorem (Hamiltonian) liquid liquid crystal Liquid drop model Liquid t of isotopes by symbol list of mathematical topics in quantum theory list of noise topics List of nuclear tests List of optical topics list of eories List of topics (scientific method) List_of_particles#Hypothetical_particles lithium local hidden variable theory local maximum local scattering Longitudinal wave loop expansion Lord Rayleigh Lords and Ladies Lords and Ladies (novel) Lorentz covariant Lorentz factor is, 7th duc de Broglie Louis-Victor de Broglie lower bound Lp space LS coupling LS-coupling LSZ formalism Lucretius Ludwig Boltzmann magnetic moment magnetic monopole magnetic potential Magnetic quantum number Magnetic Resonance Imaging magneto-optic effect theory Many-minds interpretation Many-worlds interpretation many-worlds interpretation of quantum mechanics Marshall Stone Martin aster of Mosquiton material science mathematical Mathematical analysis mathematical formulation of quantum mechanics mathematical x mechanics Matrix population models Matrix theory (physics) matrix_(mathematics) matter Max Born Max Planck Max Tegmark Maxwell 1 May 6 MCSCF mean measurable space measure measure (mathematics) measure space measure theory measurement measurement Media:Stark splitting in hydrogen.png medical imaging Meissner effect Mellin transform memories MEMS Mendeleev mental model meson black hole microscope microscopic microscopy microwave Middle Ages Mie theory Millennium (Hellsing)#HJ-Oberstammführer (Warrant xy molecular geometry molecular Hamiltonian Molecular laser isotope separation molecular physics Molecular scattering Molecular term MOSFET Mossbauer effect Mott insulator Multiple scattering multiplicative quantum number Multiverse (science) Muon muon neutrino ademy of Sciences National Bureau of Standards Natural abundance natural satellite natural unit Nearly-free electron model Neil Gaiman on matter Neutron star neutron-degenerate matter neutronium new age New Scientist New York Academy of Sciences New York Times ogen dioxide nitrous oxide no cloning theorem Nobel Prize Nobel Prize for Physics Nobel Prize in Physics Nobel Prizes for Physics nodal ng nondimensionalization nondimensionalization#Quantum harmonic oscillator nonholonomic mapping nonlinear sigma model nonlocal r normalized wavefunction normalizing constant Norman F. Ramsey normed vector space Noumenon Nova (series) November 5 NP-hard sonance Nuclear material Nuclear medicine Nuclear physics Nuclear power Nuclear power plant Nuclear proliferation Nuclear propulsion Nuclear weapon design nuclei nucleon Nucleosynthesis nucleus Nukees NUMB3RS number operator numerical O.R. Lummer Observable nd off shell One-dimensional periodic case one-loop Feynman diagram ontological Ontology open ball open quantum system operational al phenomenon Optical Society of America optical theorem Optics Orbit orbital orbital angular momentum Orbital elements orbital motion rthonormal orthonormal basis Osama bin Laden oscillation oscillator Oskar Klein Osterwalder-Schrader theorem Otto Stern outer product partial derivative partial differential equation partial differential equations partial trace particle particle accelerator particle collider particle cle physics Particle scattering particle statistics particle zoo particles partite partition function partition function (quantum field theory) ntegral Formulation patient pattern patterns Paul Adrien Maurice Dirac Paul Dirac Paul Ehrenfest Paul Gordan Paul Sophus Epstein Pauli unction periodic table periodic table of elements periodicity permittivity permittivity of free space permutation perpendicular perturbation leum Pfund series pharmaceutical Phase (matter) Phase (waves) phase distribution phase noise phase shift phase space phase transition Philosophical interpretation of classical physics Philosophy Philosophy of science phonon Phosphorescence Photocurrent Photodiode mology physical limits to computing physical paradox physical phenomenon Physical property physical quantity Physical Review Letters er Pieter Zeeman pilot wave pin group pion Pioneer plaque planck constant Planck Length Planck mass Planck time Planck units Planck's ding model plum-pudding model Plutonium Poet Laureate Poincare group Poincare symmetry Poincaré symmetry poison Poisson bracket ded Pontryagin duality Portable Document Format position position manifold position operator positive definite positive linear functional gy surface potential theory Potential well potential_energy#Graphical_representation Pound-Rebka falling photon experiment POVM power Darkness (1987 film) Princeton University Princeton University Press Principal quantum number principle principle of complementarity ity distribution probability flux probability measure probability theory Proca equation Process physics processes Product (mathematics) ive space proof of the existence of God propagator property Prophecy (Stargate SG-1) proportionality constant Propositional calculus ychology pudding pure mathematics pure state purification of quantum state Pyotr Leonidovich Kapitsa Pythagoreans Q factor QCD QCD ation of gauge theories quantization_(physics) quantized Quantum quantum entanglement Quantum annealing quantum chaos Quantum puting quantum cosmology Quantum cryptography quantum cybernetics quantum decoherence quantum dot Quantum electrochemistry n quantum gravity quantum gyroscope quantum Hall effect quantum harmonic oscillator Quantum harmonic oscillator#Ladder operator ent quantum mechanic quantum mechanical Quantum Mechanics Quantum Mechanics - simplified Quantum mechanics#Description of the quantum numbers quantum operation Quantum optics quantum particle Quantum Physics quantum process tomography quantum state eory Quantum thermodynamics quantum trajectory representation theories of quantum mechanics quantum tunneling Quantum tunnelling vel) Quark quark matter quark model quark star quark-degenerate matter Quark-gluon plasma quarks quartz quasar quasiparticle Qubit io frequency radio wave radioactive Radioactive decay radioactive nuclei Radioactive waste Radioactivity radiobiology radiofrequency k (mathematics) Rapid single flux quantum Rate distortion theory rationalism ray ray tracing Rayleigh Scattering Rayleigh-Jeans law real m reference frame Reflection (physics) reflexive space refraction Reginald Cahill region Relationship between string theory and quantum d theory relativistic wave equations relativity Relativity physics relativity theory Relic particles remote viewing Renaissance Renninger kernel Hilbert space residue theorem resolution resolution of the identity resonance resonance (disambiguation) Resonant frequency ction#Values at the integers Riemannian manifold Riesz representation theorem Rigged Hilbert space right triangle ring wave guide ripple t Millikan Robert Mills (physicist) Robert Oppenheimer Robert Serber Robertson-Schrödinger relation Rockefeller Foundation Rockefeller cess rubidium Rudolf Grimm rule of thumb rumor Rutherford Rutherford backscattering Rutherford model Rutherford scattering Rydberg c Satyendra Nath Bose scalar scalar field scalar potential scale anomaly scanning SQUID microscope scanning tunnelling microscope equation Schrödinger picture Schrödinger's cat Schrödinger's cat in fiction Schrödinger's Cat trilogy Schrödinger's equation Schumann scientific model scientific notation scientific rigor scintillation screen screw dislocation sea level sea water second second law of Self-energy semantics semiclassical semiclassical gravity semiconductor semiconductor devices semidefinite programming seminar hannon entropy Shannon entropy#Formal definitions shape resonance Sheldon Lee Glashow Shell model Shelter Island (town), New York L sine curve sine-Gordon equation singular spectrum singularity Sir Roger Penrose Skyrmion SLAC Slater determinant Slater-type orbital olid solid angle solid helium solid state physics soliton Solvay Conference Sommerfeld-Wilson-Ishiwara quantization sound sound wave oup special relativity special unitary group Special_relativity spectral line spectral measure spectral theorem spectral theory spectrometer l harmonic spherical harmonics spin Spin (physics) spin (physics)#spin multiplets spin angular momentum spin quantum number spin fission spontaneous parametric down conversion Spontaneous symmetry breaking spooky action at a distance square integrable square tion standard model standard model (basic details) standing wave standing waves Stanford Encyclopedia of Philosophy star Star Trek e statistical mechanics statistics Stefan-Boltzmann law Stellar mass black hole Stephen Donaldson Stephen Hawking Stephen Notley ated scattering Stirling's approximation stochastic stochastic process Stokes theorem Stokes' law Stone's theorem Stone's theorem on ong CP problem strong CP violation strong force strong interaction strong interactions strong nuclear force structures Stuart Kauffman nyaev Zel'dovich effect super-consciousness Super-Kamiokande supercommutator Superconducting superconductive Superconductivity n principle superpotential superselection superselection sector Supersolid superstring theory Supersymmetry supersymmetry breaking ectic space symposium synchronicity synchrotron radiation synonym system T-symmetry T. D. Lee table of Clebsch-Gordan coefficients eter-totter teleology telepathy teleportation temperature tensor tensor category tensor product tensor product#Tensor product of Hilbert uantum Cats The Compass Rose The Elegant Universe The Elementary Particles The Feynman Lectures on Physics The Gap Cycle The l chemistry theoretical physics theory Theory of Everything theory of relativity thermal de Broglie wavelength thermal equilibrium thermal of thermodynamics Thomas Kuhn Thomas Young Thomas Young (scientist) Thomas-Fermi approximation Thomson scattering Thorium Timeline of chemical element discovery Timeline of cosmic microwave background astronomy Timeline of quantum mechanics, molecular n-Oppenheimer-Volkoff limit Tomography Tonks-Girardeau gas top quark topological defect topological dimension topological entropy al angular momentum total angular momentum quantum number trace class trace-class trajectory transactional interpretation Transducer or transition rate transition rule translational invariance translationally invariant transmission coefficient (physics) Transmission electron diode tunnel_(quantum_mechanics) tunneling time Turbid media twentieth century Two Lumps two-body problem two-photon generation operator Uncertainty Uncertainty principle uncertainty principle#One of the theorems uncertainty relation uncountable uncountable set x unitary operator unitary representation unitary representation of a star Lie superalgebra unitary transformation United States National stin University of Tübingen University of Vienna Unobservables Unruh effect Unsolved problems in physics unstable unstable particle up ntum-mechanical vacuum Vaisheshika valence shell Valentine Bargmann vapour pressure variance variational method Variational method ons vector potential vector space Vector space dimension Vector_space velocity vernacular vertex renormalization Very high temperature viscous visual system Vladimir Aleksandrovich Fock Vladimir Fock Voigt volt volume von Klitzing constant von Neumann von Neumann Ritz Walther Bothe Walther Gerlach Ward-Takahashi identity water wave wave equation wave function wave functions Wave interference acket waves Wayne Itano weak decay weak force weak gauge boson Weak interaction weak interactions weak measurement weak nuclear Dwarf white dwarf material white noise whole number Wick rotation Wiener measure Wiener process Wightman axioms Wigner 3-j symbol ibson (novelist) William Rowan Hamilton Willis Lamb Willoughby Smith winding number wiretap WKB approximation Wojciech H. Zurek eraction Z boson Zagreus (Doctor Who audio) Zeeman effect Zeno's paradoxes#The arrow paradox zero-point energy zig-zag zinc sulfide

ics Bound state Bra-ket notation Breit equation t Clebsch-Gordan coefficients Coherent state ables Compton scattering Compton wavelength ion and annihilation operators Dark energy star energy level Degenerate matter Delayed choice cal space Dirac operator Double-slit experiment ctronic density Electronic Hamiltonian Electronic ess EP Quantum Mechanics Excited state Exotic gas Fermi liquid Fermi's golden rule Fermi-Dirac ock matrix Fock space Fock state Franck-Hertz ger-Marsden experiment Gibbs paradox nics) Heisenberg picture Hilbert space Hydrogen plicate and Explicate Order according to David Internal conversion Interpretation of quantum ffect Klein-Gordon equation Ladder operators d equation London moment Many-body problem nics Matrix model Maxwell-Boltzmann statistics Multiplicative quantum number Neutral particle physics Observable Oil-drop experiment Open ox Particle in Adiabatic a one-dimensional lattice (periodic invariant ic potential Path integral formulation Penrose uantum mechanics) Photoelectric effect Planck Plum pudding model Position operator Potential Zeeman effect ility current Projective Hilbert space Pure gauge ology Quantum chaos Quantum Critical Point on Quantum foam Quantum Hall effect Quantum Quantum level Quantum mechanics Quantum antum phase transition Quantum solid Quantum Quantum Theory Parallels to Consciousness uantum well Quantum Zeno effect Quasistability guide Ritz method Rutherford model Rutherford ory Schrödinger equation Schrödinger picture riational principle Selection rule Semiclassical struction Slater determinant Spin-1/2 Spin-orbital effect Stationary state Stern-Gerlach experiment upersymmetric quantum mechanics T-symmetry antum number Transformation theory (quantum To Scilesco astrophe Uncertainty principle Unitarity Unitarity rbation theory Wave packet Wave-particle duality istribution Wigner-Eckart theorem Work function

Quantum Mechanics

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Quantum Mechanics

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Adiabatic invariant -

Zeeman effect

compiled by To Scilesco

Quantum Mechanics Compiled by: To Scilesco BookId: dcchcaruqoqeqfro

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Date: 14.07.2006

All articles and pictures of this book were retrieved from the Wikipedia Project (wikipedia.org) on 02.07.2006. The articles are free to use under the terms of the GNU Free Documentation License. A copy of this license is included in the section entitled "GNU Free Documentation License". Images in this book have diverse licenses and you can find a list of figures and the corresponding licenses in the section "List of Figures". The version history of all articles can be retrieved from wikipedia.org. Each Article in this book has a reference to the original article. The principal authors of articles are referenced at the end of each article unless technical difficulties did not allow for a proper determination of the principal authors.

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Articles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Adiabatic invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Adiabatic theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Atomic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Auger electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Bargmann’s limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Bohr model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Boltzmon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Born probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Bose–Einstein condensate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Bose-Einstein statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 Bose gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Bound state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Bra-ket notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Breit equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 Canonical commutation relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Chladni’s law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Classical limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Coherent state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Complementarity (physics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Complete set of commuting observables . . . . . . . . . . . . . . . . . . . . . . . 67 Compton scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 Compton wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Conjugate variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Constraint algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 Coupling constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Creation and annihilation operators . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Dark energy star . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Davisson-Germer experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 De Broglie hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Degenerate energy level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Degenerate matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Delayed choice quantum eraser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 Diabatic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Dirac equation in the algebra of physical space . . . . . . . . . . . . . . . . . 98 Dirac operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Double-slit experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 Duru-Kleinert transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Ehrenfest theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

114 114 115 117 117 124 124 126 127 129 129 130 130 138 141 142 144 145 146 149 150 151 153 153 155 157 163 165 167 168 172 174 182 187 191 194 195 207 210 211 212

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Einselection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electronic state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elementary particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy level splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Entanglement witness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EP Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excited state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exotic hadron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Faddeev equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fano resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi-Dirac statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi liquid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fermi’s golden rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Field emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite potential well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flux quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fock matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fock state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Franck-Hertz experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Free particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Functional integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Geiger-Marsden experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gibbs paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Greenberger-Horne-Zeilinger state . . . . . . . . . . . . . . . . . . . . . . . . . Hamiltonian (quantum mechanics) . . . . . . . . . . . . . . . . . . . . . . . . . Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hydrogen-like atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hyperfine structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Imaginary time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implicate and Explicate Order according to David Bohm . . . . . . . Incompleteness of quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . Interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Internal conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpretation of quantum mechanics . . . . . . . . . . . . . . . . . . . . . . .

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Introduction to quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . Josephson effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Klein-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ladder operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Laplace-Runge-Lenz vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Area Neutron Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lindblad equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . London moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mathematical formulation of quantum mechanics . . . . . . . . . . . . Matrix mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell-Boltzmann statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Measurement in quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . Molecular Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Multiplicative quantum number . . . . . . . . . . . . . . . . . . . . . . . . . . . . Neutral particle oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normalisable wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nuclear physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Oil-drop experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Open quantum system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optical theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Parity (physics) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle in a box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle in a one-dimensional lattice (periodic potential) . . . . . . . Particle in a ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle in a spherically symmetric potential . . . . . . . . . . . . . . . . . Path integral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Penrose Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Peres-Horodecki criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbation theory (quantum mechanics) . . . . . . . . . . . . . . . . . . . Photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planck particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planck postulate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Planck’s law of black body radiation . . . . . . . . . . . . . . . . . . . . . . . . Plum pudding model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential energy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Potential well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

393 399 400 403 404 404 409 410 410 411 413 414 421 436 438 440 442 451 458 459 459 477 485 486 486 487 490 494 495 497 497 503 504 505 506 507 507 508 509 512 514

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POVM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probability current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Projective Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pure gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum 1/f noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum chaos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Critical Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum fluctuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum foam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum indeterminacy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum leap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum mechanics, philosophy and controversy . . . . . . . . . . . . Quantum mineralogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum solid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum statistical mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum superposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Theory Parallels to Consciousness . . . . . . . . . . . . . . . . . Quantum tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Zeno effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . QWiki . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Range criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ring wave guide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ritz method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

516 517 518 521 522 524 531 532 540 545 546 548 549 552 555 556 557 559 560 564 571 576 581 582 588 592 594 596 601 603 607 608 609 616 617 619 630 630 631 633 634

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Rutherford model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rutherford scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rydberg formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scattering theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schrödinger picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schrödinger’s cat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schrödinger’s cat in fiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schwinger’s variational principle . . . . . . . . . . . . . . . . . . . . . . . . . . . Selection rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semiclassical . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Separable states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shelter Island Conference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single particle reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slater determinant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-1/2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-orbital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Squashed entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Squeezed coherent state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SQUID . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stark effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stationary state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stern-Gerlach experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subatomic particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superdense coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Superselection sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Supersymmetric quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . Thermal de Broglie wavelength . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological quantum number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transformation theory (quantum mechanics) . . . . . . . . . . . . . . . . T-symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two interfering electron wave-packets . . . . . . . . . . . . . . . . . . . . . . Ultraviolet catastrophe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uncertainty principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unitarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unitarity bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variational method (quantum mechanics) . . . . . . . . . . . . . . . . . . . Variational perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

640 644 652 656 656 661 663

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List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Wave packet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wave-particle duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wien’s displacement law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wigner-Eckart theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wigner quasi-probability distribution . . . . . . . . . . . . . . . . . . . . . . . Work function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Adiabatic invariant

FT

An adiabatic invariant in general is a property of motion which is conserved to exponential accuracy in the small parameter representing the typical rate of change of the gross properties of the body. For periodic motion, the adiabatic H invariants are the action integrals p dq taken over a period of the motion. These are constants of the motion and remain so even when changes are made in the system, as long as the changes are slow compared to the period of motion. In plasma physics there are three adiabatic invariants of charged particle motion.

The first adiabatic invariant, µ The magnetic moment of a gyrating particle, µ=

1 2 2 mv⊥

B

,

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is a constant of the motion (as long as q/m does not change). In fact, it is invariant to all orders in an expansion in ω/ωc , so the magnetic moment remains nearly constant even for changes at rates approaching the gyrofrequency. There are some important situations in which the magnetic moment is not invariant:

Magnetic pumping: When µ is constant, the perpendicular particle energy is proportional to B, so the particles can be heated by increasing B, but this is a ’one shot’ deal because the field cannot be increased indefinitely. On the other hand, if the collision frequency is larger than the pump frequency, µ is no longer conserved. In particular, collisions allow net heating by transferring some of the perpendicular energy to parallel energy. • Cyclotron heating: If B is oscillated at the cyclotron frequency, the condition for adiabatic invariance is violated and heating is possible. In particular, the induced electric field rotates in phase with some of the particles and continuously accelerates them. • Magnetic cusps: The magnetic field at the center of a cusp vanishes, so the cyclotron frequency is automatically smaller than the rate of any changes. Thus the magnetic moment is not conserved and particles are scattered relatively easily into the loss cone. •

Adiabatic invariant

2

The second adiabatic invariant, J

FT

The longitudinal invariant of a particle trapped in a magnetic mirror, Rb J = a v|| ds, where the integral is between the two turning points, is also an adiabatic invariant. This guarantees, for example, that a particle in the ionosphere moving around the Earth will always return to the same line of force. The adiabatic condition is violated in transit-time magnetic pumping, where the length of a magnetic mirror is oscillated at the bounce frequency, resulting in net heating.

The third adiabatic invariant, Φ

The total magnetic flux Φ enclosed by a drift surface is the third adiabatic invariant, associated with the periodic motion of mirror-trapped particles drifting around the axis of the system. Because this drift motion is relatively slow, Φ is often not conserved in practical applications.

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External links • •

lecture notes on the second adiabatic invariant 1 lecture notes on the third adiabatic invariant 2

Source: http://en.wikipedia.org/wiki/Adiabatic_invariant

Principal Authors: Art Carlson, MathMartin, SimonP, Gurch, Linas

Adiabatic theorem

The adiabatic theorem is an important theorem in quantum mechanics which provides the foundation for perturbative quantum field theory. There are different versions of this theorem. Max Born and V. A. Fock proved the original version in 1928: A physical system remains in its instantaneous eigenstate if a given perturbation is acting on it slowly enough and if there is a gap between the eigenvalue and the rest of the Hamiltonian’s spectrum.

1 http://farside.ph.utexas.edu/teaching/plasma/lectures/node24.html 2 http://farside.ph.utexas.edu/teaching/plasma/lectures/node25.html

Adiabatic theorem

3

FT

To be more precise, the adiabatic theorem does not tell us that there is any finite lower bound for the duration over which we have to perform a perturbation on the system in order to keep it in its instantaneous eigenstate. It just tells that this is the case if the rate of change approaches zero! In 1990 J. E. Evron and A. Elgart found a new version of the adiabatic theorem that does not require gaps.

External links and references •

J. E. Evron, A. Elgart: Adiabatic Theorem without a Gap Condition 3

Source: http://en.wikipedia.org/wiki/Adiabatic_theorem

Principal Authors: Artur adib, Deco, SeventyThree, Conscious, Charles Matthews, BeteNoir

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Aharonov-Bohm effect

The Aharonov-Bohm effect, sometimes called the Ehrenberg-SidayAharonov-Bohm effect, is a quantum mechanical phenomenon by which a charged particle is affected by electromagnetic fields in regions from which the particle is excluded. The earliest form of this effect was predicted by Werner Ehrenberg and R.E. Siday in 1949, and similar effects were later rediscovered by Aharonov and Bohm in 1959. Such effects are predicted to arise from both magnetic fields and electric fields, but the magnetic version has been easier to observe. In general, the profound consequence of Aharonov-Bohm effects is that knowledge of the classical electromagnetic field acting locally on a particle is not sufficient to predict its quantum-mechanical behavior. After the 1959 paper was published, Bohm was informed that the effect had been predicted by Rory E. Siday and Werner Ehrenberg a decade earlier; Bohm and Aharonov duly cited this in their second paper (Peat, 1997, p. 192). The most commonly described case, sometimes called the Aharonov-Bohm solenoid effect, is when the wave function of a charged particle passing around a long solenoid experiences a phase shift as a result of the enclosed magnetic field, despite the magnetic field being zero in the region through which the particle passes. This phase shift has been observed experimentally by its effect on interference fringes. (There are also magnetic Aharonov-Bohm

3 http://www.arxiv.org/abs/math-ph/9805022/

Aharonov-Bohm effect

4

FT

effects on bound energies and scattering cross sections, but these cases have not been experimentally tested.) An electric Aharonov-Bohm phenomenon was also predicted, in which a charged particle is affected by regions with different electrical potentials but zero electric field, and this has also seen experimental confirmation. A separate "molecular" Aharonov-Bohm effect was proposed for nuclear motion in multiply-connected regions, but this has been argued to be essentially different, depending only on local quantities along the nuclear path (Sjöqvist, 2002). A general review can be found in Peshkin and Tonomura (1989).

Magnetic Aharonov-Bohm effect

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The magnetic Aharonov-Bohm effect can be seen as a result of the requirement that quantum physics be invariant with respect to the gauge choice for the vector potential A. This implies that a particle with charge q travelling along some path P in a region with zero magnetic field (B = 0 = ∇ × A) must acquire a phase φ; given in SI units by R φ = ~q P A · dx, with a phase difference ∆φ between any two paths with the same endpoints therefore determined by the magnetic flux Φ through the area between the paths (via Stokes theorem and ∇ × A = B), and given by: ∆φ =

qΦ ~ .

This phase difference can be observed by placing a shielded solenoid between the slits of a double-slit experiment (or equivalent). A shielded solenoid encloses a magnetic field B, but does not produce any magnetic field outside of its cylinder, and thus the charged particle (e.g. an electron) passing outside experiences no classical effect. However, there is a (curl-free) vector potential outside the solenoid with an enclosed flux, and so the relative phase of particles passing through one slit or the other is altered by whether the magnetically shielded solenoid current is turned on. This corresponds to an observable shift of the interference fringes on the observation plane. The same phase effect is responsible for the quantized-flux requirement in superconducting loops. This quantization is due to the fact that the superconducting wave function must be single valued: its phase difference ∆φ around a closed loop must be an integer multiple of 2π (with the charge q=2e for the electron Cooper pairs), and thus the flux Φ must be a multiple of h /2e. The superconducting flux quantum was actually predicted prior to Aharonov and Bohm, by London (1948) using a phenomenological model. Aharonov-Bohm effect

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5

Figure 1 Schematic of double-slit experiment in which Aharonov-Bohm effect can be observed: electrons pass through two slits, interfering at an observation screen, with the interference pattern shifted when a magnetic field B is turned on in the shielded cylindrical solenoid.

The magnetic Aharonov-Bohm effect is also closely related to Dirac’s argument that the existence of a magnetic monopole necessarily implies that both electric and magnetic charges are quantized. A magnetic monopole implies a mathematical singularity in the vector potential, which can be expressed as an infinitely long Dirac string of infinitesimal diameter that contains the equivalent of all of the 4πg flux from a monopole "charge" g. Thus, assuming the absence of an infinite-range scattering effect by this arbitrary choice of singularity, the requirement of single-valued wave functions (as above) necessitates chargequantization: 2qg/c~ must be an integer (in cgs units) for any electric charge q and magnetic charge g. The magnetic Aharonov-Bohm effect was experimentally confirmed by Osakabe et al. (1986), following earlier work summarized in Olariu and Popèscu (1984). Its scope and application continues to expand. Webb et al. (1985) demonstrated Aharonov-Bohm oscillations in ordinary, non-superconducting metallic rings; for a discussion, see Schwarzschild (1986) and Imry & Webb (1989). Bachtold et al. (1999) detected the effect in carbon nanotubes; for a discussion, see Kong et al. (2004).

Aharonov-Bohm effect

6

Electric Aharonov-Bohm effect

FT

Just as the phase of the wave function depends upon the magnetic vector potential, it also depends upon the scalar electric potential. By constructing a situation in which the electrostatic potential varies for two paths of a particle, through regions of zero electric field, an observable Aharonov-Bohm interference phenomenon from the phase shift has been predicted; again, the absence of an electric field means that, classically, there would be no effect.

From the →Schrödinger equation, the phase of an eigenfunction with energy E goes as exp(−iEt/~). The energy, however, will depend upon the electrostatic potential V for a particle with charge q. In particular, for a region with constant potential V (zero field), the electric potential energy qV is simply added to E, resulting in a phase shift: ∆φ = − qV~ t ,

where t is the time spent in the potential.

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The initial theoretical proposal for this effect suggested an experiment where charges pass through conducting cylinders along two paths, which shield the particles from external electric fields in the regions where they travel, but still allow a varying potential to be applied by charging the cylinders. This proved difficult to realize, however. Instead, a different experiment was proposed involving a ring geometry interrupted by tunnel barriers, with a bias voltage V relating the potentials of the two halves of the ring. This situation results in an Aharonov-Bohm phase shift as above, and was observed experimentally in 1998.

Mathematical interpretation

In the terms of modern differential geometry, the Aharonov-Bohm effect can be understood to be the holonomy of the complex-valued line bundle representing the electromagnetic field. The connection on the line bundle is given by the electromagnetic potential A, and thus the electromagnetic field strength is the curvature of the line bundle F =dA. The integral of A around a closed loop is the holonomy, which, by Stokes theorem, is the magnetic field threading the loop. Thus the wave function of the electron can be seen to be directly coupled to the complex line bundle representing the electromagnetic field. See also a related effect, the Berry phase.

Aharonov-Bohm effect

7

References

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•

• • •

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Aharonov, Y. and D. Bohm, "Significance of electromagnetic potentials in quantum theory," Phys. Rev. 115, 485–491 (1959). Bachtold, A., C. Strunk, J. P. Salvetat, J. M. Bonard, L. Forro, T. Nussbaumer and C. Schonenberger 4, “Aharonov-Bohm oscillations in carbon nanotubes”, Nature 397, 673 (1999). Ehrenberg, W. and R. E. Siday, "The Refractive Index in Electron Optics and the Principles of Dynamics," Proc. Phys. Soc. London Sect. B 62, 8–21 (1949). Imry, Y. and R. A. Webb, "Quantum Interference and the Aharonov-Bohm Effect," Scientific American, 260(4), April 1989. Kong, J., L. Kouwenhoven, and C. Dekker, "Quantum change for nanotubes", Physics Web 5 (July 2004). London, F. "On the problem of the molecular theory of superconductivity," Phys. Rev. 74, 562–573 (1948). Murray, M. Line Bundles 6, (2002). Olariu, S. and I. Iovitzu Popèscu, "The quantum effects of electromagnetic fluxes," Rev. Mod. Phys. 57, 339–436 (1985). Osakabe, N., T. Matsuda, T. Kawasaki, J. Endo, A. Tonomura, S. Yano, and H. Yamada, "Experimental confirmation of Aharonov-Bohm effect using a toroidal magnetic field confined by a superconductor." Phys Rev A. 34(2): 815-822 (1986). Abstract and full text. 7 Peat, F. David 8, Infinite Potential: The Life and Times of David Bohm (Addison-Wesley: Reading, MA, 1997). ISBN 0-201-40635-7. Peshkin, M. 9 and Tonomura, A., The Aharonov-Bohm effect (SpringerVerlag: Berlin, 1989). ISBN 3-540-51567-4. Schwarzschild, B. "Currents in Normal-Metal Rings Exhibit Aharonov-Bohm Effect." Phys. Today 39, 17–20, Jan. 1986. Sjöqvist, E. "Locality and topology in the molecular Aharonov-Bohm effect," Phys. Rev. Lett. 89 (21), 210401/1–3 (2002).

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•

•

4 http://pages.unibas.ch/phys-meso/ 5 http://physicsweb.org/articles/world/17/7/3/1

6 http://www.maths.adelaide.edu.au/people/mmurray/dg99/line_bundles.pdf 7 http://prola.aps.org/abstract/PRA/v34/i2/p815_1 8 http://www.fdavidpeat.com/ 9 http://www.phy.anl.gov/theory/staff/mp.html

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van Oudenaarden, A., M. H. Devoret, Yu. V. Nazarov, and J. E. Mooij, "Magneto-electric Aharonov-Bohm effect in metal rings," Nature 391, 768– 770 (1998). • Webb, R., S. Washburn, C. Umbach, and R. Laibowitz. Phys. Rev. Lett. 54, 2696 (1985).

Source: http://en.wikipedia.org/wiki/Aharonov-Bohm_effect Principal Authors: Stevenj, Linas, Liontooth, Reddi, CYD

Atomic theory

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In chemistry and physics, atomic theory is a theory of the nature of matter. It states that all matter is composed of atoms. The philosophical background of the atomic theory is called atomism. The theory applies to the common phases of matter, namely solids, liquids and gases, as directly experienced on Earth. Strictly speaking, it is not the appropriate theory for plasmas or neutron stars where unusual environments such as extremes of temperature or density prevent atoms from forming.

Importance

Arguably, the atomic theory is one of the most important theories in the history of science, with wide-ranging implications for both pure and applied science. The theory is largely credited to John Dalton, an 18th- and 19th century British chemist and physicist. Modern chemistry (and biochemistry) is based upon the theory that all matter is made up of atoms of different elements, which cannot be transmuted by chemical means. In turn, chemistry has allowed for the development of the pharmaceutical industry, the petrochemical industry, and many others. Much of thermodynamics is understandable in terms of kinetic theory, whereby gases are considered to be made up of either atoms or molecules, behaving in accordance with Newton’s laws of motion. This was, in turn, a large driving force behind the industrial revolution. Indeed, many macroscopic properties of matter are best understood in terms of atoms. Other examples include friction, material science and semiconductor theory. The latter is particularly important, as it is the foundation of electronics.

Atomic theory

9

Historical precursors Main article: Atomism

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Early atomism

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From the 6th century BC, Hindu, Buddhist and Jaina philosophers in ancient India developed the earliest atomic theories. The first philosopher who formulated ideas about the atom in a systematic manner was Kanada who lived in the 6th century BC. Another Indian philosopher, Pakudha Katyayana who also lived in the 6th century BC and was a contemporary of Gautama Buddha, had also propounded ideas about the atomic constitution of the material world. Indian atomists believed that an atom could be one of up to six elements, with each element having up to 24 properties. They developed detailed theories of how atoms could combine, react, vibrate, move, and perform other actions, and had particularly elaborate theories of how atoms combine, which explains how atoms first combine in pairs, and then group into trios of pairs, which are the smallest visible units of matter. This parallels with the structure of modern atomic theory, in which pairs or triplets of supposedly fundamental quarks combine to create most typical forms of matter. They had also suggested the possibility of splitting an atom which, as we know today, is the source of atomic energy. (See Indian atomism for more details.) Democritus and Leucippus, Greek philosophers in the 5th century BC, presented a theory of atoms. (See Atomism for more details.) The Greeks believed that atoms were all made of the same material but had different shapes and sizes, which determined the physical properties of the material. For instance, the atoms of a liquid were thought to be smooth, allowing them to slide over each other. None of these ideas, however, were founded in scientific experimentation. During the Middle Ages (the Islamic Golden Age), Islamic atomists develop atomic theories that represent a synthesis of both Greek and Indian atomism. (See Islamic atomism for more details.) Older Greek and Indian ideas were further developed by Islamic atomists, along with new Islamic ideas, such as the possibility of there being particles smaller than an atom. As Islamic influence began spreading through Europe, the ideas of Islamic atomism, along with the older ideas of Greek and Indian atomism, spread throughout Europe by the end of the Middle Ages, where modern atomic theories began taking shape.

Birth of modern atomic theory

In 1808, John Dalton proposed that an element is composed of atoms of a single, unique type, and that although their shape and structure was immutable, Atomic theory

10

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atoms of different elements could combine to form more complex structures (chemical compounds). He deduced this after the experimental discovery of the law of multiple proportions — that is, if two elements form more than one compound between them, then the ratios of the masses of the second element which combine with a fixed mass of the first element will be ratios of small whole numbers.

The experiment in question involved combining nitrous oxide (NO) with oxygen (O 2). In one combination, these gases formed dinitrogen trioxide (N 2O 3), but when he repeated the combination with double the amount of oxygen (a ratio of 1:2), they instead formed nitrogen dioxide (NO 2). 4NO + O 2 → 2N 2O 3 4NO + 2O 2 → 4NO 2

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Atomic theory conflicted with the theory of infinite divisibility, which states that matter can always be divided into smaller parts. In 1827, biologist Robert Brown observed that pollen grains floating in water constantly jiggled about for no apparent reason. In 1905, Albert Einstein theorised that this Brownian motion was caused by the water molecules continuously knocking the grains about, and developed a mathematical theory around it. This theory was validated experimentally in 1911 by French physicist Jean Perrin.

Discovery of subatomic particles

For much of this time, atoms were thought to be the smallest possible division of matter. However, in 1897, J.J. Thomson published his work proving that cathode rays are made of negatively charged particles (electrons). Since cathode rays are emitted from matter, this proved that atoms are made up of subatomic particles and are therefore divisible, and not the indivisible atomos postulated by Democritus. Physicists later invented a new term for such indivisible units, "elementary particles", since the word atom had come into its common modern use.

Study of atomic structure

At first, it was believed that the light electrons were distributed in rings or other orbits in a more or less uniform sea or cloud of positive charge (the plum pudding model). However, an experiment conducted in 1909 by colleagues of Ernest Rutherford demonstrated that atoms have a most of their mass and also their positive charge concentrated in a very small fraction of their volume, a region which Rutherford assumed to be at the very center of the atom. In the gold foil experiment, alpha particles (emitted by polonium) were shot through a sheet of gold (striking a fluorescent screen on the other side). The

Atomic theory

11

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experimenters expected all the alpha particles to pass through without significant deflection, given the uniform distribution of positive charge in the plum pudding model. On the contrary, about 1 in 8000 of the alpha particles were heavily deflected (by more than 90 degrees). This led Rutherford to propose the planetary model of the atom in which pointlike electrons orbited in the space around a massive compact nucleus like planets orbiting the Sun. The nucleus was later discovered to contain protons, and further experimentation by Rutherford found that the nuclear mass of most atoms surpassed that of the protons it possessed; this led him to postulate the existence of neutrons, whose existence would be proven in 1932 by James Chadwick.

The planetary model of the atom still had shortcomings. First, a moving electric charge emits electromagnetic waves; according to classical electromagnetism, an orbiting charge would steadily lose energy and spiral towards the nucleus, colliding with it in a tiny fraction of a second. Second, the model did not explain why excited atoms emit light only in certain discrete spectra.

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Quantum theory revolutionized physics at the beginning of the 20 th century when Max Planck and Albert Einstein postulated that light energy is emitted or absorbed in fixed amounts known as quanta. In 1913, Niels Bohr used this idea in his →Bohr model of the atom, in which the electrons could only orbit the nucleus in particular circular orbits with fixed angular momentum and energy. They were not allowed to spiral into the nucleus, because they could not lose energy in a continuous manner; they could only make quantum leaps between fixed energy levels. Bohr’s model was extended by Arnold Sommerfeld in 1916 to include elliptical orbits, using a quantization of generalized momentum. The ad hoc Bohr-Sommerfeld model was extremely difficult to use, but it made impressive predictions in agreement with certain spectral properties. However, the model was unable to explain multielectron atoms, predict transition rates or describe fine and hyperfine structure. In 1925, Erwin Schrödinger developed a full theory of quantum mechanics, described by the →Schrödinger equation. Together with Wolfgang Pauli’s exclusion principle, this allowed study of atoms with great precision when digital computers became available. Even today, these theories are used in the Hartree-Fock quantum chemical method to determine the energy levels of atoms. Further refinements of quantum theory such as the Dirac equation and quantum field theory made smaller impacts on the theory of atoms. Another model of historical interest, proposed by Gilbert N. Lewis in 1916, had cubical atoms with electrons statically held at the corners. The cubes could share edges or faces to form chemical bonds. This model was created to account for chemical phenomena such as bonding, rather than physical phenomena such as atomic spectra. Atomic theory

12

See also History of thermodynamics Kinetic theory Development of Quantum Theory Quantum Chemistry John Dalton

Related lists • •

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• • • • •

Timeline of chemical element discovery Timeline of quantum mechanics, molecular physics, atomic physics, nuclear physics, and particle physics • Timeline of thermodynamics, statistical mechanics, and random processes

References

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External links •

Ancient Atomism 10

Source: http://en.wikipedia.org/wiki/Atomic_theory

Principal Authors: Vsmith, Brighterorange, Ragesoss, Voyajer, Karol Langner, Linas, Rho, Timmy2, Dustimagic, Eric Forste

Auger electron

Auger emission (pronounced [o e]) is a phenomenon in physics in which the emission of an electron from an atom causes the emission of a second electron. This second ejected electron is called an Auger electron. The name Auger electron comes from one of its discoverers, Pierre Victor Auger. The name does not come from the similarly-named device, the auger. When an electron is removed from a core level of an atom, leaving a vacancy, an electron from a higher energy level may fall into the vacancy, resulting in a release of energy. Although sometimes this energy is released in the form of an

10 http://plato.stanford.edu/entries/atomism-ancient/

Auger electron

13 emitted photon, the energy can also be transferred to another electron, which is then ejected from the atom.

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Upon ejection the kinetic energy of the Auger electron corresponds to the difference between the energy of the initial electronic transition and the ionization energy for the shell from which the Auger electron was ejected. These energy levels depend on the type of atom and the chemical environment in which the atom was located. Auger electron spectroscopy stimulates the emission of Auger electrons by bombarding a sample with either X-rays or energetic electrons and measures the intensity of Auger electrons as a function of the Auger electron energy. The resulting spectra can be used to determine the identity of the emitting atoms and some information about their environment. A similar Auger effect occurs in semiconductors. An electron and electron hole can recombine giving up their energy to an electron in the conduction band, increasing its energy. The reverse effect is known as impact ionization.

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History

The Auger emission process was discovered in the 1920s by Lise Meitner, an Austrian physicist. Subsequently Pierre Victor Auger, a French Physicist, also discovered the process. Auger reported the discovery in the journal Radium in 1925 and it was Auger that had the process named after him.

Source: http://en.wikipedia.org/wiki/Auger_electron

Principal Authors: AjAldous, Keenan Pepper, Srleffler, Stokerm, Tristanb

Bargmann’s limit

In quantum mechanics, Bargmann’s limit, named for Valentine Bargmann, provides an upper bound on the number N l of bound states in a system. It takes the form R 1 2m ∞ Nl ≤ 2l+1 r|V (r)|V <0 dr ~2 0 Professor Hagen says, "The Bargmann limit provides, if not the best bound, a pretty darn good one." Note that the delta function potential attains this limit.

Bargmann’s limit

14

References Bargmann, Proc. Nat. Acad. Sci. 38 961 (1952) Schwinger, Proc. Nat. Acad. Sci. 47 122 (1961)

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Source: http://en.wikipedia.org/wiki/Bargmann%27s_limit

Principal Authors: TobinFricke, Covington, Amalas, Charles Matthews, Pjacobi

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Bohr model

Figure 2

The Bohr model of the atom

In atomic physics, the Bohr model depicts the atom as a small, positively charged nucleus surrounded by waves of electrons in orbit — similar in structure to the solar system, but with electrostatic forces providing attraction, rather than gravity, and with waves spread over entire orbit instead of localized planets. Bohr model

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Introduced by Niels Bohr in 1913, the model’s key success was in explaining the →Rydberg formula for the spectral emission lines of atomic hydrogen; while the Rydberg formula had been known experimentally, it did not gain a theoretical underpinning until the Bohr model was introduced. The Bohr model is a primitive model of the hydrogen atom which cannot explain the fine structure of the hydrogen atom nor any of the heavier atoms. As a theory, it can be derived as a first-order approximation of the hydrogen atom in the broader and much more accurate quantum mechanics, and thus may be considered to be an obsolete scientific theory. However, because of its simplicity, the Bohr model is still commonly taught to introduce students to quantum mechanics.

History

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In the early 20th century, experiments by Ernest Rutherford and others had established that atoms consisted of a diffuse cloud of negatively charged electrons surrounding a small, dense, positively charged nucleus. Given this experimental data, it is quite natural to consider a planetary model for the atom, with electrons orbiting a sun-like nucleus. However, a naive planetary model has several difficulties, the most serious of which is the loss of energy by synchrotron radiation.That is, an accelerating electric charge emits electromagnetic waves which carry energy; thus, with each orbit around the nucleus, the electron would radiate away a bit of its orbital energy, gradually spiralling inwards to the nucleus until the atom was no more. A quick calculation shows that this would happen almost instantly; thus, the naive planetary theory cannot explain why atoms are extremely long-lived. The naive planetary model also failed to explain atomic spectra, the observed discrete spectrum of light emitted by electrically excited atoms. Late 19th century experiments with electric discharges through various low-pressure gasses in evacuated glass tubes had shown that atoms will emit light (that is, electromagnetic radiation), but only at certain discrete frequencies. A naive planetary model cannot explain this. To overcome these difficulties, Niels Bohr proposed, in 1913, what is now called the Bohr model of the atom. The key ideas were:

• • •

The orbiting electrons existed in orbits that had discrete quantized energies. That is, not every orbit is possible but only certain specific ones. The laws of classical mechanics do not apply when electrons make the jump from one allowed orbit to another. When an electron makes a jump from one orbit to another the energy difference is carried off (or supplied) by a single quantum of light (called a Bohr model

16

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photon) which has an energy equal to the energy difference between the two orbitals. • The allowed orbits depend on quantized (discrete) values of orbital angular momentum, L according to the equation h L = n · ~ = n · 2π Where n = 1,2,3,· · · and is called the principal quantum number, and h is Planck’s constant.

Assumption (4) states that the lowest value of n is 1. This corresponds to a smallest possible radius of 0.0529 nm. This is known as the Bohr radius. Once an electron is in this lowest orbit, it can get no closer to the proton.

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The Bohr model is sometimes known as the semiclassical model of the atom, as it adds some primitive quantization conditions to what is otherwise a classical mechanics treatment. The Bohr model is certainly not a full quantum mechanical description of the atom. Assumption 2) states that the laws of classical mechanics don’t apply during a quantum jump, but it doesn’t state what laws should replace classical mechanics. Assumption 4) states that angular momentum is quantised but does not explain why.

Refinements

Several enhancements to the Bohr model were proposed; most notably the Sommerfeld model or Bohr-Sommerfeld model, which attempted to add support for elliptical orbits to the Bohr model’s circular orbits. This model supplemented condition (4) with an additional radial quantization condition, the Sommerfeld-Wilson quantization condition H pdq = nh where p is the generalized momentum conjugate to the angular generalized coordinate q; the integral is the action of action-angle coordinates. The Bohr-Sommerfeld model proved to be extremely difficult and unwieldy when its mathematical treatment was further fleshed out. In particular, the application of traditional perturbation theory from classical planetary mechanics led to further confusions and difficulties. In the end, the model was abandoned in favour of the full quantum mechanical treatment of the hydrogen atom, in 1925, using Schrödinger’s wave mechanics. However, this is not to say that the Bohr model was without its successes. Calculations based on the Bohr-Sommerfeld model were able to accurately explain a number of more complex atomic spectral effects. For example, up to firstorder perturbation, the Bohr model and quantum mechanics make the same predictions for the spectral line splitting in the →Stark effect. At higher-order Bohr model

17 perturbations, however, the Bohr model and quantum mechanics differ, and measurements of the Stark effect under high field strengths helped confirm the correctness of quantum mechanics over the Bohr model.

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The Bohr-Sommerfeld quantization condition as first formulated can be viewed as a rough early draft of the more sophisticated condition that the symplectic form of a classical phase space M be integral; that is, that it lie in the image of

ˇ 2 (M, Z) → H ˇ 2 (M, R) → H 2 (M, R) , where the first map is the homoH DR

ˇ morphism of Cech cohomology groups induced by the inclusion of the integers ˇ in the reals, and the second map is the natural isomorphism between the Cech cohomology and the de Rham cohomology groups. This condition guarantees that the symplectic form arise as the curvature form of a connection of a Hermitian line bundle. This line bundle is then called a prequantization in the theory of geometric quantization.

Electron energy levels in hydrogen

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The Bohr model is accurate only for one-electron systems such as the hydrogen atom or singly-ionized helium. This section uses the Bohr model to derive the energy levels of hydrogen. The derivation starts with three simple assumptions:

1) All particles are wavelike, and an electron’s wavelength λ, is related to its velocity v by: λ=

h me v

where h is Planck’s Constant, and me is the mass of the electron. Bohr did not make this assumption (known as the de Broglie hypothesis) in his original derivation, because it hadn’t been proposed at the time. However it allows the following intuitive statement.

2) The circumference of the electron’s orbit must be an integer multiple of its wavelength: 2πr = nλ

where r is the radius of the electron’s orbit, and n is a positive integer.

Bohr model

18 3) The electron is held in orbit by the coulomb force. That is, the coulomb force is equal to the centripetal force: =

me v 2 r

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kqe2 r2

where k = 1/(4π0 ), and qe is the charge of the electron.

These are three equations with three unknowns: λ, r, v. After solving this system of equations to find an equation for just v, it is placed into the equation for the total energy of the electron: Because of the virial theorem, the total energy simplifies to

E = − 12 me v 2

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Substituting, one obtains the energy of the different levels of hydrogen: Or, after plugging in values for the constants,

Thus, the lowest energy level of hydrogen (n = 1) is about -13.6 eV. The next energy level (n = 2) is -3.4 eV. The third (n = 3) is -1.51 eV, and so on. Note that these energies are less than zero, meaning that the electron is in a bound state with the proton. Positive energy states correspond to the ionized atom where the electron is no longer bound, but is in a scattering state.

Energy in terms of other constants Starting with what we found above, En =

−me qe4 1 8h2 20 n2

We can multiply top and bottom by c2 , and we’ll arrive at En =

−me c2 qe4 1 8h2 c2 20 n2

or re-grouping them to make it more clear:  4  q En = − 21 me c2 4h2 ce2 2 n12 0

From here we can now write the energy level equation in terms of other constants to: Bohr model

19 En =

−Er α2 2n2

where,

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En is the energy level

Er is the rest energy of the electron α is the fine structure constant

n is the principal quantum number.

Rydberg formula

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The →Rydberg formula describes the transitions or quantum jumps between one energy level and another. When the electron moves from one energy level to another, a photon is given off. Using the derived formula for the different ’energy’ levels of hydrogen one may determine the ’wavelengths’ of light that a hydrogen atom can give off. The energy of photons that a hydrogen atom can give off are given by the difference of two hydrogen energy levels:   me e4 1 1 E = Ei − Ef = 8h − 2 2  n2 n2 0

f

i

where nf means the final energy level, and ni means the initial energy level. It is assumed that the final energy level is less than the initial energy level.

Since the energy of a photon is E=

hc λ

the wavelength of the photon given off is   me e4 1 1 1 = − λ 8ch3 2 n2 n2 0

f

i

The above is known as the →Rydberg formula. This formula was known in the nineteenth century to scientists studying spectroscopy, but there was no theoretical justification for the formula until Bohr derived it, more or less along the lines above.

Bohr model

20

Shortcomings

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The Bohr model gives an incorrect value L = ~ for the ground state orbital angular momentum. The angular momentum in the true ground state is known to be zero. The Bohr model also has difficulty with or fails to explain: •

The spectra of larger atoms. At best, it can make some approximate predictions about the emission spectra for atoms with a single outer-shell electron (atoms in the lithium group.) • The relative intensities of spectral lines; although in some simple cases, it was able to provide reasonable estimates (for example, calculations by Kramers for the →Stark effect). • The existence of fine structure and hyperfine structure in spectral lines. • The →Zeeman effect - changes in spectral lines due to external magnetic fields.

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See also • • • •

→Franck-Hertz experiment provided early support for the Bohr model. Inert pair effect is adequately explained by means of the Bohr model. Lyman series →Schrödinger equation

References Historical •

Niels Bohr (1913). " On the Constitution of Atoms and Molecules (Part 1 of 3) 11". Philosophical Magazine 26: 1-25. • Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus". Philosophical Magazine 26: 476-502. • Niels Bohr (1913). "On the Constitution of Atoms and Molecules, Part III". Philosophical Magazine 26: 857-875. • Niels Bohr (1914). "The spectra of helium and hydrogen". Nature 92: 231232.

11 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Bohr/Bohr-1913a.html

Bohr model

21 Niels Bohr (1921). " Atomic Structure 12". Nature. A. Einstein (1917). "Zum Quantensatz von Sommerfeld und Epstein". Verhandlungen der Deutschen Physikalischen Gesellschaft 19: 82-92. Reprinted in The Collected Papers of Albert Einstein, A. Engel translator, (1997) Princeton University Press, Princeton. 6 p.434. (Provides an elegant reformulation of the Bohr-Sommerfeld quantization conditions, as well as an important insight into the quantization of non-integrable (chaotic) dynamical systems.)

Modern •

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Paul Tipler and Ralph Llewellyn (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0716743450.

Source: http://en.wikipedia.org/wiki/Bohr_model

Principal Authors: Linas, JabberWok, Christopher Thomas, Tim Starling, MathKnight, Munchkinguy,

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GoldenBoar, El C, Glenn

Boltzmon

A boltzmon (named after the nineteenth-century thermodynamicist Ludwig Boltzmann) is a theoretical subatomic particle postulated to be created after the explosion of a black hole. The boltzmon was proposed as a means of explaining what happens to the information of objects consumed by black holes while still preserving purity. One theory, proposed by the Dutch researcher Gerard ’t Hooft, is that information is contained in the particles that Hawking-radiate from the black hole. The other theory includes the boltzmon particle.

This theory postulates that a black hole leaves behind a remnant when it explodes—a single particle that has been dubbed the boltzmon. A boltzmon would be about the size of the Planck-Wheeler area, or 10 -66 cm 2, which is supposedly about as small as anything can be. It would contain the sum total of all the information ever consumed by the black hole, so each boltzmon would be unique in the universe. While a typical particle has a few states (positive or negative electrical charge, integral or fractional spin, etc.), a boltzmon

12 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Bohr-Nature-1921.html

Boltzmon

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References •

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would have an infinite number of states and as a result, would be highly unstable. If disturbed, it might make a hole in spacetime and vanish into it, thus departing from our universe.

Ferris, Timothy. The Whole Shebang, 1997 Simon & Schuster.

Source: http://en.wikipedia.org/wiki/Boltzmon

Born probability

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In quantum mechanics, the Born probability is a probability of an event calculated from a wavefunction or more generally from the density matrix. The probability (or its density) equals the squared modulus of the complex amplitude an : P = |an |2

The interpretation that the physical meaning of the wavefunction is probabilistic was proposed by Max Born, and it became a pillar of the Copenhagen interpretation of quantum mechanics.

Source: http://en.wikipedia.org/wiki/Born_probability

Principal Authors: Jag123, Lumidek, Karol Langner, Conscious

Bose–Einstein condensate

A Bose–Einstein condensate is a phase of matter formed by bosons cooled to temperatures very near to absolute zero. The first such condensate was produced by Eric Cornell and Carl Wieman in 1995 at the University of Colorado at Boulder, using a gas of rubidium atoms cooled to 170 nanokelvins (nK). Under such conditions, a large fraction of the atoms collapse into the lowest quantum state, at which point quantum effects become apparent on a macroscopic scale.

Bose–Einstein condensate

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Figure 3 Velocity-distribution data confirming the discovery of a new phase of matter, the Bose– Einstein condensate, out of a gas of rubidium atoms. The false colors indicate the number of atoms at each velocity, with red being the fewest and white being the most. The areas appearing white and light blue are at the lowest velocities. Left: just before the appearance of the Bose–Einstein condensate. Center: just after the appearance of the condensate. Right: after further evaporation, leaving a sample of nearly pure condensate. The peak is not infinitely narrow because of the Heisenberg uncertainty principle: since the atoms are trapped in a particular region of space, their velocity distribution necessarily possesses a certain minimum width.

Introduction

Condensates are extremely low temperature fluids with properties that are currently not completely understood, such as spontaneously flowing out of their container. The effect is the consequence of quantum mechanics, which states that systems can only acquire energy in discrete steps. Now, if a system is at such a low temperature that it is in the lowest energy state, it is no longer possible for it to reduce its energy, not even by friction. Therefore, without friction, the fluid will easily overcome gravity because of adhesion between the fluid and the container wall, and it will take up the most favorable position, i.e. all around the container.

Theory

The collapse of the atoms into a single quantum state is known as Bose condensation or Bose–Einstein condensation. This phenomenon was predicted in 1925 by Albert Einstein, by generalizing Satyendra Nath Bose’s work on the statistical mechanics of (massless) photons to (massive) atoms. (The Einstein manuscript, believed to be lost, was found in a library at Leiden University in Bose–Einstein condensate

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2005.) The result of the efforts of Bose and Einstein is the concept of a →Bose gas, governed by the →Bose-Einstein statistics, which describes the statistical distribution of identical particles with integer spin, now known as bosons. Bosonic particles, which include the photon as well as atoms such as helium-4, are allowed to share quantum states with each other. Einstein speculated that cooling bosonic atoms to a very low temperature would cause them to fall (or "condense") into the lowest accessible quantum state, resulting in a new form of matter. This transition occurs below a critical temperature, which for a uniform threedimensional gas consisting of non-interacting particles with no apparent internal degrees of freedom is given by:  2/3 n h2 Tc = ζ(3/2) 2πmkB where:

Tc

is the critical temperature, the particle density,

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n m

the mass per boson,

h

Planck’s constant,

kB

the Boltzmann constant, and

ζ

the Riemann zeta function; ζ(3/2) ≈ 2.6124.

Discovery

In 1938, Pyotr Kapitsa, John Allen and Don Misener discovered that helium-4 became a new kind of fluid, now known as a superfluid, at temperatures below 2.17 kelvins (K) (lambda point). Superfluid helium has many unusual properties, including zero viscosity (the ability to flow without dissipating energy) and the existence of quantized vortices. It was quickly realized that the superfluidity was due to Bose–Einstein condensation of the helium-4 atoms, which are bosons. In fact, many of the properties of superfluid helium also appear in the gaseous Bose–Einstein condensates created by Cornell, Wieman and Ketterle (see below). However, superfluid helium-4 is not commonly referred to as a "Bose–Einstein condensate" because it is a liquid rather than a gas, which means that the interactions between the atoms are relatively strong. The original theory of Bose–Einstein condensation must be heavily modified in order to describe it. The first "true" Bose–Einstein condensate was created by Eric Cornell, Carl Wieman, and co-workers at JILA on June 5, 1995. They did this by cooling a dilute Bose–Einstein condensate

25

FT

vapor consisting of approximately 2000 rubidium-87 atoms to below 170 nK using a combination of laser cooling (a technique that won its inventors Steven Chu, Claude Cohen-Tannoudji, and William D. Phillips the 1997 Nobel Prize in Physics) and magnetic evaporative cooling. About four months later, an independent effort led by Wolfgang Ketterle at MIT created a condensate made of sodium-23. Ketterle’s condensate had about a hundred times more atoms, allowing him to obtain several important results such as the observation of quantum mechanical interference between two different condensates. Cornell, Wieman and Ketterle won the 2001 Nobel Prize for their achievement.

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The Bose–Einstein condensation also applies to quasiparticles in solids. A magnon in an antiferromagnet carries spin 1 and thus obeys the Bose–Einstein statistics. The density of magnons is controlled by an external magnetic field, which plays the role of the magnon chemical potential. This technique provides access to a wide range of boson densities from the limit of a dilute Bose gas to that of a strongly interacting Bose liquid. A magnetic ordering observed at the point of condensation is the analog of superfluidity. In 1999 Bose condensation of magnons was demonstrated in the antiferromagnet TlCuCl3 by Oosawa et al. The condensation was observed at temperatures as large as 14 K. Such a high transition temperature (relative to that of atomic gases) is due to a greater density achievable with magnons and a smaller mass (roughly equal to the mass of an electron).

Unusual characteristics

Further experimentation by the JILA team in 2000 uncovered a hitherto unknown property of Bose–Einstein condensate. Cornell, Wieman, and their coworkers originally used rubidium-87, an isotope whose atoms naturally repel each other making a more stable condensate. The JILA team instrumentation now had better control over the condensate so experimentation was made on naturally attracting atoms of another rubidium isotope, rubidium-85 (having negative atom-atom scattering length). Through a process called Feshbach resonance involving a sweep of the magnetic field causing spin flip collisions, the JILA researchers lowered the characteristic, discrete energies at which the rubidium atoms bond into molecules making their Rb-85 atoms repulsive and creating a stable condensate. The reversible flip from attraction to repulsion stems from quantum interference among condensate atoms which behave as waves. When the scientists raised the magnetic field strength still further, the condensate suddenly reverted back to attraction, imploded and shrank beyond detection, and then exploded, blowing off about two-thirds of its 10,000 or so atoms. About half of the atoms in the condensate seemed to have disappeared from Bose–Einstein condensate

26

FT

the experiment altogether, not being seen either in the cold remnant or the expanding gas cloud. Carl Wieman explained that under current atomic theory this characteristic of Bose–Einstein condensate could not be explained because the energy state of an atom near absolute zero should not be enough to cause an implosion; however, subsequent mean-field theories have been proposed to explain it. Due to the fact that supernovae explosions are implosions, the explosion of a collapsing Bose–Einstein condensate was named "bosenova."

The atoms that seem to have disappeared are almost certainly still around in some form, just not in a form that could be detected in that current experiment. Two likely possibilities are that they have formed into molecules consisting of two bonded rubidium atoms, or they received enough energy from somewhere to fly away fast enough that they are out of the observation region before being observed.

Current research

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Compared to more commonly-encountered states of matter, Bose–Einstein condensates are extremely fragile. The slightest interaction with the outside world can be enough to warm them past the condensation threshold, forming a normal gas and losing their interesting properties. It is likely to be some time before any practical applications are developed.

Nevertheless, they have proved to be useful in exploring a wide range of questions in fundamental physics, and the years since the initial discoveries by the JILA and MIT groups have seen an explosion in experimental and theoretical activity. Examples include experiments that have demonstrated interference between condensates due to wave-particle duality 13, the study of superfluidity and quantized vortices 14, and the slowing of light pulses to very low speeds using electromagnetically induced transparency 15. Experimentalists have also realized "optical lattices", where the interference pattern from overlapping lasers provides a periodic potential for the condensate. These have been used to explore the transition between a superfluid and a Mott insulator 16, and may be useful in studying Bose–Einstein condensation in less than three dimensions, for example the Tonks-Girardeau gas. Bose–Einstein condensates composed of a wide range of isotopes have been produced 17.

13 http://cua.mit.edu/ketterle_group/Projects_1997/Interference/Interference_BEC.htm 14 http://www.aip.org/pt/vol-53/iss-8/p19.html 15 http://www.europhysicsnews.com/full/26/article1/article1.html 16 http://qpt.physics.harvard.edu/qptsi.html 17 http://physicsweb.org/articles/world/18/6/1

Bose–Einstein condensate

27

FT

Related experiments in cooling fermions rather than bosons to extremely low temperatures have created degenerate gases, where the atoms do not congregate in a single state due to the Pauli exclusion principle. To exhibit Bose– Einstein condensate, the fermions must "pair up" to form compound particles (e.g. molecules or Cooper pairs) that are bosons. The first molecular Bose– Einstein condensates were created in November 2003 by the groups of Rudolf Grimm at the University of Innsbruck, Deborah S. Jin at the University of Colorado at Boulder and Wolfgang Ketterle at MIT. Jin quickly went on to create the first fermionic condensate composed of Cooper pairs 18.

See also

Atomic coherence →Bose gas Electromagnetically induced transparency Fermionic condensate Gas in a box Slow glass Slow light Superconductivity Superfluid Supersolid Super-heavy atom Tonks-Girardeau gas

DR A

• • • • • • • • • • • •

External links

• •

•

• •

BEC Homepage 19 General introduction to Bose–Einstein condensation Nobel Prize in Physics 2001 20 - for the achievement of Bose–Einstein condensation in dilute gases of alkali atoms, and for early fundamental studies of the properties of the condensates Physics Today: Cornell, Ketterle, and Wieman Share Nobel Prize for Bose– Einstein Condensates 21 Bose–Einstein Condensates at JILA 22 The Bose–Einstein Condensate at Utrecht University, the Netherlands 23

18 http://physicsweb.org/articles/news/8/1/14/1 19 http://www.colorado.edu/physics/2000/bec/index.html

20 http://nobelprize.org/physics/laureates/2001/index.html 21 http://www.physicstoday.org/pt/vol-54/iss-12/p14.html 22 http://jilawww.colorado.edu/bec/ 23 http://www.bec.phys.uu.nl/

Bose–Einstein condensate

28

•

Alkali Quantum Gases at MIT 24 Atom Optics at UQ 25 Einstein’s manuscript on the Bose–Einstein condensate discovered at Leiden University 26 The revolution that has not stopped 27 PhysicsWeb article from June 2005

References • • • • •

S. N. Bose, Z. Phys. 26, 178 (1924) A. Einstein, Sitz. Ber. Preuss. Akad. Wiss. (Berlin) 1, 3 (1925) L.D. Landau, J. Phys. USSR 5, 71 (1941) L. Landau (1941). "Theory of the Superfluidity of Helium II". Physical Review 60: 356-358. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman, and E.A. Cornell (1995). "Observation of Bose–Einstein Condensation in a Dilute Atomic Vapor" 28. Science 269: 198-201. K.B. Davis, M.-O. Mewes, M.R. Andrews, N.J. van Druten, D.S. Durfee, D.M. Kurn, and W. Ketterle (1995). "Bose–Einstein condensation in a gas of sodium atoms". Physical Review Letters 75: 3969-3973.. D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell (1996). "Collective Excitations of a Bose–Einstein Condensate in a Dilute Gas". Physical Review Letters 77: 420-423. M. R. Andrews, C. G. Townsend, H.-J. Miesner, D. S. Durfee, D. M. Kurn, and W. Ketterle (1997). "Observation of interference between two Bose condensates". Science 275: 637-641.. M. R. Matthews, B. P. Anderson, P. C. Haljan, D. S. Hall, C. E. Wieman, and E. A. Cornell (1999). "Vortices in a Bose–Einstein Condensate". Physical Review Letters 83: 2498-2501. E.A. Donley, N.R. Claussen, S.L. Cornish, J.L. Roberts, E.A. Cornell, and C.E. Wieman (2001). "Dynamics of collapsing and exploding Bose–Einstein condensates". Nature 412: 295-299. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, I. Bloch (2002). "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms". Nature 415: 39-44..

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FT

• • •

•

•

•

•

•

24 http://cua.mit.edu/ketterle_group/home.htm 25 http://www.physics.uq.edu.au/atomoptics/

26 http://www.lorentz.leidenuniv.nl/history/Einstein_archive/ 27 http://physicsweb.org/articles/world/18/6/8/1

28 http://links.jstor.org/sici?sici=0036-8075%2819950714%293%3A269%3A5221%3C198%3AOOBCIA

%3E2.0.CO%3B2-G

Bose–Einstein condensate

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•

•

• •

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•

S. Jochim, M. Bartenstein, A. Altmeyer, G. Hendl, S. Riedl, C. Chin, J. Hecker Denschlag, and R. Grimm (2003). "Bose–Einstein Condensation of Molecules". Science 302: 2101-2103. Markus Greiner, Cindy A. Regal and Deborah S. Jin (2003). "Emergence of a molecular Bose-Einstein condensate from a Fermi gas". Nature 426: 537-540. M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, S. Gupta, Z. Hadzibabic, and W. Ketterle (2003). "Observation of Bose–Einstein Condensation of Molecules". Physical Review Letters 91: 250401. C. A. Regal, M. Greiner, and D. S. Jin (2004). "Observation of Resonance Condensation of Fermionic Atom Pairs". Physical Review Letters 92: 040403. C. J. Pethick and H. Smith, "Bose–Einstein Condensation in Dilute Gases", Cambridge University Press, Cambridge, 2001. Lev P. Pitaevskii and S. Stringari, "Bose–Einstein Condensation", Clarendon Press, Oxford, 2003. Mackie M, Suominen KA, Javanainen J., "Mean-field theory of Feshbachresonant interactions in 85Rb condensates." Phys Rev Lett. 2002 Oct 28;89(18):180403. Oxford Experimental BEC Group. http://www-matterwave.physics.ox.ac. uk/bec/bec.html T. Nikuni, M. Oshikawa, A. Oosawa, and H. Tanaka, (1999). "Bose–Einstein Condensation of Dilute Magnons in TlCuCl3" 29. Physical Review Letters 84: 5868.

FT

•

• •

Source: http://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate Principal Authors: Voyajer, CYD, Brian Jackson, Schneelocke, Michael Hardy, PAR, Matt Gies, Hfastedge, Fangz, R. Koot

29 http://dx.doi.org/10.1103/PhysRevLett.84.5868

Bose–Einstein condensate

30

Bose-Einstein statistics

FT

For other topics related to Einstein see Einstein (disambiguation).

In statistical mechanics, Bose-Einstein statistics (or more colloquially B-E statistics) determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium.

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Fermi-Dirac and Bose-Einstein statistics apply when quantum effects have to be taken into account and the particles are considered "indistinguishable". The quantum effects appear if the concentration of particles (N/V) ≥ n q (where n q is the quantum concentration). The quantum concentration is when the interparticle distance is equal to the thermal de Broglie wavelength i.e. when the wavefunctions of the particles are touching but not overlapping. As the quantum concentration depends on temperature; high temperatures will put most systems in the classical limit unless they have a very high density e.g. a White dwarf. Fermi-Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), Bose-Einstein statistics apply to bosons. Both Fermi-Dirac and Bose-Einstein become Maxwell-Boltzmann statistics at high temperatures or low concentrations.

Maxwell-Boltzmann statistics are often described as the statistics of "distinguishable" classical particles. In other words the configuration of particle A in state 1 and particle B in state 2 is different from the case where particle B is in state 1 and particle A is in state 2. When this idea is carried out fully, it yields the proper (Boltzmann) distribution of particles in the energy states, but yields non-physical results for the entropy, as embodied in →Gibbs paradox. These problems disappear when it is realized that all particles are in fact indistinguishable. Both of these distributions approach the Maxwell-Boltzmann distribution in the limit of high temperature and low density, without the need for any ad hoc assumptions. Maxwell-Boltzmann statistics are particularly useful for studying gases F-D statistics are most often used for the study of electrons in solids. As such, they form the basis of semiconductor device theory and electronics. Bosons, unlike fermions, are not subject to the Pauli exclusion principle: an unlimited number of particles may occupy the same state at the same time. This explains why, at low temperatures, bosons can behave very differently than fermions; all the particles will tend to congregate together at the same lowest-energy state, forming what is known as a Bose-Einstein condensate.

Bose-Einstein statistics

31 B-E statistics was introduced for photons in 1920 by Bose and generalized to atoms by Einstein in 1924. ni =

gi e(i −µ)/kT −1

with i > µ and where:

FT

The expected number of particles in an energy state i for B-E statistics is:

n i is the number of particles in state i g i is the degeneracy of state i

 i is the energy of the i -th state µ is the chemical potential

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k is Boltzmann’s constant T is absolute temperature

exp is the exponential function

This reduces to M-B statistics for energies (  i-µ ) » kT.

A Derivation of the Bose-Einstein distribution

Suppose we have a number of energy levels, labelled by index i, each level having energy  i and containing a total of n i particles. Suppose each level contains g i distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of g i associated with level i is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel. Let w(n,g) be the number of ways of distributing n particles among the g sublevels of an energy level. There is only one way of distributing n particles with one sublevel, therefore w(n,1) = 1. It’s easy to see that there are n + 1 ways of distributing n particles in two sublevels which we will write as: w(n, 2) =

(n+1)! n!1! .

Bose-Einstein statistics

32

FT

With a little thought it can be seen that the number of ways of distributing n particles in three sublevels is w(n,3) = w(n,2) + w(n-1,2) + ... + w(0,2) so that P P (n−k+1)! (n+2)! w(n, 3) = nk=0 w(n − k, 2) = nk=0 (n−k)!1! = n!2! where we have used the following theorem involving binomial coefficients: Pn (k+a)! (n+a+1)! k=0 k!a! = n!(a+1)! .

Continuing this process, we can see that w(n,g) is just a binomial coefficient w(n, g) =

(n+g−1)! . n!(g−1)!

The number of ways that a set of occupation numbers n i can be realized is the product of the ways that each individual energy level can be populated: Q Q (n +g −1)! Q (n +g )! W = i w(ni , gi ) = i n i!(g i−1)! ≈ i n i!(g i)! i

i

i

i

DR A

where the approximation assumes that gi >> 1. Following the same procedure used in deriving the →Maxwell-Boltzmann statistics, we wish to find the set of n i for which W is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of W and ln(W ) occur at the value of Ni and, since it is easier to accomplish mathematically, we will maximise the latter function instead. We constrain our solution using Lagrange multipliers forming the function: P P f (ni ) = ln(W ) + α(N − ni ) + β(E − ni i ) Using the gi >> 1 approximation and using Stirling’s approximation for the factorials (ln(x!) ≈ x ln(x) − x) gives: P P f (ni ) = i (ni + gi ) ln(ni + gi ) − ni ln(ni ) − gi ln(gi ) + α(N − ni ) + β(E − P ni i )

Taking the derivative with respect to n i, and setting the result to zero and solving for n i yields the Bose-Einstein population numbers: ni =

gi eα+βi −1

It can be shown thermodynamically that β = 1/kT where k is Boltzmann’s constant and T is the temperature, and that α = -µ/kT where µ is the chemical potential, so that finally:

Bose-Einstein statistics

33 ni =

gi e(i −µ)/kT −1

ni =

gi ei /kT /z−1

FT

Note that the above formula is sometimes written:

where z = exp(µ/kT ) is the absolute activity.

History

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In the early 1920s Satyendra Nath Bose was intrigued by Einstein’s theory of light waves being made of particles called photons. Bose was interested in deriving Planck’s radiation formula, which Planck obtained largely by guessing. In 1900 Max Planck had derived his formula by manipulating the math to fit the empirical evidence. Using the particle picture of Einstein, Bose was able to derive the radiation formula by systematically developing a statistics of massless particles without the constraint of particle number conservation. Bose derived Planck’s Law of Radiation by proposing different states for the photon. Instead of statistical independence of particles, Bose put particles into cells and described statistical independence of cells of phase space. Such systems allow two polarization states, and exhibit totally symmetric wavefunctions. He was quite successful in that he developed a statistical law governing the behaviour pattern of photons. However he was not able to publish his work, because no journals in Europe would accept his paper being unable to understand it. Bose sent his paper to Einstein who saw the significance of it and he used his influence to get it published.

See also

• →Maxwell-Boltzmann statistics • →Fermi-Dirac statistics • Parastatistics • →Planck’s law of black body radiation

Source: http://en.wikipedia.org/wiki/Bose-Einstein_statistics

Principal Authors: PAR, Michael Hardy, Stevenj, Voyajer, Mct mht, Salix alba, Phys, Youandme

Bose-Einstein statistics

34

Bose gas

FT

An ideal Bose gas is a quantum-mechanical version of a classical ideal gas. It is composed of bosons, which have an integral value of spin, and obey →Bose-Einstein statistics. The statistical mechanics of bosons were developed by Satyendra Nath Bose for photons, and extended to massive particles by Albert Einstein who realized that an ideal gas of bosons would form a condensate at a low enough temperature, unlike a classical ideal gas. This condensate is known as a Bose-Einstein condensate.

The Thomas-Fermi approximation

The thermodynamics of an ideal Bose gas is best calculated using the grand partition function. The grand partition function for a Bose gas is given by:

Z(z, β, V ) =

Q

i

1 − ze−βi


−gi

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where each term in the product corresponds to a particular energy  i , g i is the number of states with energy  i , z is the absolute activity, which may also be expressed in terms of the chemical potential µ by: z(β, µ) ≡ eβµ

and β defined as: β≡

1 kT

where k is Boltzmann’s constant and T is the temperature. All thermodynamic quantities may be derived from the grand partition function and we will consider all thermodynamic quantities to be functions of only the three variables z , β (or T ), and V . All partial derivatives are taken with respect to one of these three variables while the other two are held constant. It is more convenient to deal with the dimensionless grand potential defined as:
P

Ω = − ln(Z) =

i

gi ln 1 − ze−βi

Following the procedure described in the gas in a box article, we can apply the Thomas-Fermi approximation which assumes that the average energy is large compared to the energy difference between levels so that the above sum may be replaced by an integral:
R∞ Ω ≈ 0 ln 1 − ze−βE dg

Bose gas

35 The degeneracy dg may be expressed for many different situations by the general formula: dE

FT

1 E α−1 Γ(α) Ecα

dg =

where α is a constant, Ec is a "critical energy", and Γ is the Gamma function. For example, for a massive Bose gas in a box, α=3/2 and the critical energy is given by: 1 (βEc )α

=

Vf Λ3

where is the thermal wavelength. For a massive Bose gas in a harmonic trap we will have α=3 and the critical energy is given by: 1 (βEc )α

=

f (~ωβ)3

where V(r) =mω 2r 2/2 is the harmonic potential. It is seen that E c is a function of volume only.

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We can solve the equation for the grand potential by integrating the Taylor series of the integrand term by term, or by realizing that it is proportional to the Mellin transform of the Li 1(z exp(-β E)) where Li s(x) is the polylogarithm function. The solution is: α+1 (z) Ω ≈ − Li(βE α c)

The problem with this continuum approximation for a Bose gas is that the ground state has been effectively ignored, giving a degeneracy of zero for zero energy. This inaccuracy becomes serious when dealing with the Bose-Einstein condensate and will be dealt with in the next section.

Inclusion of the ground state

The total number of particles is found from the grand potential by

N = −z ∂Ω ∂z ≈

Liα (z) (βEc )α

The polylogarithm term must remain real and positive, and the maximum value it can possibly have is at z=1 where it is equal to ζ(α) where ζ is the Riemann zeta function. For a fixed N , the largest possible value that β can have is a critical value β c where N=

ζ(α) (βc Ec )α

Bose gas

36

FT

This corresponds to a critical temperature T c=1/kβ c below which the ThomasFermi approximation breaks down. The above equation can be solved for the critical temperature:  1/α Ec N Tc = ζ(α) k For example, for α = 3/2 and using the above noted value of Ec yields  2/3 2 N h Tc = V f ζ(3/2) 2πmk

Again, we are presently unable to calculate results below the critical temperature, because the particle numbers using the above equation become negative. The problem here is that the Thomas-Fermi approximation has set the degeneracy of the ground state to zero, which is wrong. There is no ground state to accept the condensate and so the equation breaks down. It turns out, however, that the above equation gives a rather accurate estimate of the number of particles in the excited states, and it is not a bad approximation to simply "tack on" a ground state term: Liα (z) (βEc )α

DR A N = N0 +

where N 0 is the number of particles in the ground state condensate: N0 =

g0 z 1−z

This equation can now be solved down to absolute zero in temperature. Figure 1 shows the results of the solution to this equation for α=3/2, with k= c=1 which corresponds to a gas of bosons in a box. The solid black line is the fraction of excited states 1-N 0/N for N =10,000 and the dotted black line is the solution for N =1000. The blue lines are the fraction of condensed particles N 0/N The red lines plot values of the negative of the chemical potential µ and the green lines plot the corresponding values of z . The horizontal axis is the normalized temperature τ defined by τ=

T Tc

It can be seen that each of these parameters become linear in τ α in the limit of low temperature and, except for the chemical potential, linear in 1/τ α in the limit of high temperature. As the number of particles increases, the condensed and excited fractions tend towards a discontinuity at the critical temperature. The equation for the number of particles can be written in terms of the normalized temperature as: Bose gas

FT

37

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Figure 4 Figure 1: Various Bose gas parameters as a function of normalized temperature τ . The value of α is 3/2. Solid lines are for N=10,000, dotted lines are for N=1000. Black lines are the fraction of excited particles, blue are the fraction of condensed particles. The negative of the chemical potential µ is shown in red, and green lines are the values of z. It has been assumed that k = c=1.

N=

g0 z 1−z

+N

Liα (z) ζ(α)

τα

For a given N and τ , this equation can be solved for τ α and then a series solution for z can be found by the method of inversion of series, either in powers of τ α or as an asymptotic expansion in inverse powers of τ α. From these expansions, we can find the behavior of the gas near T =0 and in the Maxwell-Boltzmann as T approaches infinity. In particular, we are interested in the limit as N approaches infinity, which can be easily determined from these expansions.

Thermodynamics

Adding the ground state to the equation for the particle number corresponds to adding the equivalent ground state term to the grand potential:

Ω = g0 ln(1 − z) −

Liα+1 (z) (βEc )α

All thermodynamic properties may now be computed from the grand potential. The following table lists various thermodynamic quantities calculated in

Bose gas

38

Quantity

General

z Vapor fraction 0 1− N N

Equation of state PV β Ω N = −N

=

Liα (z) ζ(α)

=

Liα+1 (z) ζ(α)

τα τα

Gibbs Free Energy = ln(z) G = ln(z)

FT

the limit of low temperature and high temperature, and in the limit of infinite particle number. An equal sign (=) indicates an exact result, while an approximation symbol indicates that only the first few terms of a series in τ α is shown. T  Tc

T  Tc

=1

≈ τ α − α 2α 2 τ =1

= τα

ζ 2 (α)

ζ(α)

ζ(α+1)

ζ(α)

= ζ(α) τ α ≈ 1 − α+1 α 2 τ =0

≈ ln



ζ(α) τα



ζ(α)

− 2α τ α

It is seen that all quantities approach the values for a classical ideal gas in the limit of large temperature. The above values can be used to calculate other thermodynamic quantities. For example, the relationship between internal energy and the product of pressure and volume is the same as that for a classical ideal gas over all temperatures: ∂Ω ∂β

= αP V

DR A

U=

A similar situation holds for the specific heat at constant volume Cv =

∂U ∂T

= k(α + 1) U β

The entropy is given by: TS = U + PV − G

Note that in the limit of high temperature, we have  α  τ T S = (α + 1) + ln ζ(α)

which, for α=3/2 is simply a restatement of the Sackur-Tetrode equation.

See also

• • •

Gas in a box Debye model Bose-Einstein condensate

Bose gas

39

References •

FT

Huang, Kerson, "Statistical Mechanics", John Wiley and Sons, New York, 1967. • A. Isihara, "Statistical Physics", Academic Press, New York, 1971. • L. D. Landau and E. M. Lifshitz, "Statistical Physics, 3rd Edition Part 1", Butterworth-Heinemann, Oxford, 1996. • C. J. Pethick and H. Smith, "Bose-Einstein Condensation in Dilute Gases", Cambridge University Press, Cambridge, 2004. • Zijun Yan, "General Thermal Wavelength and its Applications", Eur. J. Phys, 21 (2000), 625-631. online 30

Source: http://en.wikipedia.org/wiki/Bose_gas

Principal Authors: PAR, Schneelocke, Tom davis, SimonP

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Bound state

In physics, a bound state is a composite of two or more building blocks (particles or bodies) that behaves as a single object. In quantum mechanics (where the number of particles is conserved), a bound state is a state in the →Hilbert space that corresponds to two or more particles whose interaction energy is negative, and therefore these particles cannot be separated unless energy is spent. The energy spectrum of a bound state is discrete, unlike the continuous spectrum of isolated particles. (Actually, it is possible to have unstable bound states with a positive interaction energy provided that there is a "energy barrier" that has to be tunnelled through in order to decay. This is true for some radioactive nuclei.) In general, a stable bound state is said to exist in a given potential of some dimension if stationary wavefunctions exist (normalized in the range of the potential). The energy of these wavefunctions is negative.

In relativistic quantum field theory, a stable bound state of n particles with masses m 1, ..., m n shows up as a pole in the S-matrix with a center of mass energy which is less than m 1+...+m n. An unstable bound state (see resonance) shows up as a pole with a complex center of mass energy.

30 http://www.iop.org/EJ/article/0143-0807/21/6/314/ej0614.pdf

Bound state

40

Examples •

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See also

FT

A proton and an electron can move separately; the total center-of-mass energy is positive, and such a pair of particles can be described as an ionized atom. Once the electron starts to "orbit" the proton, the energy becomes negative, and bound states - namely the hydrogen atom - is formed. Only the lowest energy bound state, the ground state is stable. The other excited states are unstable and will decay into bound states with less energy by emitting a photon. • A nucleus is a bound state of protons and neutrons (nucleons). • A positronium "atom" is an unstable bound state of an electron and a positron. It decays into photons. • The proton itself is a bound state of three quarks (two up and one down; one red, one green and one blue). However, unlike the case of the hydrogen atom, the individual quarks can never be isolated. See confinement.

• •

Composite field →Resonance

Source: http://en.wikipedia.org/wiki/Bound_state

Principal Authors: Phys, SeventyThree, Dmr2, Conscious, Tony Sidaway

Bra-ket notation

Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract vectors and linear functionals in pure mathematics. It is so called because the inner product of two states is denoted by a bracket, hφ|ψi, consisting of a left part, hφ|, called the bra, and a right part, |ψi, called the ket. The notation was invented by Paul Dirac, and is also known as Dirac notation. It is also the notation of choice in quantum computing.

Bra-ket notation

41

Bras and kets

|ψi

FT

In quantum mechanics, the state of a physical system is identified with a vector in a complex →Hilbert space, H. Each vector is called a "ket", and written as

where ψ denotes the particular ket, read as "psi ket." Every ket |ψi has a dual bra, written as hψ|

This is a continuous linear function from H to the complex numbers C, defined by:   hψ|ρi = |ψi , |ρi for all kets |ρi

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where ( , ) denotes the inner product defined on the Hilbert space. The bra is simply the conjugate transpose (also called the Hermitian conjugate) of the ket and vice versa. The notation is justified by the Riesz representation theorem, which states that a Hilbert space and its dual space are isometrically isomorphic. Thus, each bra corresponds to exactly one ket, and vice versa. This is not always the case; on page 111 of Quantum Mechanics by Cohen-Tannoudji et al. it is clarified that there is such a relationship between bras and kets, so long as the defining functions used are square integrable. Consider a continuous basis and a Dirac delta function or a sine or cosine wave as a wave function. Such functions are not square integrable and therefore it arises that there are bras that exist with no corresponding ket. This does not hinder quantum mechanics because all physically realistic wave functions are square integrable.

Bra-ket notation can be used even if the vector space is not a Hilbert space. In any Banach space B, the vectors may be notated by kets and the continuous linear functionals by bras. Over any vector space without topology, we may also notate the vectors by kets and the linear functionals by bras. In these more general contexts, the bracket does not have the meaning of an inner product, because the Riesz representation theorem does not apply.

Applying the bra hφ| to the ket |ψi results in a complex number, called a "braket" or "bracket", which is written as hφ|ψi.

In quantum mechanics, this is the probability amplitude for the state ψ to collapse into the state φ. Bra-ket notation

42

Properties

FT

Because each ket is a vector in a complex →Hilbert space and each bra-ket is an inner product, it follows directly that bras and kets can be manipulated in the following ways:

Given any bra hφ|, kets |ψ1 i and |ψ2 i, and complex numbers c 1 and c 2, then, since bras are linear functionals,   hφ| c1 |ψ1 i + c2 |ψ2 i = c1 hφ|ψ1 i + c2 hφ|ψ2 i.

•

Given any ket |ψi, bras hφ1 | and hφ2 |, and complex numbers c 1 and c 2, then, by the definition of addition and scalar multiplication of linear functionals,   c1 hφ1 | + c2 hφ2 | |ψi = c1 hφ1 |ψi + c2 hφ2 |ψi.

•

Given any kets |ψ1 i and |ψ2 i, and complex numbers c 1 and c 2, from the properties of the inner product (with c* denoting the complex conjugate of c),

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•

c1 |ψ1 i + c2 |ψ2 i is dual to c∗1 hψ1 | + c∗2 hψ2 |.

•

Given any bra hφ| and ket |ψi, an axiomatic property of the inner product gives hφ|ψi = hψ|φi∗ .

Linear operators

If A : H → H is a linear operator, we can apply A to the ket |ψi to obtain the ket (A|ψi). Linear operators are ubiquitous in the theory of quantum mechanics. For example, hermitian operators are used to represent observable physical quantities, such as energy or momentum, whereas unitary linear operators represent transformative processes such as rotation or the progression of time.

Operators can also be viewed as acting on bras from the right hand side. Composing the bra hφ| with the operator A results in the bra (hφ|A), defined as a linear functional on H by the rule

Bra-ket notation

43 

 hφ|A

 |ψi = hφ|

 A|ψi .

hφ|A|ψi.

FT

This expression is commonly written as

A convenient way to define linear operators on H is given by the outer product: if hφ| is a bra and |ψi is a ket, the outer product |φihψ|

denotes the rank one operator that maps the ket |ρi to the ket |φihψ|ρi (where hψ|ρi is a scalar multiplying the vector |φi). One of the uses of the outer product is to construct projection operators. Given a ket |ψi of norm 1, the orthogonal projection onto the subspace spanned by |ψi is |ψihψ|.

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Composite bras and kets

Two Hilbert spaces V and W may form a third space V ⊗W by a tensor product. In quantum mechanics, this is used for describing composite systems. If a system is composed of two subsystems described by V and W respectively, then the Hilbert space of the entire system is the tensor product of the two spaces. (The exception to this is if the subsystems are actually identical particles. In that case, the situation is a little more complicated.) If |ψi is a ket in V and |φi is a ket in W, the tensor product of the two kets is a ket in V ⊗ W . This is written variously as |ψi|φi or |ψi ⊗ |φi or |ψφi or |ψ, φi.

Representations in terms of bras and kets

In quantum mechanics, it is often convenient to work with the projections of state vectors onto a particular basis, rather than the vectors themselves. The reason is that the former are simply complex numbers, and can be formulated in terms of partial differential equations (see, for example, the derivation of the position-basis →Schrödinger equation). This process is very similar to the use of coordinate vectors in linear algebra. For instance, the Hilbert space of a zero-spin point particle is spanned by a position basis {|xi}, where the label x extends over the set of position vectors.

Bra-ket notation

44 Starting from any ket |ψi in this Hilbert space, we can define a complex scalar function of x, known as a wavefunction:

FT

ψ(x) ≡ hx|ψi. It is then customary to define linear operators acting on wavefunctions in terms of linear operators acting on kets, by Aψ(x) ≡ hx|A|ψi.

For instance, the momentum operator p has the following form: pψ(x) ≡ hx|p|ψi = −i~∇ψ(x).

One occasionally encounters an expression like −i~∇|ψi.

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This is something of an abuse of notation, though a fairly common one. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected into the position basis: −i~∇hx|ψi.

For further details, see rigged Hilbert space.

Further reading •

Feynman, Leighton and Sands (1965). The Feynman Lectures on Physics Vol. III. Addison-Wesley. ISBN 0201021153.

Source: http://en.wikipedia.org/wiki/Bra-ket_notation

Principal Authors: CYD, MathKnight, AxelBoldt, Trewornan, Laurascudder, Mct mht, AugPi, Theresa knott, Ancheta Wis

Bra-ket notation

45

Breit equation

Introduction

FT

The Breit equation is a relativistic wave equation derived by Gregory Breit in 1929 based on the Dirac equation, which formally describes two or more massive spin-1/2 particles (electrons, for example) interacting electromagnetically to the first order in perturbation theory. It accounts for magnetic interactions and retardation effects to the order of 1/c 2. When other quantum electrodynamic effects are negligible, this equation has been shown to give results in good agreement with experiment.

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The Breit equation is not only an approximation in terms of quantum mechanics, but also in terms of relativity theory as it is not completely invariant with respect to the Lorentz transformation. Just as does the Dirac equation, it treats nuclei as point sources of an external field for the particles it describes. For N particles, the Breit equation has the form (r ij is the distance between particle i and j ): nP o P P 1 ˆ ˆ i HD (i) + i>j rij − i>j Bij Ψ = EΨ, where

h i 2 ˆ D (i) = qi φ(ri ) + c P H s=x,y,z αs (i)πs (I) + α0 (I)m0 c

is the Dirac hamiltonian (see Dirac equation) for particle i at position r i and φ( r i) is the scalar potential at that position; q i is the charge of the particle, thus for electrons q i = - e. The one-electron Dirac hamiltonians of the particles, along with their instantaneous Coulomb interactions 1/r ij, form the Dirac-Coulomb operator. To this, Breit added the operator (now known as the Breit operator):   ij ) ˆij = 1 a(i) · a(j) + (a(i)·rij )(a(j)·r , B 2rij r2 ij

where the Dirac matrices for electron i : a(i) = [α x(i),α y(i),α z(i)]. The two terms in the Breit operator account for retardation effects to the first order. The wave function Ψ in the Breit equation is a spinor with 4 N elements, since each electron is described by a Dirac bispinor with 4 elements as in the Dirac equation and total wave function is the cartesian product of these.

Breit equation

46

Breit hamiltonians

FT

The total hamiltonian of the Breit equation, sometimes called the DiracCoulomb-Breit hamiltonian (H DCB) can be decomposed into the following practical energy operators for electrons in electric and magnetic fields (also called the Breit-Pauli hamiltonian) 1, which have well-defined meanings in the interaction of molecules with magnetic fields (for instance for nuclear magnetic resonance): ˆij = H ˆ0 + H ˆ 1 + ... + H ˆ6, B in which the consequitive partial operators are: •

4 ˆ 1 = − 12 P pˆi3 is connected to the dependence of mass on velocity: H i m 8c
i 2 − m c2 2 = m2 v 2 c2 . Ekin 0

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•

2 ˆ 0 = P pˆi + V is the nonrelativistic hamiltonian (m_{i} is the stationary H i 2mi mass of particle i ).

•

qi qj 2rij mi mj c2

ˆ2 = − P H i>j

 ˆi · p ˆj + p

ˆ i )·ˆ rij (rij p pj 2 rij



is a correction that partly ac-

counts for retardation and can be described as the interaction between the magnetic dipole moments of the particles, which arise from the orbital motion of charges (also called orbit-orbit interaction).   P 2qi ˆ 3 = µB P 1 si · F(rij ) × p ˆ ˆ • H + r × p is the classical interac3 i ij j i mi j>i r c ij

tion between the orbital magnetic moments (from the orbital motion of charge) and spin magnetic moments (also called spin-orbit interaction). The first term describes the interaction of a particles spin with its own orbital moment (F ( r i) is the electric field at the particles position), and the second term between two different particles.

•

ˆ4 = H

ih 8πc2

qi ˆi i m2 p i

P

· F(ri ) is a nonclassical term characteristic for Dirac

theory, sometimes called the Darwin term.   8π 1 ˆ 5 = 4µ2 P − (s · s )δ(r + • H si · sj − 3 i j ij i>j B 3 r ij

(si ·rij )(sj ·rij ) 2 rij

 is the mag-

netic moment spin-spin interaction. The first term is called the contact interaction, because it is nonzero only when the particles are at the same position; the second term is the interaction of the classical dipole-dipole type.

Breit equation

47 •

ˆ 6 = 2µB H

P h i

H(ri ) · si +

qi ˆ mi c A(ri ) · pi

i

is the interaction between spin and

Notes •

FT

orbital magnetic moments with an external magnetic field H.

Note 1: H.A. Bethe, E.E. Salpeter, Quantum Mechanics of One- and TwoElectron Atoms, Plenum Press, New York 1977, pg.181

Source: http://en.wikipedia.org/wiki/Breit_equation

Principal Authors: Karol Langner, Neilc, CambridgeBayWeather

Canonical commutation relation

In physics, the canonical commutation relation is the relation

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[x, p] = i~

among the position x and momentum p of a point particle in one dimension, where [x, p] = xp−px is the so-called commutator of x and p, i is the imaginary unit and ~ is the reduced Planck’s constant h/2π. This relation is attributed to Heisenberg, and it implies his uncertainty principle.

Relation to classical mechanics

By contrast, in classical physics all observables commute and the commutator would be zero; however, an analogous relation exists, which is obtained by replacing the commutator with the Poisson bracket and the constant i~ with 1: {x, p} = 1

This observation led Dirac to postulate that, in general, the quantum counterparts fˆ, gˆ of classical observables f, g should satisfy \ [fˆ, gˆ] = i~{f, g}.

In 1927, Hermann Weyl showed that a literal correspondence between a quantum operator and a classical distribution in phase space could not hold. However, he did propose a mechanism, Weyl quantization, that underlies a mathematical approach to quantization known as deformation quantization.

Canonical commutation relation

48

Representations

Generalizations The simple formula [x, p] = i~,

FT

According to the standard mathematical formulation of quantum mechanics, quantum observables such as x and p should be represented as self-adjoint operators on some →Hilbert space. It is relatively easy to see that two operators satisfying the canonical commutation relations cannot both be bounded. The canonical commutation relations can be made tamer by writing them in terms of the (bounded) unitary operators e−ikx and e−iap . The result is the socalled Weyl relations. The uniqueness of the canonical commutation relations between position and momentum is guaranteed by the Stone-von Neumann theorem. The group associated with the commutation relations is called the Heisenberg group.

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valid for the quantization of the simplest classical system, can be generalized to the case of an arbitrary Lagrangian L . We identify canonical coordinates (such as x in the example above, or a field φ(x) in the case of quantum field theory) and canonical momenta πx (in the example above it is p, or more generally, some functions involving the derivatives of the canonical coordinates with respect to time).

πi ≡

∂L ∂(∂xi /∂t)

This definition of the canonical momentum ensures that one of the EulerLagrange equations has the form ∂ ∂t πi

=

∂L ∂xi

The canonical commutation relations then say [xi , πj ] = i~δij

where δij is the Kronecker delta.

Gauge invariance

Canonical quantization is performed, by definition, on canonical coordinates. However, in the presence of an electromagnetic field, the canonical momentum

Canonical commutation relation

49 p is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is

FT

p − eA/c where e is the quantum of electric charge, and A is the vector potential and c is the speed of light. Although this quantity is the "physical momentum" in that it is the quantity to be identified with momentum in laboratory experiments, it does not satisfy the canonical commutation relations; only the kinetic momentum does that. This can be seen as follows. The non-relativistic Hamiltonian for a quantized charged particle of mass m in a classical electromagnetic field is  2 1 H = 2m + eφ p − eA c

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where A is the three-vector potential and φ is the scalar potential. This form of the Hamiltonian, as well as the Schroedinger equation Hψ = i~∂ψ/∂t, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation A → A0 = A + ∇Λ φ → φ0 −

1 ∂Λ c ∂t

ψ → ψ0 = U ψ

H → H 0 = U HU †

where

U = exp



ieΛ ~c



and Λ = Λ(x, t) is the gauge function. The canonical angular momentum is L=r×p

and obeys the canonical quantization relations [Li , Lj ] = i~ijk Lk

Canonical commutation relation

50 defining the Lie algebra for so(3), where ijk is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as e ~c hψ|r

× ∇Λ|ψi

FT

hψ|L|ψi → hψ 0 |L0 |ψ 0 i = hψ|L|ψi +

The gauge-invariant angular momentum (or "kinetic angular momentum") is given by   K = r × p − eA c which has the commutation relations   [Ki , Kj ] = i~ij k Kk + e~ c xk (x · B) where B =∇×A

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is the magnetic field. The inequivalence of these two formulations shows up in the →Zeeman effect and the →Aharonov-Bohm effect.

See also • • •

canonical quantization CCR algebra Lie derivative

Source: http://en.wikipedia.org/wiki/Canonical_commutation_relation Principal Authors: Miguel, Ancheta Wis, Linas, Lumidek, Michael Hardy

Chladni’s law

Chladni’s law, named after Ernst Chladni, relates the frequency of modes of vibration for flat circular surfaces with fixed center as a function of the numbers m of diametric (linear) nodes and n of radial (circular) nodes. It is stated as the equation f = C(m + 2n)p

where C and p are coefficients which depend on the properties of the plate. Chladni’s law

51

External links •

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For flat circular plates, p is roughly 2, but Chladni’s law can also be used to describe the vibrations of cymbals, handbells, and church bells in which case p can vary from 1.4 to 2.4. In fact, p can even vary for a single object, depending on which family of modes is being examined.

A Study of Vibrating Plates 31 by Derek Kverno and Jim Nolen

Source: http://en.wikipedia.org/wiki/Chladni%27s_law

Principal Authors: Laurascudder, Michael Hardy, Choster

Classical limit

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The classical limit is the ability of a physical theory to approximate or "recover" classical mechanics when considered over special values of its parameters. The classical limit is used with physical theories that predict non-classical behavior. A postulate called the correspondence principle was introduced to quantum theory by Niels Bohr; it states that, in effect, some kind of continuity argument should apply to the classical limit of quantum systems as the value of Planck’s constant tends to zero. In quantum mechanics, due to the Heisenberg’s uncertainty principle, an electron can never be at rest; it must always have a non-zero kinetic energy, a result not found in classical mechanics. For example, if we consider something very large relative to an electron, like a baseball, the uncertainty principle predicts that it cannot have zero kinetic energy, but the uncertainty in kinetic energy is so small that the baseball can appear to be at rest, and hence appears to obey classical mechanics. In general, if large energies and large objects (relative to the size and energy levels of an electron) are considered in quantum mechanics, the result will appear to obey classical mechanics. In general and special relativity, if we consider flat space, small masses, and small speeds (in comparison to the speed of light), we find that objects once again appear to obey classical mechanics.

Source: http://en.wikipedia.org/wiki/Classical_limit

31 http://www.phy.davidson.edu/StuHome/derekk/Chladni/pages/menu.htm

Classical limit

52 Principal Authors: Cyan, Michael Hardy, Charles Matthews, Salsb

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Clebsch-Gordan coefficients

In physics, the Clebsch-Gordan coefficients are sets of numbers that arise in calculations involving addition of angular momentum under the laws of quantum mechanics. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly. The name derives from the German mathematicians Alfred Clebsch (1833-1872) and Paul Gordan (1837-1912), who encountered an equivalent problem in invariant theory.

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In terms of classical mathematics, the CG coefficients, or at least those associated to the group SO(3), may be defined much more directly, by means of formulae for the multiplication of spherical harmonics. The addition of spins in quantum-mechanical terms can be read directly from this approach. The formulas below use Dirac’s bra-ket notation.

Formal definition and some results

The Clebsch-Gordan coefficients are the numerical constants that express the probability amplitude for the spins j1 , j2 with z-projections m1 , m2 to add to j with z projection m P P |j1 j2 ; jmi = jm1 1 =−j1 jm2 2 =−j2 |j1 j2 ; m1 m2 ihj1 j2 ; m1 m2 |j1 j2 ; jmi where hj1 j2 ; m1 m2 |j1 j2 ; jmi are the CG coefficients. m = m1 + m2 if hj1 j2 ; m1 m2 |j1 j2 ; jmi = 6 0.

The following holds: Pj1 Pj2 |j1 j2 ; jmi = m m2 =−j2 |j1 j2 ; m1 m2 ihj1 j2 ; m1 m2 |j1 j2 ; jmi 1 =−j1 renaming m1 to m01 and m2 to m02 and applying the J± operator P Pj2 |j1 j2 ; m01 m02 ihj1 j2 ; m01 m02 |j1 j2 ; jmi J± |j1 j2 ; jmi = (J1± +J2± ) jm1 0 =−j m0 =−j 1

1

2

2

We get a some-what long equation:

Clebsch-Gordan coefficients

53 p

(j ∓ m)(j ± m + 1)|j1 j2 ; j, m ± 1i (j1 ∓ m01 )(j1 m02 hj1 j2 ; m01 m02 |j1 j2 ; jmi P

m01

p

P

± m01 + 1)|j1 j2 ; m01 ± 1, m02 i +

p (j2 ∓ m02 )(j2 ±

FT

=

and arbitrarily choosing one particular m1 and m2 and multiplying both sides by a bra on the left (note that with the new m1 and m2 , m1 + m2 = m ± 1 when the coefficients are not 0) p (j ∓ m)(j ± m + 1)hj1 j2 ; m1 m2 |j1 j2 ; j, m ± 1i (j1 ∓ m01 )(j1 m02 hj1 j2 ; m01 m02 |j1 j2 ; jmi

=

P

m01

P

p

± m01 + 1)hj1 j2 ; m1 m2 |j1 j2 ; m01 ± 1, m02 i +

p (j2 ∓ m

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and summing, now that most elements of the sum are 0 (note that hj1 j2 ; m1 m2 |j1 j2 ; m01 ± 1, m02 i ≥ 0 if m01 6= m1 ∓ 1 or m02 6= m2 ) etc... p (j ∓ m)(j ± m + 1)hj1 j2 ; m1 m2 |j1 j2 ; j, m ± 1i

p p = (j1 ∓ m1 + 1)(j1 ± m1 )hj1 j2 ; m1 ∓1, m2 |j1 j2 ; jmi+ (j2 ∓ m2 + 1)(j2 ± m2 )h 1|j1 j2 ; jmi

replacing m with m ∓ 1 so that again m1 + m2 = m when the coefficients are not 0 p (j ∓ m + 1)(j ± m)hj1 j2 ; m1 m2 |j1 j2 ; jmi p = (j1 ∓ m1 + 1)(j1 ± m1 )hj1 j2 ; m1 ∓ 1, m2 |j1 j2 ; j, m ∓ 1i + p (j2 ∓ m2 + 1)(j2 ± m2 )hj1 j2 ; m1 , m2 ∓ 1|j1 j2 ; j, m ∓ 1i

The above formula is useful for finding the last Clebsch-Gordan coefficients, when the other one or two coefficients in the formula are known. Note that there are sometimes only two coefficients in the formula, the third being both invalid (j < |m|) and multiplied by 0. Guessing one of the coefficients, using the formula to find the rest and normalising so that the first coefficient becomes correct, one can find, for example (up to sign) 5

5  q 10 1 7 3 2 1; m1 = 2 , m2 = 1 2 1; j = 2 , m = 2 = 21

For more coefficients, see table of Clebsch-Gordan coefficients.

Clebsch-Gordan coefficients

54

See also Wigner 3-j symbol 6-j symbol Spherical harmonics Associated Legendre polynomials Angular momentum Angular momentum coupling

FT

• • • • • •

External links •

Java TM Clebsch-Gordan Coefficient Calculator 32

References •

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A.R. Edmonds, Angular Momentum in Quantum Mechanics, (1957) Princeton University Press, ISBN 0-691-07912-9. • E. U. Condon and G. H. Shortley, The Theory of Atomic Spectra, (1970) Cambridge at the University Press, ISBN 521-09209-4 See chapter 3. • Albert Messiah, Quantum Mechanics (Volume II), (1966) North Holland Publishing, ISBN ???? (something that looks the same but doesn’t mention 1966 is ISBN 0720400457)

Source: http://en.wikipedia.org/wiki/Clebsch-Gordan_coefficients

Principal Authors: Cyp, Charles Matthews, Linas, ArnoldReinhold, JabberWok

Coherent state

In quantum mechanics a coherent state is a specific kind of quantum state of the quantum harmonic oscillator whose dynamics most closely resemble the oscillating behaviour of a classical harmonic oscillator system. It was the first example of quantum dynamics when Erwin Schrödinger derived it in 1926 while searching for solutions of the →Schrödinger equation that satisfy the correspondence principle. The quantum harmonic oscillator and hence, the coherent state, arise in the quantum theory of a wide range of physical systems. For instance, a coherent state describes the oscillating motion of the particle

32 http://www.gleet.org.uk/cleb/cgjava.html

Coherent state

55

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in a quadratic potential well. In the quantum theory of light (quantum electrodynamics) and other bosonic quantum field theories they were introduced by the work of Roy J. Glauber. Here the coherent state of a field describes an oscillating field, the closest quantum state to a classical sinusoidal wave such as a continuous laser wave.

Figure 5 Figure 1: The electric field, measured by optical homodyne detection, as a function of phase for three coherent states emitted by a Nd:YAG laser. The amount of quantum noise in the electric field is completely independent of the phase. As the field strength, i.e. the oscillation amplitude α of the coherent state is increased, the quantum noise or uncertainty is constant at 1/2, and so becomes less and less significant. In the limit of large field the state becomes a good approximation of a noiseless stable classical wave. The average photon numbers of the three states from top to bottom are =4.2, 25.2, 924.5 (source: link 1 and ref. 2)

Coherent states in quantum optics

In classical optics light is thought of as electromagnetic waves radiating from a source. Specifically, coherent light is thought of as light that is emitted by many such sources that are in phase. For instance a light bulb radiates light that is the result of waves being emitted at all the points along the filament. Such light is incoherent because the process is highly random in space and time (see Coherent state

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56

Figure 6 Figure 2: The oscillating wave packet corresponding to the second coherent state depicted in Figure 1. At each phase of the light field, the distribution is a Gaussian of constant width.

thermal light). In a laser, however, light is emitted by a carefully controlled system in processes that are not random but interconnected by stimulation and the resulting light is highly ordered, or coherent. Therefore a coherent state corresponds closely to the quantum state of light emitted by an ideal laser. Semi-classically we describe such a state by an electric field oscillating as a stable wave. Contrary to the coherent state, which is the most wave-like quantum state, the →Fock state (e.g. a single photon) is the most particle-like state. It is indivisible and contains only one quanta of energy. These two states are examples of the opposite extremes in the concept of wave-particle duality. A coherent state distributes its quantum-mechanical uncertainty equally, which means that the phase and amplitude uncertainty are approximately equal. Conversely, in a single-particle state the phase is completely uncertain.

Quantum mechanical definition

Mathematically, the coherent state |αi is defined to be the eigenstate of the annihilation operator a. Formally, this reads: a|αi = α|αi

Coherent state

FT

57

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Figure 7 Figure 3: Wigner function of the coherent state depicted in Figure 2. The distribution is centered on state’s amplitude α and is symmetric around this point. The ripples are due to experimental errors.

Since a is not hermitian, α = |α|eiθ

is complex. Here |α| and θ are called the amplitude and phase of the state. Physically, this formula means that a coherent state is left unchanged by the detection (or annihilation) of a particle. Consequently, in a coherent state, one has exactly the same probability to detect a second particle. Note, this condition is necessary for the coherent state’s Poissonian detection statistics, as discussed below. Compare this to a single-particle state (→Fock state): Once one particle is detected, we have zero probability of detecting another. For the following discussion we need to define the dimensionless X and P quadratures. For a harmonic oscillator, x = (mωπ/h) -1/2X is the oscillating particle’s position and p = (mωh/π) -1/2P is its momentum. For an optical field, E R = (hω/π 0V) 1/2cosθX ; and E I = (hω/π 0V) 1/2sinθP ; are the real and imaginary components of the electric field. Erwin Schrödinger was searching for the most classical-like states when he first introduced coherent states. He described them as the quantum state of the harmonic oscillator which minimizes the uncertainty relation with uncertainty equally distributed in both X and P quadratures (ie. ∆X = ∆Y = 1/2). From Coherent state

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58

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Figure 8 Figure 4: The probability of detecting n photons, the photon number distribution, of the coherent state in Figure 3. As is necessary for a Poissonian distribution the mean photon number is equal to the variance of the photon number distribution. Bars refer to theory, dots to experimental values.

the generalized uncertainty relation, it is shown that such a state |α> must obey the equation (P − hP i)|αi = i(X − hXi)|αi

In the general case, if the uncertainty is not equally distributed in the X and P component, the state is called a squeezed coherent state. If this formula is written back in terms of a and a †, it becomes: a|αi = (hXi + ihP i)|αi

The coherent state’s location in the complex plane (phase space) is centered at the position and momentum of a classical oscillator of the same phase θ and amplitude (or the same complex electric field value for an electromagnetic wave). As shown in Figure 2, the uncertainty, equally spread in all directions, is represented by a disk with diameter 1/2. As the phase increases the coherent state circles the origin and the disk neither distorts nor spreads. This is the most similar a quantum state can be to a single point in phase space. Since the uncertainty (and hence measurement noise) stays constant at 1/2 as the amplitude of the oscillation increases, the state behaves more and more Coherent state

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59

Figure 9 Figure 5: Phase space plot of a coherent state. This shows that the uncertainty in a coherent state is equally distributed in all directions. The horizontal and vertical axes are the X and P quadratures, respectively (see text). The red dots on the x-axis trace out the boundaries of the quantum noise in Figure 1.

like a sinusoidal wave, as shown in Figure 1. Conversely, since the vacuum state |0> is just the coherent state with α=0, all coherent states have the same uncertainty as the vacuum. Therefore one can interpret the quantum noise of a coherent state as being due to the vacuum fluctuations. Furthermore, it is sometimes useful to define a coherent state simply as the vacuum state displaced to a location α in phase space. Mathematically this is done by the action of the displacement operator D(α): |αi = eαa

†

−α∗ a |0i

= D(α)|0i

This can be easily obtained, as can virtually all results involving coherent states, using the representation of the coherent state in the basis of Fock states: |α|2 P ∞ √ αn |αi = e− 2 |ni. n=0 n!

Coherent state

60

P (n) = e−hni

hnin n!

FT

A stable classical wave has a constant intensity. Consequently, the probability of detecting n photons in a given amount of time is constant with time. This condition ensures there will be shot noise in our detection. Specificially, the probability of detecting n photons is Poissonian:

Similarly, the average photon number in coherent state ==|α| 2 and the variance (∆n) 2= Var(a †a)=|α| 2, identical to the variance of the Poissonian distribution. Not only does a coherent state go to a classical sinusoidal wave in the limit of large α but the detection statistics of it are equal to that of a classical stable wave for all values of α.

DR A

This also follows from the fact that for the prediction of the detection results at a single detector (and time) any state of light can always be modelled as a collection of classical waves (see degree of coherence). However, for the prediction of higher-order measurement like intensity correlations (which measure the degree of nth-order coherence) this is not true. The coherent state is unique in the fact that all n-orders of coherence are equal to 1. It is perfectly coherent to all orders. There are other reasons why a coherent state can be considered the most classical state. Roy J. Glauber coined the term "coherent state" and proved they are produced when a classical electrical current interacts with the electromagnetic field. In the process he introduced the coherent state to quantum optics. In general when a quantum state of light is split at a beamsplitter, the two output modes are entangled. Aharonov proved that coherent states are the only pure states of light that remain unentangled (and thus classical) when split into two states.

From Figure 5, simple geometry gives ∆θ=1/2|α|. From this we can see that there is a tradeoff between number uncertainty and phase uncertainty ∆θ∆n = 1/2, the number-phase uncertainty relation. This is not a formal uncertainty relation: there is no uniquely defined phase operator in quantum mechanics.

Mathematical characteristics

The coherent state does not display all the nice mathematical features of a →Fock state; for instance two different coherent states are not orthogonal: 1

2

hβ|αi = e− 2 (|β|

+|α|2 −2β ∗ α)

6= δ(α − β)

so that if the oscillator is in the quantum state |α> it is also with nonzero probability in the other quantum state |β> (but the farther apart the states are situated in phase space, the lower the probability is). However, since they obey Coherent state

61

FT

a closure relation, any state can be decomposed on the set of coherent states. They hence form an overcomplete basis in which one can diagonally decompose any state. This is the premise for the Sudarshan-Glauber P representation. Another difficulty is that a † has no eigenket (and a has no eigenbra). The following formal equality is the closest substitute and turns out to be very useful for technical computations:   ∗ ∂ a† |αi = ∂α + α2 |αi

Coherent states of Bose-Einstein condensates

A Bose-Einstein condensate (BEC) is a collection of boson atoms that are all in the same quantum state. An approximate theoretical description of its properties can be derived by assuming the BEC is in a coherent state. However, unlike photons atoms interact with each other so it now appears that it more likely to be one of the squeezed coherent states mentioned above.

DR A

•

Generalizations •

In quantum field theory and string theory, a generalization of coherent states to the case of infinitely many degrees of freedom is used to define a vacuum state with a different vacuum expectation value from the original vacuum.

See also

• →Quantum field theory • Quantum optics • Electromagnetic field • degree of coherence • quantum coherence

External links •

Quantum states of the light field 33

33 http://gerdbreitenbach.de/gallery

Coherent state

62

References • • •

FT

E. Schrödinger, Naturwissenschaften 14 (1926) 664. R.J. Glauber, Phys. Rev. 131 (1963) 2766. Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 2000), [ISBN 0198501773] • G. Breitenbach, S. Schiller, and J. Mlynek, "Measurement of the quantum states of squeezed light", Nature, 387, 471 (1997) 34

Source: http://en.wikipedia.org/wiki/Coherent_state

Principal Authors: J S Lundeen, Gerd Breitenbach, Laussy, Charles Matthews, JerryFriedman

Complementarity (physics)

DR A

In physics, complementarity is a basic principle of quantum theory, and refers to effects such as the wave-particle duality, in which different measurements made on a system reveal it to have either particle-like or wave-like properties. Niels Bohr is usually associated with this concept; in the orthodox form, it is stated that a quantum mechanical system consisting of a boson or fermion can either behave as a particle or as wave, but never simultaneously as both. A less orthodox interpretation is the "duality condition," described by the inequality proven by Jaeger, Shimony, and Vaidman (G. Jaeger, A. Shimony, and L. Vaidman, Phys. Rev. A, Vol. 51, 54 (1995)), and later by Englert (B. Englert, Phys. Rev. Lett., Vol. 77, 2154 (1996)), which allows wave and particle attributes to co-exist, but postulates that a stronger manifestation of the particle nature leads to a weaker manifestation of the wave nature and vice versa. →Wave-particle duality is considered to be one of the distinguishing characteristics of quantum mechanics, whose theoretical and experimental development has been honoured by more than a few Nobel Prizes for Physics. It has been discussed by prominent physicists for the last 100 years, from the time of Albert Einstein, Niels Bohr and Werner Heisenberg, onwards. On the basis of Bohr’s principle of complementarity, it is indeed universally accepted that the observation of two complementary properties, such as position and momentum, requires mutually exclusive experimental measurements. The emergence of complementarity in a system occurs when one considers the circumstances under which one attempts to measure its properties; as Bohr

34 http://www.exphy.uni-duesseldorf.de/Publikationen/1997/N387/471z.htm

Complementarity (physics)

63

FT

noted, the principle of complementarity "implies the impossibility of any sharp separation between the behaviour of atomic objects and the interaction with the measuring instruments which serve to define the conditions under which the phenomena appear." It is important to distinguish, as did Bohr in his original statements, the principle of complementarity from a statement of the uncertainty principle. For a technical discussion of contemporary issues surrounding complementarity in physics, see, e.g., 35 (from which parts of this discussion were drawn.)

DR A

Various neutron interferometry experiments demonstrate the subtleness of the notions of duality and complementarity in an interesting way. In order to pass through the interferometer, the neutron must act as a wave. Yet upon passage, the neutron is subject to gravitation, which one would think only particles, and not waves, are subject to. As the neutron interferometer is rotated through Earth’s gravitational field a phase change between the two arms of the interferometer is created, resulting in a change in the constructive and destructive interference of the neutron waves on exit from the interferometer. Note that in order to understand the interference effect one must concede that a single neutron takes both paths through the interferometer at the same time: a single neutron must "be in two places at once", as it were. Since the two paths through a neutron interferometer can be as far as five to 15 cm apart, the effect is hardly microscopic. This is not in contrast to traditional double-slit experiments (or mirror interferometer) where the slits (or mirrors) can be arbitrary far apart. So in interference and diffraction experiments neutron behaves the same way as a photon (or an electron) of corresponding wavelength.

The mathematics of two-slit diffraction

This section reviews the mathematical formulation of the double-aperture experiment (see Fig.1). The formulation is in terms of the diffraction and interference of waves. The culmination of the develpment is a presentation of two numbers that characterizes the visibility of the interference fringes in the experiment, linked together as the Englert-Greenberger duality relation.The next will then discuss the orthodox quantum mechanical interpretation of the duality relation in terms of wave-particle duality. Of this experiment, Richard Feynman once said that it “has in it the heart of quantum mechanics. In reality it contains the only mystery´´. The wave function in the Young double-aperture experiment can be written as ΨTotal (x) = ΨA (x) + ΨB (x).

35 http://citebase.eprints.org/cgi-bin/citations?id=oai:arXiv.org:quant-ph/0003073

Complementarity (physics)

64 The function ΨA (x) = CA Ψ0 (x − xA )

Ψ0 (x) ∝

FT

is the wave function associated with the pinhole at A centered on xA ; a similar relation holds for pinhole B. The variable x is a position in space downstream of the slits. The constants CA and CB are proportionality factors for the corresponding wave amplitudes, and Ψ0 (x) is the single hole wave function for an aperture centered on the origin. The single-hole wave-function is taken to be that of Fraunhofer diffraction; the pinhole shape is irrelevant, and the pinholes are considred to be idealized. The wave is taken to have a fixed incident momentum p0 = h/λ: eip0 ·|x|/~ |x|

where |x| is the radial distance from the pinhole.

DR A

To distinguish which pinhole a photon passed through, one needs some measure of the distinguishability between pinholes. Such a measure is given by |C |2 −|C |2 D = |CA |2 +|CB |2 = |PA − PB |, A

B

where

PA =

|CA |2 |CA |2 +|CB |2

and

PB =

|CB |2 |CA |2 +|CB |2

are the probabilities of finding that the particle passed through aperture A or aperture B. We have in particular D = 0 for two symmetric holes and D = 1 for a single aperture (perfect distinguishability). In the far-field of the two pinholes the two waves interfere and produce fringes. The intensity of the interference pattern at a point y in the focal plane (denoted by (F) in the image) is given by I(y) ∝ 1 + V cos (py d/~ + φ)

where py = h/λ · sin(α) is the momentum of the particle along the y direction, φ = Arg(CA ) − Arg(CB ) is a fixed phase shift, and d is the separation between the two pinholes. The angle α from the horizontal is given by sin(α) ' tan(α) = y/L where L is the distance between the aperture screen and the far field analysis plane. If a lens is used to observe the fringes in the Complementarity (physics)

65 rear focal plane (F), the angle is given by sin(α) ' tan(α) = y/f where f is the focal length of the lens. The visibility of the fringes is defined by Imax −Imin Imax +Imin

FT

V =

where max and min denote the maximum and minimum intensity of the fringes. Equivalently, this can be written as |CA ·CB∗ | 2 2. A | +|CB |

V = 2 |C

In a single hole experiment, the fringe visibility will be zero (as there are no fringes); that is, V = 0. On the other hand, for a two slit configuration, where the two slits are indistinguishable, one has perfect visibility; that is, V = 1 or √ |CA | = |CB | = 1/ 2. It is straighforward to see that the duality relation V 2 + D2 = 1

DR A

is always true.

The above presentation was limited to a pure quantum state. More generally, for a mixture of quantum states, one will have V 2 + D2 ≤ 1.

For the remainder of the development, the light source will be assumed to be a laser, so that V 2 + D2 = 1 can be assumed to hold, following from the coherence properties of laser light.

Complementarity

The mathematical discussion presented above does not require quantum mechanics at its heart. In particular, the derivation is essentially valid for waves of any sort. With slight modifications to account for the squaring of amplitudes, the derivation could be applied to, for example, sound waves or water waves in a ripple tank. In order for the relation to be a precise formulation of Bohr complementarity one must introduce wave-particle duality in the discussion. This means one must consider both wave and particle behavior of light on an equal footing. Wave-particle duality implies that one must A) use the unitary evolution of the wave before the observation and B) consider the particle aspect after the detection (this is called the Heisenberg-von Neumann collapse postulate). Indeed since one could only observe the photon in one point of space (a photon can not be absorbed twice) this implies that the meaning of the wave function Complementarity (physics)

66 is essentially statistical and can not be confused with a classical wave (like it exists in air or water).

FT

In this context the direct observation of a photon in the aperture plane precludes the following recording of the same photon in (F). Reciprocally the observation in (F) means that we did not absorb the photon before. If both holes are open this implies that we don’t know where we would have detected the photon in the aperture plane. D defines thus the distinguishability of the two holes A and B.

A maximal value of distinguishability D = 1 means that only one hole (say A ) is open. If now we detect the photon at (F), we know that that photon would have been detected in A necessarily. Conversely, D = 0 means that both holes are open and play a symmetric role. If we detect the photon at (F), we don’t know where the photon would have been detected in the aperture plane and D = 0 characterizes our ignorance.

DR A

Similarly, if D = 0 then V = 1 and this means that a statistical accumulation of photons at (F) will build up an interference pattern with maximal visibility. Conversely, D = 1 implies V = 0 and thus, no fringes will appear after a statistical recording of several photons. The above treatment formalizes wave particle duality for the double-slit experiment.

See also • • • •

Afshar experiment →Wave-particle duality →Quantum entanglement →Quantum indeterminacy

Further Reading •

Bethold-Georg Englert, Marlan O Scully & Herbert Walther, Quantum Optical Tests of Complementarity , Nature, Vol 351, pp 111-116 (9 May 1991). Demonstrates that quantum interference effects are destroyed by irreversible object-apparatus correlations ("measurement"), not by Heisenberg’s uncertainty principle itself. See also The Duality in Matter and Light Scientific American, (December 1994)

Source: http://en.wikipedia.org/wiki/Complementarity_%28physics%29

Principal Authors: Danko Georgiev MD, Linas, Dewain Belgard, Enormousdude, Arcturus

Complementarity (physics)

67

Complete set of commuting observables

FT

In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system.

For example, in the case of the hydrogen atom, the Hamiltonian H, the angular momentum L and its projection L z along any arbitrary z axis form a CSCO (if one ignores the spin of the proton and of the electron as well as the movement of the proton).

Source: http://en.wikipedia.org/wiki/Complete_set_of_commuting_observables Principal Authors: TobinFricke, Conscious, Gazpacho

DR A

Compton scattering

In physics, Compton scattering or the Compton effect, is the decrease in energy (increase in wavelength) of an X-ray or gamma ray photon, when it interacts with matter. The amount the wavelength increases by is called the Compton shift. Although nuclear compton scattering exists, what is meant by Compton scattering usually is the interaction involving only the electrons of an atom. Compton effect was observed by Arthur Holly Compton in 1923, for which he earned the 1927 Nobel Prize in Physics. The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering, the classical theory of charged particles scattered by an electromagnetic wave, cannot explain any shift in wavelength. Light must behave as if it consists of particles in order to explain the Compton scattering. Compton’s experiment convinced physicists that light can behave as a stream of particles whose energy is proportional to the frequency. The interaction between high energy photons and electrons results in the electron being given part of the energy (making it recoil), and a photon containing the remaining energy being emitted in a different direction from the original, so that the overall momentum of the system is conserved. If the photon still has enough energy left, the process may be repeated.

Compton scattering

68

FT

Compton scattering occurs in all materials and predominantly with photons of medium energy, i.e. about 0.5 to 3.5 MeV. It is also observed that highenergy photons; photons of visible light of higher frequency, for example, have sufficient energy to even eject the bound electrons from the atom (photoelectric effect).

The Compton shift formula

DR A

For differential cross section of Compton scattering, see Klein-Nishina formula.

Figure 10

Compton Scattering (in the rest frame of the target)

Compton used a combination of three fundamental formulas representing the various aspects of classical and modern physics, combining them to describe the quantum behavior of light. • • •

Light as a particle, as noted previously in the photoelectric effect. Relativistic dynamics Special Theory of Relativity Trigonometry - Law of cosines

The final result gives us the Compton scattering equation: λ2 =

h me c (1 − cos θ) + λ1

where

λ1 is the wavelength of the photon before scattering,

λ2 is the wavelength of the photon after scattering,

Compton scattering

69 m e is the mass of the electron,

FT

h/(m ec) is known as the →Compton wavelength, θ is the angle by which the photon’s heading changes, h is Planck’s constant, and c is the speed of light.

Collectively, the Compton wavelength is 2.43×10 -12 meter.

Derivation We use that:

DR A

Eγ + Ee = Eγ 0 + Ee0

(Conservation of energy, where Eγ is the energy of a photon before the collision and Ee is the energy of an electron before collision — its rest mass). The variables with a prime are used for those after the collision. And: p~γ + p~e = p~γ 0 + p~e0

(Conservation of momentum, with the pe = 0 because we assume that the electron is at rest.) We then use E = hf = pc: p~e0 = p~γ − p~γ 0

p~e0 2 = (p~γ − p~γ 0 )2

p~e0 2 = p~γ 2 − 2 · p~γ · p~γ 0 + p~γ 0 2

p~e0 · p~e0 = p~γ · p~γ − 2 · p~γ · p~γ 0 + p~γ 0 · p~γ 0

pe0 2 · cos(0) = p2γ · cos(0) − 2 · pγ · pγ 0 · cos(θ) + p2γ 0 · cos(0)

The cos(θ) term appears because the momenta are spatial vectors, all of which lie in a single 2D plane, thus their inner product is the product of their norms Compton scattering

70 multiplied by the cosine of the angle between them. p2e0

=

h2 f 2 c2

+

hf c

and pγ 0 with

−

2h2 f f 0 cos θ c2

h2 f 02 c2

Now we fill in for the energy part: Eγ + Ee = Eγ 0 + Ee0 hf + mc2 = hf 0 +

hf 0 c ,

we derive

FT

substituting pγ with

p (pe0 c)2 + (mc2 )2

We solve this for p e’:

(hf + mc2 − hf 0 )2 = (pe0 c)2 + (mc2 )2 (hf +mc2 −hf 0 )2 −m2 c4 c2

= p2e0

DR A

Then we have two equations for p2e0 , which we equate: (hf +mc2 −hf 0 )2 −m2 c4 c2

=

h2 f 2 c2

+

h2 f 02 c2

−

2h2 f f 0 cos θ c2

Now it’s just a question of rewriting:

h2 f 2 + h2 f 02 − 2h2 f f 0 + 2h(f − f 0 )mc2 = h2 f 2 + h2 f 02 − 2h2 f f 0 cos θ

−2h2 f f 0 + 2h(f − f 0 )mc2 = −2h2 f f 0 cos θ hf f 0 − (f − f 0 )mc2 = hf f 0 cos θ hf f 0 (1 − cos θ) = (f − f 0 )mc2

h λc0 λc (1 − cos θ) =

c λ

h λc0 λc (1 − cos θ) =



λ0 λ c c



cλ0 λ0 λ

−

λ c

h(1 − cos θ) = h(1 − cos θ) =



λ0 c

c λ0

−

cλ0 λλ0




mc2

−

cλ λ0 λ



mc2

−

cλ λλ0



mc2



mc2

Compton scattering

71 h mc (1 − cos θ)

= λ0 − λ

FT

Applications Compton scattering is of prime importance to radiobiology, as it happens to be the most probable interaction of high energy X rays with atomic nuclei in living beings and is applied in radiation therapy.

Compton scattering has on occasion been proposed as an alternative explanation for the phenomenon of the redshift by opponents of the Big Bang theory, although this is not generally accepted because the influence of the Compton scattering would be noticeable in the spectral lines of distant objects and this is not observed. In material physics, Compton scattering can be used to probe the wave function of the electrons in matter in the momentum representation.

DR A

Compton Scatter is an important effect in Gamma spectroscopy, as it is possible for the gamma rays to scatter out of the detectors used. Compton suppression is used to detect stray scatter gamma rays to counteract this effect.

See also

• • • • • • • •

Thomson scattering →Photoelectric effect Timeline of cosmic microwave background astronomy Peter Debye Sunyaev Zel’dovich effect Walther Bothe List of astronomical topics List of physics topics

External links •

Compton Effect 36 (PDF file) by Michael Brandl for Project PHYSNET 37.

Source: http://en.wikipedia.org/wiki/Compton_scattering

Principal Authors: Pt, Eteq, Fresheneesz, Eleassar777, Omegatron, Silenced, Reddi, Pfalstad, Niven, AugPi

36 http://35.9.69.219/home/modules/pdf_modules/m219.pdf 37 http://physnet2.pa.msu.edu/

Compton scattering

72

Compton wavelength λ=

h mc

~ = 2π mc ,

where h is the Planck constant, m is the particle’s mass, c is the speed of light.

FT

The Compton wavelength λ of a particle is given by

The Compton wavelength of the electron is approximately 2.4 × 10 -12 meters.

DR A

The Compton wavelength can be thought of as a fundamental limitation on measuring the position of a particle, taking quantum mechanics and special relativity into account. This depends on the mass m of the particle. To see this, note that we can measure the position of a particle by bouncing light off it - but measuring the position accurately requires light of short wavelength. Light with a short wavelength consists of photons of high energy. If the energy of these photons exceeds mc2 , when one hits the particle whose position is being measured the collision may have enough energy to create a new particle of the same type. This renders moot the question of the original particle’s location. This argument also shows that the Compton wavelength is the cutoff below which quantum field theory– which can describe particle creation and annihilation – becomes important.

We can make the above argument a bit more precise as follows. Suppose we wish to measure the position of a particle to within an accuracy ∆x . Then the uncertainty relation for position and momentum says that ∆x ∆p ≥ ~/2 ~ so the uncertainty in the particle’s momentum satisfies ∆p ≥ 2∆x Using the relativistic relation between momentum and energy, when ∆p exceeds mc then the uncertainty in energy is greater than mc2 , which is enough energy to create another particle of the same type. So, with a little algebra, we see ~ there is a fundamental limitation ∆x ≥ 2mc So, at least to within an order of magnitude, the uncertainty in position must be greater than the Compton wavelength h/mc .

Compton wavelength

73 The Compton wavelength can be contrasted with the de Broglie wavelength, which depends on the momentum of a particle and determines the cutoff between particle and wave behavior in quantum mechanics.

FT

For fermions, the Compton wavelength sets the cross-section of interactions. For example, the cross-section for Thomson scattering of a photon from an electron is equal to (8π/3)α2 λ2e , where α is the fine-structure constant and λe is the Compton wavelength of the electron. For gauge bosons, the Compton wavelength sets the effective range of the Yukawa interaction: since the photon is massless, electromagnetism has infinite range.

The Compton wavelength of the electron is one of a trio of related units of length, the other two being the Bohr radius a0 and the classical electron radius re . The Compton wavelength is built from the electron mass me , Planck’s constant h and the speed of light c. The Bohr radius is built from me , h and the electron charge e. The classical electron radius is built from me , c and e. Any one of these three lengths can be written in terms of any other using the fine structure constant α: αλe 2π

= α2 a0

DR A

re =

The Planck mass is special because ignoring factors of 2π and the like, the Compton wavelength for this mass is equal to its Schwarzschild radius. This special distance is called the Planck length. This is a simple case of dimensional analysis: the Schwarzschild radius is proportional to the mass, whereas the Compton wavelength is proportional to the inverse of the mass.

External links •

Length Scales in Physics: the Compton Wavelength 38

Source: http://en.wikipedia.org/wiki/Compton_wavelength

Principal Authors: Joke137, John Baez, MagnaMopus, Rmrfstar, Wigie

38 http://math.ucr.edu/home/baez/lengths.html#compton_wavelength

Compton wavelength

74

Conjugate variables

FT

In physics, especially in quantum mechanics, conjugate variables are pairs of variables that share an uncertainty relation. The terminology comes from classical Hamiltonian mechanics, but also appears in quantum mechanics and engineering. Examples of canonically conjugate variables include the following: •

DR A

Time and frequency: the longer a musical note is sustained, the more precise we know its frequency (but it spans more time). Conversely, a very short musical note becomes just a click, and so one can’t know its frequency very accurately. • Position and momentum: precise measurements of position lead to ambiguity of momentum, and v.v. • Doppler and range: the more we know about how far away a radar target is, the less we can know about the exact velocity of approach or retreat, and vice versa. In this case, the two dimensional function of doppler and range is known as a radar ambiguity function or radar ambiguity diagram. A pair of conjugate variables are often Fourier transform duals of one-another, or more generally are related through Pontryagin duality. The duality relations lead naturally to an uncertainty relation between them. A more precise mathematical definition, in the context of Hamiltonian mechanics, is given in the article canonical coordinates.

Source: http://en.wikipedia.org/wiki/Conjugate_variables

Principal Authors: Linas, Charles Matthews, Glogger, Demoscn, Arthur Rubin

Constraint algebra

In theoretical physics, a constraint algebra is a linear space of all constraints and all of their polynomial functions or functionals whose action on the physical vectors of the →Hilbert space should be equal to zero. For example, in electromagnetism, the equation for the Gauss’ law ~ =ρ ∇·E

Constraint algebra

75

FT

is an equation of motion that does not include any time derivatives. This is why it is counted as a constraint, not a dynamical equation of motion. In quantum electrodynamics, one first constructs a Hilbert space in which Gauss’ law does not hold automatically. The true Hilbert space of physical states is constructed as a subspace of the original Hilbert space of vectors that satisfy ~ (∇ · E(x) − ρ(x))|ψi = 0. In more general theories, the constraint algebra may be a noncommutative algebra.

Source: http://en.wikipedia.org/wiki/Constraint_algebra

Coupling constant

DR A

In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part. The coupling constant determines the strength of the interaction part with respect to the kinetic part, or between two sectors of the interaction part. For example, the electric charge of a particle is a coupling constant. A coupling constant plays an important role in dynamics. For example, one often sets up hierarchies of approximation based on the importance of various coupling constants. In the motion of a large lump of magnetized iron, the gravitational forces are more important than the magnetic forces because of the relative coupling constants. However, in classical mechanics one usually makes these decisions directly by comparing forces.

Fine structure constant

The coupling constant comes into its own in a quantum field theory. A special role is played in relativistic quantum theories by coupling constants which are dimensionless, ie, are pure numbers. For example, the fine-structure constant, α=

e2 4π0 ~c

(where e is the charge of an electron,  0 is the permittivity of free space, is Dirac’s constant and c is the speed of light) is such a dimensionless coupling constant that determines the strength of the electromagnetic force on an electron. Coupling constant

76

Gauge coupling 1 Tr Gµν Gµν 4g 2

FT

In a non-Abelian gauge theory, the gauge coupling parameter, g, appears in the Lagrangian as

(where G is the gauge field tensor) in some conventions. In another widely used convention, G is rescaled so that the coefficient of the kinetic term is 1/4 and g appears in the covariant derivative. This should be understood to be similar to a dimensionless version of the electric charge defined as √ 4π0 α.

Weak and strong coupling

DR A

In a quantum field theory with a dimensionless coupling constant, g, if it is (much) smaller than one, then one says that the theory is weakly coupled. In this case it is well described by an expansion in powers of g, called perturbation theory. If the coupling constant is of order one or larger, the theory is said to be strongly coupled. An example of the latter is the hadronic theory of strong interactions (which is why it is called strong in the first place). In such a case non-perturbative methods have to be used to investigate the theory.

Running coupling

Figure 11

Virtual particles renormalize the coupling

Coupling constant

77

FT

One can probe a quantum field theory at short times or distances by changing the wavelength or momentum, k of the probe one uses. With a high frequency, ie, short time probe, one sees virtual particles taking part in every process. The reason this can happen, seemingly violating the conservation of energy is the uncertainty relation ∆E∆t ≥ ~

which allows such violations at short times. The previous remark only applies to some formulations of QFT, in particular, canonical quantization in the interaction picture. In other formulations, the same event is described by "virtual" particles going off the mass shell. Such processes renormalize the coupling and make it dependent on the scale, k at which one observes the coupling. The phenomenon of scale dependence of the coupling, g(k) is called running coupling in a quantum field theory.

Beta-function

DR A

The beta function β(g) of a quantum field theory measures the running of a coupling parameter. It is defined by the relation: β(g) = k

∂g ∂k

=

∂g ∂ ln k .

For most theories the beta function is positive, so that the coupling is increasing in k (equivalently, the coupling rises as the scale at which the theory is observed becomes shorter). This is also the case in quantum electrodynamics (QED). At low energy, i.e. long distances, α ≈ 1/137. At the scale of the Z boson, about 90 GeV, α ≈ 1/127. In a classical field theory in which a scale change is an invariance (symmetry) of the theory, the beta function breaks this scale invariance. Since this is a quantum effect arising directly from the uncertainty principle, a non-zero beta function implies a scale anomaly in such a quantum field theory.

Landau pole and asymptotic freedom

We noted that QED is weakly coupled at long distances, but the coupling increases at short distances. This increase was first noticed by Lev Landau who showed that QED becomes strongly coupled at high energy, and in fact the coupling becomes infinite at asympototically high energy. This phenomenon is called the Landau pole. In non-Abelian gauge theories, the beta function is negative, as first found by Frank Wilczek, David Politzer and David Gross. As a result the coupling

Coupling constant

78 decreases at short distances. Furthermore, the coupling decreases logarithmically, a phenomenon known as asymptotic freedom. The coupling decreases approximately as gs2 (k 2 ) 4π

≈

1 β0 ln(k 2 /Λ2 )

FT

αs (k 2 ) ≡

where β 0 is a constant computed by Wilczek, Gross and Politzer.

QCD scale

In quantum chromodynamics (QCD), the quantity is called the QCD scale. The value is ΛM S = 217+25 −23 MeV

This value is to be used at a scale above the bottom quark mass of about 5 GeV. The meaning of MS is given in the article on dimensional regularization.

DR A

Charge, colour charge, etc

In quantum field theory, since the size of the interaction term is absorbed into the notion of the coupling constant (more correctly coupling parameter, since it runs), the word charge is freed up for another use. One says, for example, that the electrical charge of an electron is -1 and that of any observable particle is an integer multiple of this. The notion of charge is now exactly the same as the representation of the gauge group to which the particle belongs. Thus the colour charge of a quark is fixed at 4/3 since it belongs to the fundamental representation of SU(3), and the colour charge of a gluon is 8 since it belongs to the adjoint representation. This difference in the notion of charge in classical and quantum field theory is alluded to in a shorthand phrase that is sometimes used: "charge in units of the positron charge".

String theory

A remarkably different situation exists in string theory. Each perturbative description of string theory depends on a string coupling constant. However, in the case of string theory, these coupling constants are not pre-determined, adjustable, or universal parameters; rather they are dynamical scalar fields that can depend on the position in space and time and whose values are determined dynamically.

Coupling constant

79

See also →Quantum field theory, especially quantum electrodynamics and quantum chromodynamics • Canonical quantization, renormalization and dimensional regularization • fine structure constant • gravitational coupling constant

FT

•

References and external links • •

An introduction to quantum field theory, by M.E.Peskin and H.D.Schroeder, ISBN 0201503972 The Nobel Prize in Physics 2004 – Information for the Public 39

Source: http://en.wikipedia.org/wiki/Coupling_constant

DR A

Principal Authors: Bambaiah, Phys, Xerxes314, Ricky81682, Cmdrjameson

Creation and annihilation operators In physics, an annihilation operator is an operator that lowers the number of particles in a given state by one. A creation operator is an operator that increases the number of particles in a given state by one, and it is the adjoint of the annihilation operator. Depending on the context, the identity of the particles in question varies; for example, in quantum chemistry and manybody theory the creation and annihilation operators often act on electrons. Annihilation and creation operators can also refer specifically to the ladder operators for the quantum harmonic oscillator. In the latter case, the raising operator is interpreted as a creation operator, adding a quantum of energy to the oscillator system. Vice versa for the lowering operator. In many subfields of physics and chemistry, using these operators instead of a wavefunction picture, is known as second quantization. The mathematics behind the creation and the annihilation operators is identical as the formulae for ladder operators that appear in the quantum harmonic oscillator. For example, the commutator of the annihilation and the creation operator associated with the same state equals one; all other commutators vanish.

39 http://nobelprize.org/physics/laureates/2004/public.html

Creation and annihilation operators

80

FT

While the concept of creation and annihilation operators is well defined for free field theories, in interacting QFTs, they can only be defined in the interaction picture, which does not exist according to Haag’s theorem.

Derivation of bosonic creation and annihilation operators In the context of the quantum harmonic oscillator, we reinterpret the ladder operators as creation and annihilation operators, adding or subtracting fixed quanta of energy to the oscillator system. Creation/annihilation operators are different for bosons (integer spin) and fermions (half-integer spin). This is because their wavefunctions have different symmetry properties. Suppose the wavefunctions are dependent on N properties. Then For bosons: ψ(1,2,3,4,...N ) = ψ(2,1,3,4,...N )

DR A

For fermions: ψ(1,2,3,4,...N ) = -ψ(2,1,3,4,...N )

For now let’s just consider the case of bosons because fermions are more complicated. Start with the →Schrödinger equation for the one dimensional time independent quantum harmonic oscillator  2 2  ~ d 1 2 2 ψ(x) = Eψ(x) − 2m 2 + 2 mω x dx

Make a coordinate substitution to nondimensionalize the differential equation q ~ x ≡ mω q. and the Schrödinger equation for the oscillator becomes   d2 ~ω 2 ψ(q) = Eψ(q). − dq 2 +q 2

Notice that the quantity hω = h ν is the same energy as that found for light quanta and that the parenthesis in the Hamiltonian can be written as h ih i d2 d d d d 2 − dq 2 + q = − dq + q dq + q + dq q − q dq

The last two terms in that equation form the commutator of q with its derivative. So let’s calculate that commutator [ q, ∂/∂q ]   df d d d q dq − dq q f (q) = q dq − dq (qf (q)) = −f (q)

Creation and annihilation operators

81 In other words [ q, d /dq ] = - 1 or [ d /dq, q ] = 1. Therefore  ih i  h d/dq+q 1 d2 2 = ~ω −d/dq+q √ √ + 21 ~ω 2 ~ω − dq 2 + q If we define a† ≡ a≡

√1 2 √1 2





2

FT

2

 d − dq + q as the "creation operator" or the "raising operator" and

 d + dq + q as the "annihilation operator" or the "lowering operator"

the Hamiltonian becomes
H = ~ω a† a + 12 .

This Hamiltonian is significantly simpler than the original form. Further simplifications of this equation enables one to derive all the properties listed above thus far.

DR A

Letting p = - i d /dq, where p is the nondimensionalized momentum operator a† = a=

√1 (q 2

√1 (q 2

− ip)

+ ip).

Substituting backwards, the laddering operators are recovered.

Mathematical details

The operators derived above are actually a specific instance of a more generalized class of creation and annihilation operators. The more abstract (and hence more applicable) form of the operators satisfy the properties below. Let H be the one-particle →Hilbert space. To get the bosonic CCR algebra, look at the algebra generated by a(f ) for any f in H. The operator a(f ) is called an annihilation operator and the map a(.) is antilinear. Its adjoint is a †(f ) which is linear in H. For a boson,

[a(f ), a(g)] = [a† (f ), a† (g)] = 0

[a(f ), a† (g)] = hf |gi,

where we are using bra-ket notation.

Creation and annihilation operators

82 For a fermion, the anticommutators are

{a(f ), a ∗ (g)} = hf |gi. A CAR algebra.

FT

{a(f ), a(g)} = {a ∗ (f ), a ∗ (g)} = 0

Physically speaking, a(f ) removes (i.e. annihilates) a particle in the state |f > wheareas a †(f ) creates a particle in the state |f >. The free field vacuum state is the state with no particles. In other words, a(f )|0i = 0 where |0> is the vacuum state.

If |f > is normalized so that =1, then a †(f ) a(f ) gives the number of particles in the state |f >.

DR A

Note that the creation and annihilation operators are "generalized complex conjugates" of each other. Usually, the notation is chosen in such a way that the a †(f ) is the creation operator, and a(f ) is the annihilation operator. The † reminds us that something "extra" is being added to the system. The topic can be misleadingly confusing if this is not done.

Notational caveats and considerations In quantum mechanics, Dirac bra-ket notation is often used. However, there is some ambiguity in this notation, particularly when there is the need to differentiate between these things: • • • •

The lowest energy state The zero state The vacuum state The zero ket

Often, these are all interchangeably notated as |0>, or even | >. As a result, it is necessary to read carefully, and consider the context in which the notation is used. For example, in the quantum harmonic oscillator, the ground state has the property that when the annihilation operator b is applied to it, it satisfies b|0> = 0| > = 0

The intermediate step is rarely indicated as it is considered necessary only when more conceptual/mathematical rigour is needed. Creation and annihilation operators

83

FT

In this example, the lowest energy state is denoted as |0>. It is labeled as the "zero state", but it is important to emphasize that any state can be labeled as the "zero" state. The zero state is often used as a reference state to other quantum states. Therefore, the |0> state need not be the state with the absolutely lowest energy. In the case of the harmonic oscillator, it is due to the particulars of the mathematics that the ground state is chosen to be |0>. The vacuum state is the state where no quanta is available to be extracted. This special null state is denoted by | >. This vacuum state is also known as the "zero ket" because there are zero particles in the state. Unfortunately, the lowest energy state |0> is also known as the "zero ket" for the different reason that the state is labeled as "zero". Care must be taken that the four concepts listed above are not mixed together. Sometimes, the terms "null state" and "empty state" are used interchangeably for |> and |0>. The meaning for this usage is again dependent on the context.

The vacuum state

DR A

The vacuum state is a conceptual state which has no particles. The state is usually denoted as |0>, not the "empty ket" | >. Interestingly enough, no actual function actually represents the |0> state, but for notational purposes, we define the vacuum state as being normalized such that <0|0> = 1 and that |0> is orthogonal to all other states of the form |N >, where N is any indexing of quantum states for a particular system.

Energy spectra

In a quantum mechanical system, the range of discrete energies allowed in a system may be either finite, or infinite, or "semi-infinite". In a system where the energies are confined to be semi-infinite on the interval [constant, ∞) such as the quantum harmonic oscillator, the vacuum state | >, (different from |0>) needs to be introduced in order to make the theory of creation and annihilation operators consistent. The lowest energy allowed in a semi-infinite energy system is known as the ground state. Since it is often used as the reference state, it is denoted by |0>. However, this state is not empty the vacuum state | > is introduced to disambiguate these two states. In a system where the range of energies is (-∞, ∞), the vacuum state is almost always denoted by |0>. There is no need for the "null" state | > as |0> already is sufficient to denote "emptiness". There is also no "ground state" present, which is why the notational ambiguity arises. This interpretation arises directly from the relativistic formalism of quantum mechanics by Paul Dirac, which later became one of the foundations for quantum field theory. One of the shortcomings of quantum field theory however, is its allowance of energy states Creation and annihilation operators

84 tending to infinity. The attempt to resolve this problem is very much an active part of quantum mechanical research today.

FT

In summary for infinite and semi infinite systems COMMON NOTATION FOR STATES infinite semi-infinite ground state

none

vacuum state

|0>

|0>

| >

There is no | > state needed for infinite-ranged-energy-systems in quantum mechanics.

See also

→Ladder operators Bogolibov transformations - arises in the theory of quantum optics. Also transliterated as Bogolubov transformations’

DR A

• •

Source: http://en.wikipedia.org/wiki/Creation_and_annihilation_operators Principal Authors: HappyCamper, Laurascudder, Phys, Salsb, Sunev

Dark energy star

A dark-energy star is a hypothetical compact astrophysical object which a minority of physicists feel might constitute an alternative explanation for observations of astronomical black hole candidates. The concept was proposed by physicist George Chapline. The theory states that infalling matter is converted into vacuum energy, or dark energy as the matter falls through the event horizon. The space within the event horizon would end up with a large value for the cosmological constant, and have negative pressure to exert against gravity. There would be no information destroying singularity.

Theory

In March 2005, physicist George Chapline claimed that quantum mechanics makes it a "near certainty" black holes do not exist and are instead dark energy stars. The dark energy star is a different concept than that of a gravastar. Dark energy star

85

FT

Dark-energy stars were first proposed due to the fact that in quantum physics, absolute time is required, however in general relativity, an object falling towards a black hole would to an outside observer seem to have time pass infinitely slowly at the event horizon. The object itself would feel as if time flowed normally. In order to reconcile quantum mechanics with black holes, Chapline theorized that a phase transition in the phase of space occurs at the event horizon. He based his ideas on the physics of superfluids. As a column of superfluid grows taller, at some point, density increases slowing down the speed of sound so that it approaches zero. However, at that point, quantum physics makes sound waves dissipate their energy into the superfluid, so that the zero sound speed condition is never encountered.

DR A

In the dark-energy star hypothesis, infalling matter approaching the event horizon decay into successively lighter particles. Nearing the event horizon, environmental effects accelerate proton decay. This may account for high energy cosmic ray sources and positron sources in the sky. When the matter falls through the event horizon, the energy equivalent of some or all of that matter is converted into dark energy. This negative pressure counteracts the mass the star gains, avoiding a singularity.

The negative pressure also gives a very high number for the cosmological constant. As there is no singularity to evaporate, Hawking radiation may not exist in this model of black holes. Furthermore ’primordial’ dark-energy stars could form by fluctuations of spacetime itself which is analogous to "blobs of liquid condensing spontaneously out of a cooling gas." This not only alters the understanding of black holes but has the potential to explain the dark energy and dark matter that are indirectly observed.

See also • • • •

Gravastar acoustic metric analog model of gravity Stellar mass black hole

External links • •

Dark Energy Stars 40 MPIE Galactic Center Research 41

40 http://www.llnl.gov/tid/lof/documents/pdf/317506.pdf

Dark energy star

86

Source: http://en.wikipedia.org/wiki/Dark_energy_star

FT

Principal Authors: RoyBoy, Hillman, Phys, Algri, Christopher Thomas

Davisson-Germer experiment

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow moving electrons at a crystalline nickel target. The angular dependence of the reflected electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for X-Rays. This experiment, like Arthur Compton’s proof of the particle-like nature of light, proved the wave-like nature of matter and completed the wave-particle duality hypothesis, which was a fundamental step in quantum theory.

DR A

Source: http://en.wikipedia.org/wiki/Davisson-Germer_experiment

Principal Authors: Linas, Gershwinrb, Tone, Charles Matthews, Wiccan Quagga

De Broglie hypothesis

In physics, the de Broglie hypothesis is the statement that all matter has a wave-like nature (wave-particle duality). The de Broglie relations show that the wavelength is inversely proportional to the momentum of a particle and that the frequency is directly proportional to the particle’s kinetic energy. The hypothesis was advanced by Louis de Broglie in 1923 in his PhD thesis 42; he was awarded the Nobel Prize for Physics in 1929 for this work, which made him the first person to receive a Nobel Prize on a PhD thesis.

The de Broglie relations

The first de Broglie equation relates the wavelength to the particle momentum as q λ=

h p

=

h mv

1−

v2 c2

41 http://www.mpe.mpg.de/ir/GC/index.php 42 L. de Broglie, PhD thesis, reprinted in Ann. Found. Louis de Broglie 17 (1992) p. 22.

De Broglie hypothesis

87 where λ is the particle’s wavelength, h is Planck’s constant, p is the particle’s momentum, m is the particle’s rest mass, v is the particle’s velocity, and c is the speed of light in a vacuum.

f=

Ek h

=

mc2 h

q 1

2

1− v2 c

FT

The greater the energy, the larger the frequency and the shorter (smaller) the wavelength. Given the relationship between wavelength and frequency, it follows that short wavelengths are more energetic than long wavelengths. The second de Broglie equation relates the frequency of a particle to the kinetic energy such that   − 1

where f is the frequency and Ek is the kinetic energy. The two equations are often written as p = ~k

DR A

Ek = ~ω

where ~ is Dirac’s constant, k is the wavenumber, and ω is the angular frequency. See the article on group velocity for detail on the argument and derivation of the de Broglie relations.

Experimental confirmation

In 1927 at Bell Labs, Clinton Davisson and Lester Germer fired slow-moving electrons at a crystalline nickel target. The angular dependence of the reflected electron intensity was measured, and was determined to have the same diffraction pattern as those predicted by Bragg for X-Rays. Before the acceptance of the De Broglie hypothesis, diffraction was a property that was only exhibited by waves. Therefore, the presence of any diffraction effects by matter, demonstrated the wave-like nature of matter. When the De Broglie wavelength was inserted into the Bragg condition, the observed diffraction pattern was predicted, thereby experimentally confirming the De Broglie hypothesis for electrons. This was a pivotal result in the development of quantum mechanics. Just as Arthur Compton demonstrated the particle nature of light, the →DavissonGermer experiment showed the wave-nature of matter, and completed the theory of wave-particle duality. For physicists this idea was important because it means that not only can any particle exhibit wave characteristics, but that one can use wave equations to describe phenomena in matter if one uses the de Broglie wavelength. De Broglie hypothesis

88

FT

Since the original Davisson-Germer experiment for electrons, the De Broglie hypothesis has been confirmed for other elementary particles. Recent experiments even confirm the relations for macromolecules, which are normally considered too large to undergo quantum mechanical effects. In 1999, a research team in Vienna demonstrated diffraction for molecules as large as fullerenes. 43

Wavelength of large objects

Theoretically, all objects, not just sub-atomic particles, exhibit wave properties according to the De Broglie Hypothesis. Consider the following example:

A baseball has a mass of 0.15 kg and is thrown by a professional baseball player at 40 m/s. The de Broglie wavelength of the baseball is given by: m = 0.15kg v = 40m/s (about 90 mph) λ=

where p = mv

6.626×10−34 kgm2 /s 0.15kg×40m/s 1.10 × 10−34 m

DR A

λ=

h p

λ=

This wavelength is considerably smaller than the diameter of a proton (about 10−15 m) and is approaching the Planck Length. As such, the wave-like properties of this baseball are so small as to be unobservable.

See also •

→Bohr model

References •

•

Steven S. Zumdahl, Chemical Principles 5th Edition, (2005) Houghton Mifflin Company. Tipler, Paul A. and Ralph A. Llewellyn (2003). Modern Physics. 4th ed. New York; W. H. Freeman and Company. ISBN 0-7167-4345-0. pp. 203-4, 222-3, 236.

Source: http://en.wikipedia.org/wiki/De_Broglie_hypothesis

43 Arndt, M., O. Nairz, J. Voss-Andreae, C. Keller, G. van der Zouw, A. Zeilinger (14 October 1999).

"Wave-particle duality of C60". Nature 401: 680-682.

De Broglie hypothesis

89 Principal Authors: Linas, Teply, Cdang, Blinken, Dataphile

FT

Degenerate energy level

This article refers to physical states having the same energy. For other uses of the word degeneracy, see degeneracy (disambiguation).

In physics two or more physical states are said to be degenerate if they are both at the same energy level; the energy level is said to be degenerate if it contains two or more such states. The number of occupation states available at a particular energy level is called the level’s degeneracy.

DR A

In quantum theory, this usually pertains to electronic configurations and the electron energy levels, where different possible occupation states for particles may be related by symmetry. The usage comes from the fact that degenerate eigenstates correspond to identical eigenvalues of the Hamiltonian. Since eigenvalues correspond to roots of the characteristic equation, degeneracy here has the same meaning as the common mathematical usage of the word. If the symmetry is broken by a perturbation, such as applying an external electric field, this can change the energies of the states, causing energy level splitting.

See also •

Density of states

Source: http://en.wikipedia.org/wiki/Degenerate_energy_level Principal Authors: Jheald, Tim Starling, Linas, Karol Langner

Degenerate energy level

90

Degenerate matter

FT

Degenerate matter is matter which has sufficiently high density that the dominant contribution to its pressure arises from the Pauli exclusion principle. The pressure maintained by a body of degenerate matter is called the degeneracy pressure, and arises because the Pauli principle forbids the constituent particles from occupying identical quantum states. Therefore, reducing the volume requires forcing the particles into higher-energy quantum states. The species of fermion are sometimes identified, so that we may speak of electron degeneracy pressure, neutron degeneracy pressure, and so forth.

DR A

Unlike a classical ideal gas, whose pressure is proportional to its temperature (P = nkT , where P is pressure, n is particles per unit volume, k is Boltzmann’s constant, and T is temperature), the pressure exerted by degenerate matter depends only weakly on its temperature. In particular, the pressure remains nonzero even at absolute zero temperature. At relatively low densities, the pressure of a fully degenerate gas is given by P = Kn5/3 , where K depends on the properties of the particles making up the gas. At very high densities, where most of the particles are forced into quantum states with relativistic energies, the pressure is given by P = K 0 n4/3 , where K 0 again depends on the properties of the particles making up the gas. Exotic examples of degenerate matter include neutronium, strange matter, metallic hydrogen and white dwarf matter. Degeneracy pressure contributes to the pressure of conventional solids, but these are not usually considered to be degenerate matter as a significant contribution to their pressure is provided by the interplay between the electrical repulsion of atomic nuclei and the screening of nuclei from each other by electrons allocated among the quantum states determined by the nuclear electrical potentials. In metals it is useful to treat the conduction electrons alone as a degenerate, free electron gas while the majority of the electrons are regarded as occupying bound quantum states. This contrasts with the case of the degenerate matter that forms the body of a white dwarf where all the electrons would be treated as occupying free particle momentum states.

Degenerate gases

Degenerate gases are gases composed of fermions that have a particular configuration which usually forms at high densities. Fermions are subatomic particles with half-integer spin. Their behaviour is regulated by a set of quantum mechanical rules called the →Fermi-Dirac statistics. One particular rule is the

Degenerate matter

91

FT

Pauli exclusion principle that states that there can be only one fermion occupying each quantum state which also applies to electrons that are not bound to a nucleus but merely confined to a fixed volume, such as the deep interior of a star. Such particles as electrons, protons, neutrons, and neutrinos are all fermions and obey Fermi-Dirac statistics.

A fermion gas in which all the energy states below a critical value, designated Fermi energy, are filled is called a fully degenerate fermion gas. The electron gas in ordinary metals and in the interior of white dwarf stars constitute two examples of a degenerate electron gas. Most stars are supported against their own gravitation by normal gas pressure. White dwarf stars are supported by the degeneracy pressure of the electron gas in their interior. For white dwarfs the degenerate particles are the electrons while for neutron stars the degenerate particles are neutrons.

Electron degeneracy

DR A

In ordinary gas, most of the energy levels called n-spheres of only certain discrete energy states available to electrons are unfilled and the electrons are free to move about. As particle density is increased in a fixed volume, electrons progressively fill the lower energy states and additional electrons are forced to occupy states of higher energy. Therefore, degenerate gases strongly resist further compression because the electrons cannot move to lower energy levels which are already filled due to the Pauli Exclusion Principle. The degenerate electrons are locked into place because all of the lower energy shells are filled up so they no longer move freely as in a normal gas. Even though thermal energy may be extracted from the gas, it still may not cool down, since electrons cannot give up energy by moving to a lower energy state. This increases the pressure of the fermion gas termed degeneracy pressure. In a degenerate gas, the average pressure is high enough to keep material from being compressed by gravity. Under high densities the matter becomes a degenerate gas when the electrons are all stripped from their parent atoms. In the core of a star once hydrogen burning in nuclear fusion reactions stops, it becomes a collection of positively charged ions, largely helium and carbon nuclei, floating in a sea of electrons which have been stripped from the nuclei. Degenerate gas is an almost perfect conductor of heat and does not obey the ordinary gas laws. White dwarfs are luminous not because they are generating any energy but rather because they have trapped a large amount of heat. Normal gas exerts higher pressure when it is heated and expands, but the pressure in a degenerate gas does not depend on the temperature. When gas become super-compressed, particles position right up against each other to produce degenerate gas that behaves more like Degenerate matter

92

FT

a solid. In degenerate gases the kinetic energies of electrons are quite high and the rate of collision between electrons and other particles is quite low, therefore degenerate electrons can travel great distances at velocities that approach the speed of light. Instead of temperature, the pressure in a degenerate gas depends only on the speed of the degenerate particles; however, adding heat does not increase the speed. Pressure is only increased by the mass of the particles which increases the gravitational force pulling the particles closer together. Therefore, the phenomenon is opposite of that normally found in matter where if the mass of the matter is increased, the object becomes bigger. In degenerate gas, when the mass is increased, the pressure is increased, and the particles become spaced closer together, so the object becomes smaller. Degenerate gas can be compressed to very high densities, typical values being in the range of 10 7 grams per cubic centimetre.

DR A

There is an upper limit to the mass of an electron-degenerate object, the Chandrasekhar limit, beyond which electron degeneracy pressure cannot support the object against collapse. The limit is approximately 1.44 solar masses for objects with compositions similar to the sun. The mass cutoff changes with the chemical composition of the object, as this affects the ratio of mass to number of electrons present. Celestial objects below this limit are white dwarf stars, formed by the collapse of the cores of stars which run out of fuel. During collapse, an electron-degenerate gas forms in the core, providing sufficient degeneracy pressure as it is compressed to resist further collapse. Above this mass limit, a neutron star (supported by neutron degeneracy pressure) or a black hole may be formed instead.

Proton degeneracy

Sufficiently dense matter containing protons experiences proton degeneracy pressure, in a manner similar to the electron degeneracy pressure in electrondegenerate matter. As protons and electrons occur in roughly equal numbers in most forms of matter, proton degeneracy is usually modelled as a correction to the equations of state of electron-degenerate matter, as opposed to the dominant source of degeneracy pressure (which would require proton-degenerate matter that was free of electrons).

Neutron degeneracy

Neutron degeneracy is analogous to electron degeneracy and is demonstrated in neutron stars, which are supported by the pressure from a degenerate neutron gas. This happens when a stellar core above 1.44 solar masses (the Chandrasekhar limit) collapses and is not halted by the degenerate electrons. As the star collapses, the →Fermi energy of the electrons increases to the point where it is energetically favorable for them to combine with protons to produce Degenerate matter

93 neutrons (via inverse beta decay, also termed "neutralization"). The result of this collapse, for the simplest models of neutron-degenerate matter, is an extremely compact star composed almost entirely of a degenerate neutron gas.

FT

Neutrons in a degenerate neutron gas are spaced much more closely than electrons in an electron-degenerate gas, because the more massive neutron has a much shorter wavelength at a given energy. In the case of neutron stars and white dwarf stars, this is compounded by the fact that the pressures within neutron stars are much higher than those in white dwarfs. The pressure increase is caused by the fact that the compactness of neutron stars causes gravitational forces to be much higher than in a less compact body with similar mass, resulting in a star on the order of a thousand times smaller than a white dwarf.

DR A

There is an upper limit to the mass of a neutron-degenerate object, the TolmanOppenheimer-Volkoff limit, which is analogous to the Chandrasekhar limit for electron-degenerate objects. The precise limit is unknown, as it depends on the equations of state of neutron-degenerate matter, for which a highly accurate model is not yet available. Above this limit, a neutron star may collapse into a black hole, or into other, denser forms of degenerate matter (such as quark matter) if these forms exist and have suitable properties (mainly related to degree of compressibility, or "stiffness", described by the equations of state).

Quark degeneracy

At densities greater than those supported by neutron degeneracy, quark matter is expected to occur. Several variations of this have been proposed that represent quark-degenerate states. Strange matter is a degenerate gas of quarks that is often assumed to contain strange quarks in addition to the usual up and down quarks. Color superconductor materials are degenerate gases of quarks in which quarks pair up in a manner similar to Cooper pairing in electrical superconductors. The equations of state for the various proposed forms of quark-degenerate matter vary widely, and are usually also poorly defined, due to the difficulty modelling strong force interactions.

Quark-degenerate matter may occur in the cores of neutron stars, depending on the equations of state of neutron-degenerate matter. It may also occur in quark stars, formed by the collapse of objects above the Tolman-OppenheimerVolkoff mass limit for neutron-degenerate objects. Whether quark-degenerate matter forms at all in these situations depends on the equations of state of both neutron-degenerate matter and quark-degenerate matter, both of which are poorly known.

Degenerate matter

94

Speculative types of degenerate matter Preon degeneracy

See also

Degenerate star White dwarf (degenerate dwarf) - white dwarf material - electrondegenerate matter Neutron star - neutron matter - neutron-degenerate matter Quark star - quark matter - quark-degenerate matter Preon star - preon matter - preon-degenerate matter Pauli Exclusion Principle Uncertainty Principle

DR A

• •

FT

Preons are subatomic particles proposed to be the constituents of quarks, which become composite particles in preon-based models. If preons exist, preondegenerate matter might occur at densities greater than that which can be supported by quark-degenerate matter. The properties of preon-degenerate matter depend very strongly on the model chosen to describe preons, and the existence of preons is not assumed by the majority of the scientific community, due to conflicts between the preon models originally proposed and experimental data from particle accelerators.

• • • • •

External links •

Detailed mathematical explanation of degenerate gases 44

Source: http://en.wikipedia.org/wiki/Degenerate_matter

Principal Authors: Voyajer, Christopher Thomas, Alan Peakall, Eyu100, Roadrunner

44 http://ircamera.as.arizona.edu/astr_250/Lectures/Lec17_sml.htm

Degenerate matter

95

Delayed choice quantum eraser

Introduction

FT

A delayed choice quantum eraser is a combination between a quantum eraser experiment and Wheeler’s delayed choice experiment. This experiment has actually been performed and published by Yoon-Ho Kim, R. Yu, S.P. Kulik, Y.H. Shih, and Marlon O. Scully 45 Phys.Rev.Lett. 84 1-5 (2000). This experiment was designed to investigate a very peculiar result of the well known double slit experiment of quantum mechanics, the dual wave particle nature of light, and in fact all matter.

DR A

In the double slit experiment, a photon passes through a double slit apparatus, in which the photon must pass either through one or the other of two slits, and then registers on a detector, which can determine where the photon reaches the detector, like an image projected on a screen. If one allows many photons to individually pass through either slit A or slit B and doesn’t know which slit they passed through, an interference pattern emerges on the detector. The interference pattern indicates that the light beam is in fact made up of waves. However, if one somehow observes which of the two slits each photon actually passes through, a different result will be obtained. In this case, each photon hits the detector after going through only one slit and a single concentration of hits in the middle of the detection field. This result is consistent with light behaving as individual particles, like tiny bullets. The very odd thing about this is that a different outcome results based on whether or not the photon is observed after it goes through the slit but before it hits the detector. In a quantum eraser experiment, one arranges to detect which one of the slits the photon passes through, but also construct the experiment in such a way that this information can be "erased" after the fact. It turns out that if one observes which slit the photon passes through, the "no interference" or particle behavior will result, which is what quantum mechanics predicts, but if the quantum information is "erased" regarding which slit the photon passed through, the photons revert to behaving like waves.

However, Kim, et al. have shown that it is possible to delay the choice to erase the quantum information until after the photon has actually hit the target. But, again, if the information is "erased," the photons revert to behaving like waves, even if the information is erased after the photons have hit the detector.

45 http://xxx.lanl.gov/pdf/quant-ph/9903047

Delayed choice quantum eraser

96

The experiment

FT

The experimental setup, described in much more detail at 46, is as follows. First, generate a photon and pass it through a double slit apparatus. After the photon goes through slit A or B, a special crystal (one at each slit) uses spontaneous parametric down conversion (SPDC) to convert the photon into two identical entangled photons with 1/2 the frequency of the original photon. One of these photons continues to the target detector, while the other entangled photon is deflected by a prism to bounce off a mirror some distance away. Now, if the second photon (coming from slit A or slit B) is observed, it is known which slit the original photon went through, so the photon behaves like a particle. If the second photon’s paths from slit A and B are combined, the which-way path is not observed, and the first photon behaves like a wave. The experimenter can choose to observe or not observe the which-way information by erasing (or detecting) information about the second photon’s path.

DR A

The results from Kim, et al. have shown that, in fact, observing the second photon’s path will determine the particle or wavelike behavior of the first photon at the detector, even if the second photon is not observed until after the first photon arrives at the detector. In other words, the delayed choice to observe or not observe the second photon will change the outcome of an event in the past.

Discussion

In terms of the conventional way of viewing the physical universe, this result seems to be a paradox. This experiment demonstrates the possibility of observing both particle-like and wave-like behavior of a photon using quantum entanglement. Furthermore, the behavior of the photon at the primary detector can be changed even after the registration of the event by the detector. In their paper, Kim, et al. 47 explain that the concept of complementarity is one of the most basic principles of quantum mechanics. According to the Heisenberg Uncertainty Principle, it is not possible to measure both precise position and momentum of a quantum particle at the same time. In other words, position and momentum are complementary. In 1927, Niels Bohr described complementarity as “wave-like” and “particle-like” behavior of a quantum particle. this has come to be known as the wave-particle duality of quantum mechanics. The double-slit experiment is a good example of this concept. Feynman believed that this was the basic mystery of quantum mechanics. The actual

46 http://xxx.lanl.gov/pdf/quant-ph/9903047 47 http://xxx.lanl.gov/PS_cache/quant-ph/pdf/9903/9903047.pdf

Delayed choice quantum eraser

97

FT

mechanisms that enforce complementarity vary from one experimental situation to another. In the double-slit experiment, the common wisdom is that the Heisenberg Uncertainty Principle makes it impossible to determine which slit the photon passes through without at the same time disturbing it enough to destroy the interference pattern. However, in 1982, Scully and Druhl found a way around the position-momentum uncertainty obstacle and proposed a quantum eraser to obtain which-path or particle-like information without introducing large uncontrolled phase factors to disturb the interference. They found that the interference pattern disappears when which-path information is obtained, even if this information was obtained without directly observing the original photon. Even more surprising was that, if you somehow "erase" the which-path information, the interference pattern reappears! And, perhaps most provocative of all, you can delay the "choice" to "erase" or "observe" the which-path information and still restore the interference pattern, even after the original photon has been "observed" at the primary detector!

DR A

How can this be? It would seem that the "choice" to observe or erase the whichpath information can change the position where the photon is recorded on the detector, even after it should have already been recorded. One explanation of this paradox would be that this is a kind of time travel. In other words, the delayed "choice" to "erase" or "observe" the which-path information of the original photon can change the outcome of an event in the past. Another explanation would be that in fact both outcomes occur. The universe itself exists in a superposition of states in which either the original photon goes through slit A or slit B and in which the which-path information either "observed" or "erased". This is described in detail in the Everett manyworlds interpretation of quantum mechanics.

External links

• • •

basic delayed choice experiment 48 delayed choice quantum eraser 49 the notebook of philosophy and physics 50

Source: http://en.wikipedia.org/wiki/Delayed_choice_quantum_eraser

48 http://www.bottomlayer.com/bottom/basic_delayed_choice.htm 49 http://www.bottomlayer.com/bottom/kim-scully/kim-scully-web.htm 50 http://www.bottomlayer.com/

Delayed choice quantum eraser

98

Diabatic

FT

In quantum chemistry, the potential energy surfaces are obtained within the adiabatic or Born-Oppenheimer approximation. This corresponds to a representation of the molecular wave function where the variables corresponding to the molecular geometry and the electronic degrees of freedom are separated. The non separable terms are due to the nuclear kinetic energy terms in the molecular Hamiltonian and are said to couple the potential energy surfaces. In the neighbourhood of an avoided crossing or conical intersection, these terms cannot be neglected. One therefore usually performs one unitary transformation from the adiabatic representation to the so-called diabatic representation in which the nuclear kinetic energy operator is diagonal. In this representation, the coupling is due to the electronic energy and is a scalar quantity which is much more easy to estimate numerically.

DR A

Source: http://en.wikipedia.org/wiki/Diabatic Principal Authors: TimBentley, Elfguy

Dirac equation in the algebra of physical space The Dirac equation, as the relativistic equation that describes spin 1/2 particles in quantum mechanics can be written in terms of the Algebra of physical space (APS), which is a case of a Clifford algebra or geometric algebra that is based in the use of paravectors. The Dirac equation in APS, including the electromagnetic interaction, reads ¯ 3 + eAΨ ¯ = mΨ ¯† i∂Ψe Another form of the Dirac equation in terms of the Space time algebra was given earlier by D. Hestenes. In general, the Dirac equation in the formalism of geometric algebra has the advantage of providing a direct geometric interpretation.

Dirac equation in the algebra of physical space

99

Relation with the standard form

FT

The Dirac equation can be written also as ¯ † e3 + eAΨ ¯ † = mΨ i∂ Ψ The standard form is obtained multiplying the spinor with the projector P = 1 2 (1+e3 ). Without electromagnetic interaction, the following matrix is obtained from forms  of the Dirac equation  equivalent   the two  previous

0 i∂

i∂¯ 0

¯ † P3 Ψ ΨP3

=m

¯ † P3 Ψ ΨP3

This colum of projected spinors are related to the spinors in the Weyl representation. This is more evident identifying the right and left Weyl spinors as ¯ † P3 = ψ L Ψ

DR A

ΨP3 = ψR so  that

    0 i∂0 + i∇ ψL ψL =m ψR i∂0 − i∇ 0 ψR

In the matrix representation each expression is replaced by a 2 by 2 matrix, including ψL and ψR . The nabla operator can be written in terms of the Pauli matrices as ∇ → σ · ∇.

On the other hand, only the firs column of each spinor is taken ψL,R → F irstColum(ψL,R ),

so the Dirac  equation  becomes  

0 1 0 ∂ + 1 0 0 −σ

σ ·∇ 0

ψL ψR



=m

 ψL , ψR



from which, the standard relativistic covariant form of the Dirac equation is found iγ µ ∂µ ψ = mψ.

Dirac equation in the algebra of physical space

100

Spinor expansion in a null basis The spinor Ψ can be expanded in a null basis as follows Ψ = ψ ∗ P¯3 − ψ ∗ P3 e1 + ψ3 P3 + ψ4 e1 P3 , 2

FT

1

where each coefficient is extracted from the spinor such that ¯ † P3 iS ψ1 = 2hP3 Ψ ¯ † P3 iS ψ2 = 2he1 P¯3 Ψ ψ3 = 2hP3 Ψ P3 iS ψ4 = 2he1 P¯3 ΨP3 iS

This expansion is cleanly related with the colum spinor in the Weyl representation  so that  the Weyl spinor components are

DR A

ψ1 ψ2   Ψ→ ψ3  ψ4

Similarly, the spinor components in the Pauli-Dirac representation are calculated as  

ψ3 + ψ1 ψ4 + ψ2   Ψ→ ψ3 − ψ1  ψ4 − ψ2

Current

The current is defined as J = ΨΨ† ,

which satisfies the continuity equation

 ¯ ∂J =0 S

Dirac equation in the algebra of physical space

101

Second order Dirac equation

FT

An application of the Dirac equation on itself leads to the second order Dirac equation

 ¯ − i(2e A∂¯ + eF )Ψe3 = m2 Ψ (−∂ ∂¯ + AA)Ψ S

Free particle solutions Positive energy solutions

A solution for the free particle with momentum p = p0 + p and positive energy p0 > 0 is q p Ψ= m R(0) exp(−i hp¯ xiS e3 ).

DR A

This solution is unimodular ¯ =1 ΨΨ

and the current resembles the classical proper velocity u=

p m

J = ΨΨ† =

p m

Negative energy solutions

A solution for the free particle with negative energy and momentum p = −|p0 | − p = −p0 is q p0 R(0) exp(i hp0 x ¯iS e3 ), Ψ=i m This solution is anti-unimodular ¯ = −1 ΨΨ

and the current resembles the classical proper velocity u = J = ΨΨ† =

p m

p −m ,

but with a remarkable feature: "the time runs backwards"

p dt dτ = m S < 0

Dirac equation in the algebra of physical space

102

Dirac Lagrangian

See also • • •

Paravector Algebra of physical space Geometric algebra

References

FT

The Dirac Lagrangian is ¯ † e3 Ψ ¯ − eAΨ ¯ †Ψ ¯ − mΨΨi ¯ 0 L = hi∂ Ψ

Baylis, William (2002). Classical eigenspinors and the Dirac equation, Phys. Rev. A 45, 4293–4302 (1992)

•

Hestenes D., Observables, operators, and complex numbers in the Dirac theory, J. Math. Phys. 16, 556 (1975)

DR A

•

Source: http://en.wikipedia.org/wiki/Dirac_equation_in_the_algebra_of_physical_space

Dirac operator

In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compatible with special relativity; to get the relevant Laplacian as a product of first-order operators he introduced spinors. In general, let D be a first-order differential operator acting on a vector bundle V over a Riemannian manifold M . If

D2 = ∆,

with ∆ being the Laplacian of V , D is called a Dirac operator. In high-energy physics, this requirement is often relaxed: only the second-order part of D2 must equal the Laplacian. Dirac operator

103

Examples 1. −i∂x is a Dirac operator on the tangent bundle over a line.

χ(x, y) η(x, y)

FT

2: We now consider a simple bundle of importance in physics: The configuration space of a particle with spin 1/ 2 confined to a plane, which is also the base manifold. Physicists generally think of wavefunctions ψ: R 2 → C 2 which they write   where x and y are the usual coordinate functions on R 2. χ specifies the probability amplitude for the particle to be in the spin-up state, and similarly for η. The so-called spin-Dirac operator can then be written D = −iσx ∂x − iσy ∂y ,

DR A

where σ i are the Pauli matrices. Note that the anticommutation relations for the Pauli matrices make the proof of the above defining property trivial. Those relations define the notion of a Clifford algebra. 3: The most famous Dirac operator describes the propagation of a free electron in three dimensions and is elegantly written D = γ µ ∂µ

using Einstein’s summation convention and even more elegantly as D = ∂/

using the Feynman slash notation.

See also • • • •

Clifford algebra Connection Dolbeault operator Heat kernel.

Source: http://en.wikipedia.org/wiki/Dirac_operator

Principal Authors: Phys, Charles Matthews, Dreamy224, Dcarlson, Rex the first

Dirac operator

104

Double-slit experiment

Importance to physics

FT

The double-slit experiment or two-slit experiment consists of letting light diffract through two slits producing fringes on a screen. These fringes or interference patterns have light and dark regions corresponding to where the light waves have constructively and destructively interfered. The experiment can also be performed with a beam of electrons or atoms, showing similar interference patterns; this is taken as evidence of the "wave-particle duality" predicted by quantum physics. Note, however, that a double-slit experiment can also be performed with water waves in a ripple tank; the explanation of the observed wave phenomena does not require quantum mechanics in any way. The phenomenon is quantum mechanical only when quantum particles - such as atoms, electrons, or photons - manifest as waves.

DR A

Although the double-slit experiment is now often referred to in the context of quantum mechanics, it was originally performed by the English scientist Thomas Young some time around 1805 in an attempt to resolve the question of whether light was composed of particles (the "corpuscular" theory), or rather consisted of waves traveling through some aether, just as sound waves travel in air. The interference patterns observed in the experiment seemed to discredit the corpuscular theory, and the wave theory of light remained well accepted until the early 20th century, when evidence began to accumulate which seemed instead to confirm the particle theory of light.

The double-slit experiment, and its variations, then became a classic Gedankenexperiment (thought experiment) for its clarity in expressing the central puzzles of quantum mechanics; although in this form the experiment was not actually performed with anything other than light until 1961, when Claus Jönsson of the University of Tübingen performed it with electrons. (C Jönsson, Zeitschrift für Physik 161, 454; C. Jönsson 1974 "Electron diffraction at multiple slits", American Journal of Physics 42 4-11), and not until 1974 in the form of "one electron at a time", in a laboratory at the University of Milan, by researchers led by Pier Giorgio Merli, of LAMEL-CNR Bologna. The results of the 1974 experiment were published and even made into a short film, but did not receive wide attention. The experiment was repeated in 1989 by Tonomura et al at Hitachi in Japan. Their equipment was better, reflecting 15 years of advances in electronics and a dedicated development effort by the Hitachi team. Their methodology was more precise and elegant, and their Double-slit experiment

105

FT

results agreed with the results of Merli’s team. Although Tonomura asserted that the Italian experiment had not detected electrons one at a time - a key to demonstrating the wave-particle paradox - single electron detection is clearly visible in the photos and film taken by Merli and his group. In September 2002, the double-slit experiment of Claus Jönsson was voted "the most beautiful experiment" by readers of Physics World.

DR A

Explanation of experiment

In Young’s original experiment, sunlight passes first through a single slit, and then through two thin vertical slits in otherwise solid barriers, and is then viewed on a rear screen. When either slit is covered, a single peak is observed on the screen from the light passing through the other slit. But when both slits are open, instead of the sum of these two singular peaks that would be expected if light were made of particles, a pattern of light and dark fringes is observed. This pattern of fringes was best explained as the interference of the light waves as they recombined after passing through the slits, much as waves in water recombine to create peaks and swells. In the brighter spots, there is "constructive interference", where two "peaks" in the light wave coincide as they reach the screen. In the darker spots, "destructive interference" occurs where a peak and a trough occur together.

Double-slit experiment

FT

106

Figure 12

Replicating Young’s experiment

DR A

This experiment can easily be demonstrated in just the way that Young demonstrated it to the Royal Society of London. An assistant outside used mirrors to direct sunlight at a pinhole opening. The beam from the opening was then bisected by "a slip of card". To make things easier, a modern experimenter could replace the sunlight and mirrors with a laser pointer covered, except for a pinhole, by black paper. Splitting the beam with a small strip of notecard will produce a visible interference pattern when the beam is projected across the room. 51

Quantum version of experiment

By the 1920s, various other experiments (such as the photoelectric effect) had demonstrated that light interacts with matter only in discrete, "quantum"-sized packets called photons. If sunlight is replaced with a light source that is capable of producing just one photon at a time, and the screen is sensitive enough to detect a single photon, Young’s experiment can, in theory, be performed one photon at a time – with identical results. If either slit is covered, the individual photons hitting the screen, over time, create a pattern with a single peak. But if both slits are left open, the pattern of photons hitting the screen, over time, again becomes a series of light and dark fringes. This result seems to both confirm and contradict the wave theory. On the one hand, the interference pattern confirms that light still behaves much like a wave, even though we send it one particle at a time. On the other hand, each time a photon with a certain energy is emitted, the screen detects a photon with the same energy. Under the Copenhagen Interpretation of quantum

51 http://www.cavendishscience.org/phys/tyoung/tyoung.htm

Double-slit experiment

107 theory, an individual photon is seen as passing through both slits at once, and interfering with itself, producing the interference pattern.

FT

A remarkable refinement of the double-slit experiment consists of putting a detector at each of the two slits, to determine which slit the photon passes through on its way to the screen (If the photon or electron passes through only one slit - which it must do, as, by definition, a photon or an electron is a quantum, or "packet" of energy which cannot be subdivided - then logically it cannot interfere with itself and produce an interference pattern). When the experiment is arranged in this way, the fringes disappear.

DR A

The Copenhagen interpretation posits the existence of probability waves which describe the likelihood of finding the particle at a given location. Until the particle is detected at any location along this probability wave, it effectively exists at every point. Thus, when the particle could be passing through either of the two slits, it will actually pass through both, and so an interference pattern results. But if the particle is detected at one of the two slits, then it can no longer be passing through both - it must exist at one or the other, and so no interference pattern appears. The many worlds interpretation states that the particle not only goes through both slits but that it is detected at every possible final location as well – but in different, mutually unobservable worlds.

This is similar to the path integral formulation of quantum mechanics provided by Richard Feynman (although Feynman stresses that this is merely a mathematical description, not an attempt to describe some "real" process that we cannot see), in which a particle such as a photon takes every possible path through space-time to get from point A to point B. In the double-slit experiment, point A might be the emitter, and point B the screen upon which the interference pattern appears, and a particle takes every possible path - through both slits at once - to get from A to B. When a detector is placed at one of the slits, the situation changes, and we now have a different point B at the detector, and a new path between the detector and the screen - upon which the interference pattern no longer appears).

Conditions for interference

A necessary condition for obtaining an interference pattern in a double-slit experiment concerns the difference in pathlength between two paths that light can take to reach a zone of constructive interference on the viewing screen. This difference must be the wavelength of the light that is used, or a multiple of this wavelength. (See illustration. 52) If a beam of sunlight is let in, and that

52 http://schools.matter.org.uk/Content/Interference/formula.html

Double-slit experiment

108

FT

beam is allowed to fall immediately on the double slit, then the fact that the Sun is not a point source degrades the interference pattern. The light from a source that is not a point source behaves like the light of many point sources side by side. Each can create an interference pattern, but the interference patterns of each of the many-side-by-side sources does not coincide on the screen, so they average each other out, and no interference pattern is seen. The presence of the first slit is necessary to ensure that the light reaching the double slit is light from a single point source. The path length from the single slit to the double slit is equally important for obtaining the interference pattern as the path from the double slit to the screen.

Newton’s rings show that light does not have to be coherent in order to produce an interference pattern. Newton’s rings can be readily obtained with plain sunlight. 53 More rings are discernible if for example light from a Sodium lamp is used, since Sodium lamp light is only a narrow band of the spectrum. Light from a Sodium lamp is incoherent. Other examples of interference patterns from incoherent light are the colours of soap bubbles and of oil films on water.

DR A

In general, interference patterns are clearer when monochromatic or nearmonochromatic light is used. Laserlight is as monochromatic as light can be made, therefore laserlight is used to obtain an interference pattern. If the two slits are illuminated by coherent waves, but with polarizations perpendicular with respect to each other, the interference pattern disappears.

Results observed

The bright bands observed on the screen happen when the light has interfered constructively – where a crest of a wave meets a crest. The dark regions show destructive interference – a crest meets a trough. nλ d

=

nλ =

x L

xd L

where

is the wavelength of the light

d is the separation of the slits

53 Newton’s rings. Newton’s Rings from Eric Weisstein’s World of Physics(http://scienceworld.wolfram.

com/physics/NewtonsRings.html)

Double-slit experiment

109 x is the distance between the bands of light (also called fringe distance)

FT

L is the distance from the slits to the screen n is the order of maxima observed (Central Maximum is n=1) This is only an approximation and depends on certain conditions.

It is possible to work out the wavelength of light using this equation and the above apparatus. If d and L are known and x is observed, then can be easily calculated. A detailed treatment of the mathematics of double-slit interference in the context of quantum mechanics is given in the article on Englert-Greenberger duality.

Shape of interference fringes

DR A

The theoretical shapes of the interference fringes observed in Young’s double slit experiment are straight lines which is easily proved. In case two pinholes are used instead of slits, as in the original Young’s experiment, hyperbolic fringes are observed. This is because the difference in paths travelled by the light from the two sources is a constant for a fringe which is the property of a hyperbola.

If the two sources are placed on a line perpendicular to the screen, the shape of the interference fringes is circular as the individual paths travelled by light from the two sources are always equal for a given fringe. This can be done in simpler way by placing a mirror parallel to a screen at a distance and a source of light just above the mirror. (Note the extra phase difference of π due to reflection at the interface of a denser medium)

See also • • •

Elitzur-Vaidman bomb-testing problem Quantum eraser experiment Quantum coherence

Video Demonstration •

From Movie "Down the Rabbit Hole" (sequel to What the Bleep Do We Know!?) Clip of Double Split Experiment

Double-slit experiment

110

References •

FT

http://www.whatthebleep.com/trailer/doubleslit.wm.low.html (The download time over a modem connection is very slow, and playback is interrupted so frequently that it may be impossible to understand.)

Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics, 5th ed., W. H. Freeman. ISBN 0716708108. • Gribbin, John (1999). Q is for Quantum: Particle Physics from A to Z. Weidenfeld & Nicholson. ISBN 0753806851. • Feynman, Richard P. (1988). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 0691024170.

External links

Simple Derivation of Interference Conditions 54 The Double Slit Experiment 55 Double-Slit in Time 56 Keith Mayes explains the Double Slit Experiment in plain English 57

• • • •

Carnegie Mellon department of physics, photo images of Newton’s rings 58 Java demonstration of double slit experiment 59 Double-slit experiment animation 60 Double-slit experiment cartoon animation 61

DR A

• • • •

Source: http://en.wikipedia.org/wiki/Double-slit_experiment

Principal Authors: Eequor, Linas, The Anome, Reddi, Samboy, Pfalstad, Laurascudder, Afshar, Cleonis, Lumidek

54 http://schools.matter.org.uk/Content/Interference/formula.html 55 http://physicsweb.org/article/world/15/9/1 56 http://physicsweb.org/articles/news/9/3/1/1?rss=2.0 57 http://www.thekeyboard.org.uk/Quantum%20mechanics.htm 58 http://physdemo.phys.cmu.edu/newton_rings.htm 59 http://www.falstad.com/ripple/ex-2slit.html

60 http://homepage.univie.ac.at/Franz.Embacher/KinderUni2005/waves.gif 61 http://video.google.com/videoplay?docid=-4237751840526284618&q=quantum

Double-slit experiment

111

Duru-Kleinert transformation

Papers

FT

The Duru-Kleinert transformation, named after H. Duru and Hagen Kleinert, is a mathematical method for solving path integrals of physical systems with singular potentials, which is necessary for the solution of all atomic path integrals due to the presence of Coulomb potentials (singular like 1/r). The Duru-Kleinert transformation replaces the diverging time-sliced path integral of Richard Feynman (which thus does not exist) by a well-defined convergent one.

H. Duru and H. Kleinert, Solution of the Path Integral for the H-Atom, Phys. Letters B 84, 185 (1979) 62

•

H. Duru and H. Kleinert, Quantum Mechanics of H-Atom from Path Integrals, Fortschr. d. Phys. 30, 401 (1982) 63

DR A

•

•

H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets 3. ed., World Scientific (Singapore, 2004) 64 ( read book here 65)

Source: http://en.wikipedia.org/wiki/Duru-Kleinert_transformation Principal Authors: Tobias Bergemann, Michael Hardy, Phys, Conscious

62 http://www.physik.fu-berlin.de/~kleinert/65/65.pdf 63 http://www.physik.fu-berlin.de/~kleinert/83/83.pdf 64 http://www.worldscibooks.com/physics/5057.html 65 http://www.physik.fu-berlin.de/~kleinert/b5

Duru-Kleinert transformation

112

Ehrenfest theorem

d dt hAi

FT

The Ehrenfest theorem, named after Paul Ehrenfest, relates the time derivative of the expectation value for a quantum mechanical operator to the commutator of that operator with the Hamiltonian of the system. It is = ~i h[H, A]i

where A is some QM operator and is its expectation value. Notice how neatly Ehrenfest’s theorem fits into the →Heisenberg picture of quantum mechanics.

Ehrenfest’s theorem is closely related to Liouville’s theorem from Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. In fact, it is a general rule of thumb that a theorem in quantum mechanics which contains a commutator can be turned into a theorem in Classical mechanics by changing the commutator into a Poisson bracket and multiplying by i~.

DR A

The theorem can be shown to follow from the →Lindblad equation, a master equation for the time evolution of a mixed state.

Derivation

Suppose some system is presently in a quantum state Φ. If we want to know the instantaneous time derivative of the expectation value of A, that is, by definition   R ∗ R  dΦ∗  R d d = AΦ dV + Φ∗ dA dt hAi = dt Φ AΦ dV dt dt Φ dV + R ∗  dΦ  Φ A dt dV =

R  dΦ∗  dt

AΦ dV +

R

Φ∗ A



dΦ dt



dV,

where we are integrating over all space, and we have assumed the operator A is time independent, so that its derivative is zero. If we apply the →Schrödinger equation, we find that dΦ dt

=

1 i~ HΦ

and

dΦ∗ dt

=

−1 ∗ ∗ i~ Φ H

=

−1 ∗ i~ Φ H.

Ehrenfest theorem

113

FT

Notice H = H ∗ because the Hamiltonian is hermitian. Placing this into the above equation we have R ∗ d 1 1 i dt hAi = i~ Φ (AH − HA)Φ dV = i~ h[A, H]i = ~ h[H, A]i.

General example

For the very general example of a massive particle moving in a potential, the Hamiltonian is simply H=

p2 2m

+ V (r)

where r is just the location of the particle. Suppose we wanted to know the instantaneous change in momentum p. Using Ehrenfest’s theorem, we have d dt hpi

=

1 i~ h[p, H]i

=

1 i~ h[p, V (r)]i

DR A

since p commutes with itself. When represented in coordinate space, the momentum operator p = −i~∇, so R ∗ R ∗ d dt hpi = Φ V (r)∇Φ dV − Φ ∇(V (r)Φ) dV. After applying a product rule, we have d dt hpi

= h−∇V (r)i = hF i,

but we recognize this as Newton’s second law. This is an example of the correspondence principle, the result manifests as Newton’s second law in the case of having so many particles that the net motion is given exactly by the expectation value of a single particle.

Notes •

↑ In →Bra-ket notation d dt hφ|xi

=

−1 ˆ i~ hφ|H|xi

=

−1 i~ hφ|xiH

=

−1 ∗ i~ Φ H

ˆ is the Hamiltonian operator, and H is the Hamiltonian representwhere H ed in coordinate space (as is the case in the derivation above). In other words, we applied the adjoint operation to the entire Schrödinger equation, which flipped the order of operations for H and Φ.

Source: http://en.wikipedia.org/wiki/Ehrenfest_theorem

Ehrenfest theorem

114 Principal Authors: Michael Hardy, Hyandat, WMDickson, Pathfinder, Brienanni

FT

Einselection

DR A

Einselection is short for environmentally-induced superselection, a nickname coined by Wojciech H. Zurek. Einselection is the quantum process whereby the environment persistently monitors a quantum system, causing decoherence between its states. The decoherence process selects a certain subset of states from the enormous →Hilbert space. These ’pointer states’ are stable despite environmental interaction, which explains the emergence of a preferred basis in quantum measurement. The einselected states lack coherence, and therefore do not exhibit the quantum behaviours of entanglement and superposition. Since only quasi-local, essentially classical states survive the decoherence process, einselection can in many ways explain the emergence of a (seemingly) classical reality in a fundamentally quantum universe (at least to local observers).

Source: http://en.wikipedia.org/wiki/Einselection

Principal Authors: Conscious, Mo0, Icairns, ShaneKing, Jag123

Electronic density

In quantum mechanics, and in particular in quantum chemistry, the electronic density ρ corresponding to an N -electron wavefunction Ψ(N ) is the oneelectron function given by R ρ(x) = dx2 ... dxN |Ψ(N ) (x, x2 , ..., xN )|2 In the case Ψ(N ) is a →Slater determinant made of N spin orbitals ϕk : PN 2 ρ(x) = N1 k=1 |ϕk (x)|

The two-electron electronic density is given by R ρ(x, x0 ) = dx3 ... dxN |Ψ(N ) (x, x0 , x3 , ..., xN )|2

Those quantities are particularly important in the context of density functional theory. The coordinates x used here are the spin-spatial coordinates. Electronic density

115

Source: http://en.wikipedia.org/wiki/Electronic_density

FT

Principal Authors: Vb, Gsp

Electronic Hamiltonian

The Electronic Hamiltonian is an operator in quantum mechanics (and in particular quantum chemistry) which describes the motions of electrons and nuclei in a polyatomic molecule. The terminology is sometimes used interchangeably to mean either the Electronic molecular Hamiltonian or the full electronic Hamiltonian. The latter includes a kinetic energy operator corresponding to the contributions from the nuclei. There are a number of interrelated concepts associated with the term "Electronic Hamiltonian". These include the following: Full electronic Hamiltonian Electronic Hamiltonian Nuclear Hamiltonian Clampled Hamiltonian

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• • • •

Depending on the context, "full electronic Hamiltonian" may be used interchangeably with "electronic Hamiltonian". Furthermore, the "Clamped Hamiltonian" or may be used interchangeably with "Electronic Hamiltonian". The latter is usually used only when discussing various methods associated with the Born-Oppenheimer approximation.

Full electronic Hamiltonian

Let R denote the vector of nuclear coordinates, and r the vector of electronic coordinates. The full electronic Hamiltonian consists of 5 terms. They are • • •

The kinetic energy operators for each nuclei in the system; The kinetic energy operators for each electron in the system; The potential energy between the electrons and nuclei - the total electronnucleus Coulombic attraction in the system; • The potential energy arising from Coulombic electron-electron repulsions • The potential energy arising from Coulombic nuclei-nuclei repulsions - also known as the nuclear repulsion energy. See electric potential for more details. Electronic Hamiltonian

116

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Hence, the full electronic Hamiltonian is ˆ el = Tˆn + Tˆe + U ˆ en + U ˆ ee + U ˆ nn H

Electronic Hamiltonian and potential energy surfaces Very often, the electronic Hamiltonian is defined to be ˆ e = Tˆe + U ˆ en + U ˆ nn + U ˆ ee H so that the full Hamiltonian would be written as ˆ el = H ˆ e + Tˆn . H

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In this case, the kinetic energy operator Tˆn would be known as the nuclear Hamiltonian. The electronic Hamiltonian contains all three potential terms because their sum ˆ en + U ˆ nn + U ˆ ee Vˆ (r, R) = U is the expression which gives rise to the physically meaningful potential energy surfaces ubiquitous in chemistry, for fixed nuclear geometry R.

Written in atomic units, the electronic Hamiltonian becomes:

ˆ el = P − 1 ∇2 −P P H i i a 2 i Za Zb 1 P P

2

a

Za 1 |ri −da | + 2

P P i

1 j6=i |ri −rj | +

b6=a |da −db |

where • • •

ri is the vector position of electron i with vector components in Bohr radii, Za is the charge of fixed nucleus a in units of the elementary charge, da is the vector position of nucleus a with vector components in Bohr radii.

Electronic molecular Hamiltonian

The electronic molecular Hamiltonian is the term of the molecular Hamiltonian obtained when the molecular geometry is frozen. This is also known as the clamped Hamiltonian or clamped Hamiltonian approximation. Within the Born-Oppenheimer approximation, the electronic Hamiltonian is said to depend adiabatically on the molecular geometry. Its discrete eigenvalues are called potential energy surfaces and the corresponding eigenstates the electronic states of the molecule. The electronic states are labelled according to their group representation and spin multiplicity.

Electronic Hamiltonian

117

Adiabatic and diabatic states

FT

By definition, the adiabatic states are diagonal in the electronic Hamiltonian. A consequence is that it is not diagonal in the kinetic energy operator. The off diagonal terms of this operator are known as the nonadiabatic operator.

Strictly diabatic states do not exist in general, although in the ideal case, it is diagonal in the kinetic energy operator, and off diagonal in the electronic Hamiltonian. In other words, the diabatic states minimize the magnitude of the contributions of the nonadiabatic operator.

Source: http://en.wikipedia.org/wiki/Electronic_Hamiltonian Principal Authors: HappyCamper, Ian Pitchford

Electronic state

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Electronic state is a quantum state of a system consisting of electrons (usually orbitals or chemical bonds in crystals or molecules). The state with lowest energy is called ground state, states with higher energy are excited states.

Source: http://en.wikipedia.org/wiki/Electronic_state

Elementary particle

For the novel by Michel Houellebecq, see The Elementary Particles.

In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic particles of the universe from which all larger particles are made. In the modern theory of particle physics, the Standard Model, the quarks, leptons, and gauge bosons are elementary particles. 6667 Historically, the hadrons (mesons and baryons such as the proton and neutron) and even whole atoms were once regarded as elementary particles.

66 Gribbon, John (2000). Q is for Quantum - An Encyclopedia of Particle Physics. Simon & Schuster.

ISBN 068485578X.

67 Clark, John, E.O. (2004). The Essential Dictionary of Science. Barnes & Noble. ISBN 0760746168.

Elementary particle

118

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Elementary particles are grouped according to spin: particles normally associated with matter are fermions, having spin 1/2; they are divided into twelve flavours. Particles associated with fundamental forces are bosons, having spin 1 (or 2 for gravity). 68 Fermions:

Quarks — up, down, strange, charm, bottom, top

Leptons — electron, muon, tau, electron neutrino, muon neutrino, tau neutrino

•

Bosons:

Gauge bosons – gluon, W and Z bosons, photon

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Other bosons — Higgs boson, graviton

Standard Model

Main article: Standard Model

The Standard Model of particle physics contains 12 flavours of elementary fermions ("matter particles"), plus their corresponding antiparticles, as well as elementary bosons that mediate the forces and the still undiscovered Higgs boson. However, the Standard Model is widely considered to be a provisional theory rather than a truly fundamental one, since it is fundamentally incompatible with Einstein’s general relativity. There are likely to be hypothetical elementary particles not described by the Standard Model, such as the graviton, the particle that would carry the gravitational force or the sparticles, supersymmetric partners of the ordinary particles.

Fundamental fermions Main article: fermion

The 12 fundamental fermionic flavours are divided into three generations of four particles each. Six of the particles are quarks. The remaining six are

68 Veltman, Martinus (2003). Facts and Mysteries in Elementary Particle Physics. World Scientific. ISBN

981238149X.

Elementary particle

119 leptons, three of which are neutrinos, and the remaining three of which have an electric charge of -1: the electron and its two cousins, the muon and the tau lepton.

• • • •

Second generation

electron: e -

Third generation

FT

First generation

muon: µ -

• electron-neutrino: ν e • up quark: u • down quark: d •

• muon-neutrino: ν µ • charm quark: c • strange quark: s •

Table 1

Particle Generations

Antiparticles Main article: antimatter

tau lepton: τ tau-neutrino: ν τ top quark: t bottom quark: b

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There are also 12 fundamental fermionic antiparticles which correspond to these 12 particles. The positron e + corresponds to the electron and has an electric charge of +1 and so on: First generation

• • • •

Second generation

positron: e + • electron-antineutrino: ν¯e • up antiquark: u ¯ • down antiquark: d¯ •

positive muon: µ + muon-antineutrino: ν¯µ charm antiquark: c¯ strange antiquark: s¯

Table 2

Third generation • • • •

positive tau lepton: τ + tau-antineutrino: ν¯τ top antiquark: t¯ bottom antiquark: ¯b

Antiparticles

Quarks

Main article: quark

Quarks and antiquarks have never been detected to be isolated, a fact explained by confinement. Every quark carries one of three color charges of the strong interaction; antiquarks similarly carry anticolor. Color charged particles interact via gluon exchange in the same way that charged particles interact via photon exchange. However, gluons are themselves color charged, resulting in an amplification of the strong force as color charged particles are separated. Unlike the electromagnetic force which diminishes as charged particles separate, color charged particles feel increasing force; effectively, they can never separate from one another. However, color charged particles may combine to form color neutral composite particles called hadrons. A quark may pair up to an antiquark: the quark has a color and the antiquark has the corresponding anticolor. The color and

Elementary particle

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FT

anticolor cancel out, forming a color neutral meson. Or three quarks can exist together: one quark is "red", another "blue", another "green". These three colored quarks together form a color neutral baryon. Or three antiquarks can exist together: one antiquark is "antired", another "antiblue", another "antigreen". These three anticolored antiquarks form a color neutral antibaryon. Quarks also carry fractional electric charges, but since they are confined within hadrons whose charges are all integral, fractional charges have never been isolated. Note that quarks have electric charges of either +2/3 or -1/3, whereas antiquarks have corresponding electric charges of either -2/3 or +1/3.

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Evidence for the existence of quarks comes from deep inelastic scattering: firing electrons at nuclei to determine the distribution of charge within nucleons (which are baryons). If the charge is uniform, the electric field around the proton should be uniform and the electron should scatter elastically. Low-energy electrons do scatter in this way, but above a particular energy, the protons deflect some electrons through large angles. The recoiling electron has much less energy and a jet of particles is emitted. This inelastic scattering suggests that the charge in the proton is not uniform but split among smaller charged particles: quarks.

Fundamental bosons Main article: boson

In the Standard Model, vector (spin-1) bosons (gluons, photons, and the W and Z bosons) mediate forces, while the Higgs boson (spin-0) is responsible for particles having intrinsic mass.

Gluons

Main article: gluon

Gluons are the mediators of the strong interaction and carry both colour and anticolour. Although gluons are massless, they are never observed in detectors due to colour confinement; rather, they produce jets of hadrons, similar to single quarks. The first evidence for gluons came from annihilations of electrons and positrons at high energies which sometimes produced three jets — a quark, an antiquark, and a gluon.

Electroweak bosons

Main article: W and Z bosons

Elementary particle

121 There are three weak gauge bosons: W +, W -, and Z 0; these mediate the weak interaction. The massless photon mediates the electromagnetic interaction.

Main article: higgs boson

FT

Higgs boson

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Although the weak and electromagnetic forces appear quite different to us at everyday energies, the two forces are theorized to unify as a single electroweak force at high energies. This prediction was clearly confirmed by measurements of cross-sections for high-energy electron-proton scattering at the HERA collider at DESY. The differences at low energies is a consequence of the high masses of the W and Z bosons, which in turn are a consequence of the Higgs mechanism. Through the process of spontaneous symmetry breaking, the Higgs selects a special direction in electroweak space that causes three electroweak particles to become very heavy (the weak bosons) and one to remain massless (the photon). Although the Higgs mechanism has become an accepted part of the Standard Model, the Higgs boson itself has not yet been observed in detectors. Indirect evidence for the Higgs boson suggests its mass lies below about 200 GeV. In this case, the LHC experiments will be able to discover this last missing piece of the Standard Model.

Beyond the Standard Model

Although all experimental evidence confirms the predictions of the Standard Model, many physicists find this model to be unsatisfactory due to its many undetermined parameters, many fundamental particles, the non-observation of the Higgs boson and other more theoretical considerations such as the hierarchy problem. There are many speculative theories beyond the Standard Model which attempt to rectify these deficiencies.

Grand unification

Main article: grand unification theory

One extension of the Standard Model attempts to combine the electroweak interaction with the strong interaction into a single ’grand unified theory’ (GUT). Such a force would be spontaneously broken into the three forces by a Higgslike mechanism. The most dramatic prediction of grand unification is the existence of X bosons, which cause proton decay. However, the non-observation of proton decay at Super-Kamiokande rules out the simplest GUTs, including SU(5) and SO(10). Elementary particle

122

Supersymmetry Main article: supersymmetry

String theory

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Main article: string theory

FT

Supersymmetry extends the Standard Model by adding an additional class of symmetries to the Lagrangian. These symmetries exchange fermionic particles with bosonic ones. Such a symmetry predicts the existence of supersymmetric particles, abbreviated as sparticles, which include the sleptons, squarks, neutralinos and charginos. Each particle in the Standard Model would have a superpartner whose spin differs by 1/2 from the ordinary particle. Due to the breaking of supersymmetry, the sparticles are much heavier than their ordinary counterparts; they are so heavy that existing particle colliders would not be powerful enough to produce them. However, some physicists believe that sparticles will be detected when the Large Hadron Collider at CERN begins running.

According to string theorists, each kind of fundamental particle corresponds to a different pattern of fundamental string. All strings are essentially the same, although they may be open (lines) or closed (loops). Different particles differ in the coordination of their strings. Modern string theories include supersymmetry, making them superstring theories. One particular prediction of string theory is the existence of extremely massive counterparts of ordinary particles due to vibrational excitations of the fundamental string. Another important prediction of string theory is the existence of a massless spin-2 particle behaving like the graviton. By predicting gravity, string theory unifies quantum mechanics with general relativity, making it the first consistent theory of quantum gravity. One problem with string theory is that it predicts that the number of dimensions for spacetime much greater than 4 (the number of observed dimensions). These extra dimensions are supposedly compactified or rolled-up. Other related theories such as brane theories contain extended extra dimensions, which are hidden from us by our confinement to a brane.

Preon theory

Main article: preon

Elementary particle

123

See also • • •

→Subatomic particle Particle physics List of particles

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References

FT

According to preon theory there are one or more orders of particles more fundamental than those (or most of those) found in the Standard Model. The most fundamental of these are normally called preons, which is derived from "prequarks". In essence, preon theory tries to do for the Standard Model what the Standard Model did for the particle zoo that came before it. Most models assume that almost everything in the Standard Model can be explained in terms of three to half a dozen more fundamental particles and the rules that govern their interactions. Interest in preons has waned since the simplest models were experimentally ruled out in the 1980’s.

•

Brian Greene, The Elegant Universe, W.W.Norton & Company, 1999, ISBN 0-393-05858-1.

External links

Greene, Brian, " Elementary particles 69". The Elegant Universe, NOVA (PBS) particleadventure.org: The Standard Model 70, * Unsolved Mysteries. Beyond The Standard Model 71, * What is the World Made of? The Naming of Quarks 72 • University of California: Particle Data Group 73 • particleadventure.org: Particle chart 74 • CERNCourier: Season of Higgs and melodrama 75 • Pentaquark information page 76

• •

Source: http://en.wikipedia.org/wiki/Elementary_particle

69 http://www.pbs.org/wgbh/nova/elegant/part-flash.html 70 http://particleadventure.org/particleadventure/frameless/standard_model.html 71 http://particleadventure.org/particleadventure/frameless/beyond_start.html 72 http://particleadventure.org/particleadventure/frameless/quarknaming.html 73 http://pdg.lbl.gov/ 74 http://particleadventure.org/particleadventure/frameless/chart.html 75 http://www.cerncourier.com/main/article/41/2/17 76 http://plato.phy.ohiou.edu/~hicks/thplus.htm

Elementary particle

124 Principal Authors: AugPi, Xerxes314, Glenn, Sadi Carnot, Reddi

FT

Energy level splitting

Energy level splitting occurs in physics when the degenerate energy levels of two or more states are split because of external fields or other effects. The term is most commonly used in quantum theory in reference to the electron configuration in atoms or molecules.

Examples

The →Zeeman effect - the splitting of electronic levels in an atom because of an external magnetic field. • The →Stark effect - splitting because of an external electric field. • The Jahn-Teller effect - splitting of electronic levels in a molecule because breaking the symmetry lowers the energy when the degenerate orbitals are partially filled.

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•

See also •

Energy level

Source: http://en.wikipedia.org/wiki/Energy_level_splitting

Principal Authors: AnonUser, GangofOne, Donbert, Karol Langner, Itub

Entanglement witness

In quantum information theory, an entanglement witness is an object of geometric nature which distinguishes an entangled state from separable ones.

Details

We first recall a few preliminary facts before giving the main result which shows the existence of entanglement witnesses. Let a composite quantum system have state space HA ⊗ HB . A mixed state ρ is then a trace-class positive operator on the state space which has trace 1. We can view the family of states as a subset of the real Banach space generated Entanglement witness

125

FT

by the Hermitian trace-class operators, with the trace norm. A mixed state ρ is separable if it can be approximated, in the trace norm, by states of the form P B ξ = ki=1 pi ρA i ⊗ ρi B ,where ρA i ’s and ρi ’s are pure states on the subsystems A and B respectively. So the family of separable states is the closed convex hull of pure product states. We will make use of the following variant of Hahn-Banach theorem:

Theorem Let S1 and S2 be convex closed sets in a real Banach space and one of them is compact, then there exists a bounded functional f separating the two sets.

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This is a generalization of the fact that, in real Euclidean space, given a convex set and a point outside, there always exists an affine subspace separating the two. The affine subspace manifests itself as the functional f. In the present context, the family of separable states is a convex set in the space of trace class operators. If ρ is an entangled state (thus lying outside the convex set), then by theorem above, there is a functional f separating ρ from the separable states. It is this functional f, or its identification as an operator, that we call an entanglement witness. There are more than one hyperplane separating a closed convex set and a point lying outside of it. So for an entangled state there are more than one entanglement witnesses. Recall the fact that the dual space of the Banach space of trace-class operators is isomorphic to the set of bounded operators. Therefore we can identify f with a Hermitian operator A. Therefore, modulo a few details, we have shown the existence of an entanglement witness given an entangled state: Theorem For every entangled state ρ, there exists a Hermitian operator A such that T r(A ρ) < 0 and T r(Aσ) ≥ 0, for all separable state σ.

When both HA and HB have finite dimensions, there is no difference between trace-class and Hilbert-Schmidt operators. So in that case A can be given by Riesz representation theorem. As an immediate corollary, we have: Theorem A mixed state σ is separable if and only if T r(Aσ) ≥ 0

for any bounded operator A satisfying T r(A · P ⊗ Q) ≥ 0, for all product pure state P ⊗ Q. If a state is separable, clearly the desired implication from the theorem must hold. On the other hand, given an entangled state, one of its entanglement witnesses will violate the given condition.

Entanglement witness

126 Thus if a bounded functional f of the trace-class Banach space and f is positive on the product pure states, then f, or its identification as a Hermitian operator, is an entanglement witness. Such a f indicates the entanglement of some state.

FT

Using the isomorphism between entanglement witnessnes and non-completlely positive maps, it was shown (by the Horodecki’s) that

Theorem A mixed state σ ∈ L(HA ) ⊗ L(HB ) is separable if for every positive map from bounded operators on HB to bounded operators on HA , the operator (IA ⊗ Λ)(σ) is positive, where IA is the identity map on L(HA ), the bounded operators on HA .

References

R.B. Holmes. Geometric Functional Analysis and Its Applications, SpringerVerlag, 1975.

•

M. Horodecki, P. Horodecki, R. Horodecki, Separability of Mixed States: Necessary and Sufficient Conditions, Physics Letters A 210, 1996.

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•

Source: http://en.wikipedia.org/wiki/Entanglement_witness

EP Quantum Mechanics

In physics, EP quantum mechanics is a theory of motion of point particles, partly included in the framework of quantum trajectory representation theories of quantum mechanics, based upon an equivalence postulate similar in content to the equivalence principle of general relativity, rather than on the traditional Copenhagen interpretation of quantum mechanics. The equivalence postulate states that all one-particle systems can be connected by a non-degenerate coordinate transformation, more precisely by a map over the cotangent bundle of the position manifold, so that there exists a quantum action function S(q) transforms as a scalar field. Here, the action is defined as dS(q) = pi (q)dq i

is the canonical one-form. This property is the heart of the EP formulation of quantum mechanics. An immediate consequence of the EP is the removal of the rest frame. The theory is based on symmetry properties of Schwarzian derivative and on the quantum stationary Hamilton-Jacobi equation (QSHJE), which is a partial differential equation for the quantum action function S(q), EP Quantum Mechanics

127 the quantum version of the Hamilton-Jacobi equations differing from the classical one for the presence of a quantum potential term ~2 4m {S(q), q}

FT

Q(q) =

with {, } denoting the Schwarzian derivative. The QSHJE can be demonstrated to imply the →Schrödinger equation with square-summability of the wave function, and thus quantization of energy, due to continuity conditions of the quantum potential, without any assumption on the probabilistic interpretation of the wave function. The theory, which is a work in progress, may or may not include probabilistic interpretation as a consequence OR a hidden variable description of trajectories.

References

Alon E. Faraggi, M. Matone (2000) "The Equivalence Postulate of Quantum Mechanics", International Journal of Modern Physics A, Volume 15, Issue 13, pp. 1869-2017. arXiv hep-th/9809127 77

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G. Bertoldi, Alon E. Faraggi, M. Matone (2000) "Equivalence principle, higher dimensional Mobius group and the hidden antisymmetric tensor of Quantum Mechanics", Class. Quantum Grav. 17 (2000) 3965–4005. arXiv hep-th/9909201 78

Source: http://en.wikipedia.org/wiki/EP_Quantum_Mechanics Principal Authors: Matteoeo, Pjacobi, Enochlau

Excited state

In quantum mechanicsan excited state of a system (such as an atom, molecule or nucleus) is any quantum state of the system that has a higher energy than the ground state (that is, more energy than the absolute minimum). The lifetime (see resonance) of a system in an excited state is usually short: spontaneous or induced emission of a quantum of energy (such as a photon or a phonon) usually occurs shortly after the system is promoted to the excited

77 http://arxiv.org/abs/hep-th/9809127 78 http://arxiv.org/abs/hep-th/9909201

Excited state

128 state, returning the system to a state with lower energy (a less excited state or the ground state).

FT

Example: the hydrogen atom

A simple example of this concept comes by considering the hydrogen atom.

The ground state of the hydrogen atom corresponds to having the atom’s single electron in the lowest possible orbit (that is, the spherically symmetric "1s" wavefunction, which has the lowest possible quantum numbers). By giving the atom additional energy (for example, by the absorption of a photon of an appropriate energy), the electron is able to move into an excited state (one with one or more quantum numbers greater than the minimum possible). If the photon has too much energy, the electron will cease to be bound to the atom, and the atom will become ionised.

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Once the electron is in its excited state, we deem the hydrogen atom to be in its excited state. The atom may return to a lower excited state, or the ground state, by emitting a photon with a characteristic energy. Emission of photons from atoms in various excited states leads to an electromagnetic spectrum showing a series of characteristic emission lines (including, in the case of the hydrogen atom, the Lyman series, the Balmer series, the Paschen series, the Brackett series.)

External links •

Picture of a hydrogen atom changing from ground state to an excited state 79

See also • •

→Rydberg formula →Quantum state

Source: http://en.wikipedia.org/wiki/Excited_state

Principal Authors: Andrewwall, Bensaccount, ALoan, Vogon, Conscious

79 http://www.klimaforschung.net/kernreaktion/Orbital01.gif

Excited state

129

Exotic hadron

FT

Exotic hadrons are subatomic particles made of quarks and bound by the strong interaction that are not predicted by the simple quark model. That is, they do not have the same quark content as ordinary hadrons: exotic baryons have more than just the three quarks of ordinary baryons, and exotic mesons do not have one quark and one antiquark like ordinary mesons. Experimental signatures for exotic hadrons have been seen recently but remain a topic of controversy in particle physics.

Source: http://en.wikipedia.org/wiki/Exotic_hadron

Faddeev equations

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The Faddeev equations are equations that describes, at once, all the possible exchanges/interactions in a system of three particles in a fully quantum mechanical formulation. They can be solved iteratively with powerful computer codes. In general, Faddeev equations need as input a potential that describes the interaction between two individual particles. It is also possible to introduce a term in the equation in order to take also three-body forces into account. The Faddeev equations are the most often used non-perturbative formulation of the quantum-mechanical three-body problem. Unlike the three body problem in classical mechanics, the quantum three body problem is uniformly soluble. In nuclear physics, the off the energy shell nucleon-nucleon interaction has been studied by analyzing (n,2n) and (p,2p) reactions on deuterium targets, using the Faddeev Equations. The nucleon-nucleon interaction is expanded (approximated) as a series of separable potentials. The Coulomb interaction between two protons is a special problem, in that its expansion in separable potentials does not converge, but this is handled by matching the Faddeev solutions to long range coulomb solutions, instead of to plain waves.

Separable potentials are interactions that do not preserve a particle’s location. Ordinary local potentials can be expressed as sums of separable potentials. The physical nucleon-nucleon interaction, which involves exchange of mesons, is not expected to be either local or separable.

Faddeev equations

130 Source: http://en.wikipedia.org/wiki/Faddeev_equations

FT

Principal Authors: David R. Ingham, Charles Matthews, Freakofnurture, Conscious, Philipum

Fano resonance

In physics, a Fano resonance, in contrast with a Breit-Wigner resonance, is a resonance for which the corresponding profile in the cross-section has the so-called Fano shape, i.e. it can be fitted with a function proportional to: (qΓres /2+E−Eres )2 . (E−Eres )2 +(Γres /2)2

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The Eres and Γres parameters are the standard Breit-Wigner parameters (position and width of the resonance, respectively). The q parameter is the socalled Fano parameter. It is interpreted (within the Feshbach-Fano partioning theory) as the ratio between the resonant and direct (background) scattering probability. In the case the direct scattering probability is vanishing, the q parameter becomes infinite and the Fano formula is boiling down to the usual Breit-Wigner (Lorentzian) formula: 1 (E−Eres )2 +(Γres /2)2 .

The classical reference is U. Fano, Phys. Rev. 124, 1866 (1961).

Source: http://en.wikipedia.org/wiki/Fano_resonance Principal Authors: Michael Hardy, Vipul

Fermi-Dirac statistics

In statistical mechanics, Fermi-Dirac statistics is a particular case of particle statistics developed by Enrico Fermi and Paul Dirac that determines the statistical distribution of fermions over the energy states for a system in thermal equilibrium. In other words, it is a probability of a given energy level to be occupied by a fermion. Fermions are particles which are indistinguishable and obey the Pauli exclusion principle, i.e., no more than one particle may occupy the same quantum state at the same time. Statistical thermodynamics is used to describe the behaviour of large numbers of particles. A collection of non-interacting fermions is called a →Fermi gas. Fermi-Dirac statistics

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FT

131

Figure 13 Fermi-Dirac distribution as a function of /µ plotted for 4 different temperatures. Occupancy transitions are smoother at higher temperatures.

F-D statistics was introduced in 1926 by Enrico Fermi and Paul Dirac and applied in 1927 by Arnold Sommerfeld to electrons in metals. For F-D statistics, the expected number of particles in states with energy i is ni =

gi e(i −µ)/kT +1

where:

ni is the number of particles in state i, i is the energy of state i,

gi is the degeneracy of state i (the number of states with energy i ), µ is the chemical potential (Sometimes the →Fermi energy EF instead, as a low-temperature approximation),

Fermi-Dirac statistics

is used

132 k is Boltzmann’s constant, and

FT

T is absolute temperature. In the case where µ is the →Fermi energy EF and gi = 1 , the function is called the Fermi function: 1 e(i −EF )/kT +1

DR A

F (E) =

Figure 14 Fermi-Dirac distribution as a function of temperature. More states are occupied at higher temperatures.

Which distribution to use

Fermi-Dirac and Bose-Einstein statistics apply when quantum effects have to be taken into account and the particles are considered "indistinguishable". The quantum effects appear if the concentration of particles (N/V) ≥ n q (where n q is the quantum concentration). The quantum concentration is when the interparticle distance is equal to the thermal de Broglie wavelength i.e. when the wavefunctions of the particles are touching but not overlapping. As the quantum concentration depends on temperature; high temperatures will put Fermi-Dirac statistics

133

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most systems in the classical limit unless they have a very high density e.g. a White dwarf. Fermi-Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), Bose-Einstein statistics apply to bosons. Both Fermi-Dirac and Bose-Einstein become Maxwell-Boltzmann statistics at high temperatures or low concentrations.

DR A

Maxwell-Boltzmann statistics are often described as the statistics of "distinguishable" classical particles. In other words the configuration of particle A in state 1 and particle B in state 2 is different from the case where particle B is in state 1 and particle A is in state 2. When this idea is carried out fully, it yields the proper (Boltzmann) distribution of particles in the energy states, but yields non-physical results for the entropy, as embodied in →Gibbs paradox. These problems disappear when it is realized that all particles are in fact indistinguishable. Both of these distributions approach the Maxwell-Boltzmann distribution in the limit of high temperature and low density, without the need for any ad hoc assumptions. Maxwell-Boltzmann statistics are particularly useful for studying gases F-D statistics are most often used for the study of electrons in solids. As such, they form the basis of semiconductor device theory and electronics.

A derivation

Consider a single-particle state of a multiparticle system, whose energy is . For example, if our system is some quantum gas in a box, then a state might be a particular single-particle wave function. Recall that, for a grand canonical ensemble in general, the grand partition function is P Z = s e−(E(s)−µN (s))/kT where

E(s) is the energy of a state s,

N (s) is the number of particles possessed by the system when in the state s,

µ denotes the chemical potential, and

s is an index that runs through all possible microstates of the system.

In the present context, we take our system to be a fixed single-particle state (not a particle). So our system has energy n ·  when the state is occupied by n particles, and 0 if it is unoccupied. Consider the balance of single-particle states to be the reservoir. Since the system and the reservoir occupy the same Fermi-Dirac statistics

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134

Figure 15 Fermi-Dirac distribution as a function of . High energy states are less probable. Or, low energy states are more probable.

physical space, there is clearly exchange of particles between the two (indeed, this is the very phenomenon we are investigating). This is why we use the grand partition function, which, via chemical potential, takes into consideration the flow of particles between a system and its thermal reservoir.

For fermions, a state can only be either occupied by a single particle or unoccupied. Therefore our system has multiplicity two: occupied by one particle, or unoccupied, called s1 and s2 respectively. We see that E(s1 ) = , N (s1 ) = 1, and E(s2 ) = 0, N (s2 ) = 0. The partition function is therefore P Z = 2i=1 e−(E(si )−µN (si ))/kT = e−(−µ)/kT + 1. For a grand canonical ensemble, probability of a system being in the microstate sα is given by P (sα ) =

e−(E(sα )−µN (sα ) . Z

Our state being occupied by a particle means the system is in microstate s1 , whose probability is Fermi-Dirac statistics

135 e−(E(s1 )−µN (s1 ))/kT Z

n ¯ = P (s1 ) =

=

e−(−µ)/kT e−(−µ)/kT +1

=

1 . e(−µ)/kT +1

FT

n ¯ is called the Fermi-Dirac distribution. For a fixed temperature T, n ¯ () is the probability that a state with energy  will be occupied by a fermion. Notice n ¯ is a decreasing function in . This is consistent with our expectation that higher energy states are less likely to be occupied. Note that if the energy level  has degeneracy g , then we would make the simple modification: n ¯ = g ·

1 . e(−µ)/kT +1

This number is then the expected number of particles in the totality of the states with energy .

For all temperature T, n ¯ (µ) = 21 , that is, the states whose energy is µ will always have equal probability of being occupied or unoccupied.

DR A

In the limit T → 0, n ¯ becomes a step function (see graph above). All states whose energy is below the chemical potential will be occupied with probability 1 and those states with energy above µ will be unoccupied. The chemical potential at zero temperature is called →Fermi energy, denoted by EF , i.e. EF = µ(T = 0).

It may be of interest here to note that, in general the chemical potential is temperature-dependent. However, for systems well below the Fermi temperature TF = EkF , it is often sufficient to use the approximation µ ≈ EF .

Another derivation

In the previous derivation, we have made use of the grand partition function (or Gibbs sum over states). Equivalently, the same result can be achieved by directly analysing the multiplicities of the system. Suppose there are two fermions placed in a system with four energy levels. There are six possible arrangements of such a system, which are shown in the diagram below.

A B C D E F

1 * * *

2 *

3

4

*

*

* *

* *

* *

Fermi-Dirac statistics

136

FT

Each of these arrangements is called a microstate of the system. Assume that, at thermal equilibrium, each of these microstates will be equally likely, subject to the constraints that there be a fixed total energy and a fixed number of particles. Depending on the values of the energy for each state, it may be that total energy for some of these six combinations is the same as others. Indeed, if we assume that the energies are multiples of some fixed value , the energies of each of the microstates become: A: 3 B: 4 C: 5 D: 5

DR A

E: 6 F: 7

So if we know that the system has an energy of 5, we can conclude that it will be equally likely that it is in state C or state D. Note that if the particles were distinguishable (the classical case), there would be twelve microstates altogether, rather than six. Now suppose we have a number of energy levels, labelled by index i , each level having energy  i and containing a total of n i particles. Suppose each level contains g i distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of g i associated with level i is called the "degeneracy" of that energy level. The Pauli exclusion principle states that only one fermion can occupy any such sublevel.

Let w(n, g) be the number of ways of distributing n particles among the g sublevels of an energy level. Its clear that there are g ways of putting one particle into a level with g sublevels, so that w(1, g) = g which we will write as: w(1, g) =

g! 1!(g−1)!

Fermi-Dirac statistics

137

FT

We can distribute 2 particles in g sublevels by putting one in the first sublevel and then distributing the remaining n - 1 particles in the remaining g - 1 sublevels, or we could put one in the second sublevel and then distribute the remaining n - 1 particles in the remaining g - 2 sublevels, etc. so that w’(2, g) = w(1, g - 1) + w(1,g - 2) + ... + w(1, 1) or Pg−1 P (g−k)! g! w(2, g) = k=1 w(1, g − k) = g−1 k=1 1!(g−k−1)! = 2!(g−2)! where we have used the following theorem involving binomial coefficients: Pg (k+1)! k! k=n n!(k−n)! = (n+1)!(k−n)!

Continuing this process, we can see that w(n, g) is just a binomial coefficient w(n, g) =

g! n!(g−n)!

The number of ways that a set of occupation numbers n i can be realized is the product of the ways that each individual energy level can be populated: Q Q ! W = i w(ni , gi ) = i n !(ggi−n )! i

i

DR A

i

Following the same procedure used in deriving the Maxwell-Boltzmann distribution, we wish to find the set of n i for which W is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. We constrain our solution using Lagrange multipliers forming the function: P P f (ni ) = ln(W ) + α(N − ni ) + β(E − ni i )

Again, using Stirling’s approximation for the factorials and taking the derivative with respect to n i, and setting the result to zero and solving for n i yields the Fermi-Dirac population numbers: ni =

gi eα+βi +1

It can be shown thermodynamically that β = 1/kT where k is Boltzmann’s constant and T is the temperature, and that α = -µ/kT where µ is the chemical potential, so that finally: ni =

gi e(i −µ)/kT +1

Note that the above formula is sometimes written: ni =

gi ei /kT /z+1

where z = exp(µ/kT ) is the absolute activity.

Fermi-Dirac statistics

138

See also →Maxwell-Boltzmann statistics →Bose-Einstein statistics Parastatistics

FT

• • •

Source: http://en.wikipedia.org/wiki/Fermi-Dirac_statistics

Principal Authors: Mct mht, PAR, Unc.hbar, Fresheneesz, Tim Starling

Fermi energy

DR A

In physics and →Fermi-Dirac statistics, the Fermi energy (E F) of a system of non-interacting fermions is the smallest possible increase in the ground state energy when exactly one particle is added to the system. It is equivalent to the chemical potential of the system in its ground state at absolute zero. It can also be interpreted as the maximum energy of an individual fermion in this ground state. The Fermi energy is one of the central concepts of condensed matter physics.

Fermi level

The Fermi level is the top of the collection of electron energy levels at absolute zero temperature. Since fermions cannot exist in identical energy states (see the exclusion principle), at absolute zero, electrons pack into the lowest available energy states and build up a "Fermi sea" of electron energy states. 80 In this state (at 0 K), the average energy of an electron is given by: Eav = 35 Ef

where Ef is the fermi energy.

The Fermi momentum is the momentum of fermions at the Fermi surface. The Fermi momentum is given by: p pF = 2me Ef where me is the mass of the electron.

80 http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html

Fermi energy

139 This concept is usually applied in the case of dispersion relations between the energy and momentum that do not depend on the direction. In more general cases, one must consider the Fermi energy.

FT

The Fermi velocity is the average velocity of an electron in an atom at absolute zero. This average velocity corresponds to the average energy given above. The Fermi velocity is defined by: q 2Ef Vf = me where me is the mass of the electron.

Below the Fermi temperature, a substance gradually expresses more and more quantum effects of cooling. The Fermi temperature is defined by: T.f =

Ef k

where k is the Boltzmann constant.

DR A

Quantum mechanics

According to quantum mechanics, fermions – particles with a half-integer spin, usually 1/2, such as electrons – follow the Pauli exclusion principle, which states that multiple particles may not occupy the same quantum state. Consequently, fermions obey →Fermi-Dirac statistics. The ground state of a noninteracting fermion system is constructed by starting with an empty system and adding particles one at a time, consecutively filling up the lowest-energy unoccupied quantum states. When the desired number of particles has been reached, the Fermi energy is the energy of the highest occupied molecular orbital (HOMO). Within conductive materials, this is equivalent to the lowest unoccupied molecular orbital (LUMO), however within other materials there will be a significant gap between the HOMO and LUMO on the order of 2-3 eV. This gap does exist in conductors, however it is infinitesimally small.

Free electron gas

In the free electron gas, the quantum mechanical version of an ideal gas of fermions, the quantum states can be labelled according to their momentum. Something similar can be done for periodic systems, such as electrons moving in the atomic lattice of a metal, using something called the "quasi-momentum" (see Bloch wave). In either case, the Fermi energy states reside on a surface in momentum space known as the Fermi surface. For the free electron gas, the Fermi surface is the surface of a sphere; for periodic systems, it generally has a contorted shape (see Brillouin zones). The volume enclosed by the Fermi surface defines the number of electrons in the system, and the topology is directly Fermi energy

140 related to the transport properties of metals, such as electrical conductivity. The study of the Fermi surface is sometimes called Fermiology. The Fermi surfaces of most metals are well studied both theoretically and experimentally.

FT

The Fermi energy of the free electron gas is related to the chemical potential by the equation    2  4 2 kT π 4 kT µ = EF 1 − π12 E + + · · · 80 E F F

where E F is the Fermi energy, k is the Boltzmann constant and T is temperature. Hence, the chemical potential is approximately equal to the Fermi energy at temperatures of much less than the characteristic Fermi temperature E F/k. The characteristic temperature is on the order of 10 5 K for a metal, hence at room temperature (300 K), the Fermi energy and chemical potential are essentially equivalent. This is significant since it is the chemical potential, not the Fermi energy, which appears in →Fermi-Dirac statistics.

DR A

See also • • • • •

fermi gas semiconductors electrical engineering electronics thermodynamics

References

• •

Table of fermi energies, velocities, and temperatures for various elements 81. a discussion of fermi gases and fermi temperatures 82.

Source: http://en.wikipedia.org/wiki/Fermi_energy

Principal Authors: Fresheneesz, Tim Starling, The Anome, CYD, Tantalate

81 http://hyperphysics.phy-astr.gsu.edu/hbase/tables/fermi.html 82 http://physicsweb.org/articles/world/15/4/7

Fermi energy

141

Fermi gas

FT

A Fermi gas, or Free electron gas, is a collection of non-interacting fermions. It is the quantum mechanical version of an ideal gas, for the case of fermionic particles. Electrons in metals and semiconductors and neutrons in a neutron star can be approximately considered Fermi gases. The energy distribution of the fermions in a Fermi gas in thermal equilibrium is determined by their density, the temperature and the set of available energy states, via →FermiDirac statistics. By the Pauli principle, no quantum state can be occupied by more than one fermion, so the total energy of the Fermi gas at zero temperature is larger than the product of the number of particles and the single-particle ground state energy. For this reason, the pressure of a Fermi gas is nonzero even at zero temperature, in contrast to that of a classical ideal gas. This socalled degeneracy pressure stabilizes a neutron star (a Fermi gas of neutrons) or a White Dwarf star (a Fermi gas of electrons) against the inward pull of gravity.

DR A

It is possible to define a Fermi temperature below which the gas can be considered degenerate. This temperature depends on the mass of the fermions and the energy density of states. For metals, the electron gas’s Fermi temperature is generally many thousands of kelvins, so they can be considered degenerate. The maximum energy of the fermions at zero temperature is called the →Fermi energy. The Fermi energy surface in momentum space is known as the Fermi surface. Since interactions are neglected by definition, the problem of treating the equilibrium properties and dynamical behaviour of a Fermi gas reduces to the study of the behaviour of single independent particles. As such, it is still relatively tractable and forms the starting point for more advanced theories (such as Fermi liquid theory or perturbation theory in the interaction) which take into account interactions to some degree of accuracy.

See also • •

Gas in a box →Bose gas

Source: http://en.wikipedia.org/wiki/Fermi_gas

Principal Authors: PAR, SimonP, Tantalate, Schneelocke, Tom davis

Fermi gas

142

Fermi liquid

FT

A Fermi liquid is a generic term for a quantum mechanical liquid of fermions that arises under certain physical conditions—when the temperature is sufficiently low, and when the system is translationally invariant. The interaction between the particles of the many-body system does not need to be small (see e.g. electrons in a metal). The phenomenological theory of Fermi liquids, which was introduced by the Russian physicist Lev Davidovich Landau in 1956, explains why some of the properties of an interacting fermion system are very similar to those of the →Fermi gas (i.e. non-interacting fermions), and why other properties differ.

DR A

Liquid He-3 is a Fermi liquid at low temperatures (but not low enough to be in its superfluid phase.) He-3 is an isotope of Helium, with 2 protons, 1 neutron and 2 electrons per atom; because there is an odd number of fermions inside the atom, the atom itself is also a fermion. The electrons in a normal (nonsuperconducting) metal also form a Fermi liquid.

The Fermi liquid is qualitatively analogous to the non-interacting Fermi gas, in the following sense: The system’s dynamics and thermodynamics at low excitation energies and temperatures may be described by substituting the noninteracting fermions with so-called quasiparticles, each of which carries the same spin, charge and momentum as the original particles. Physically these may be thought of as being particles whose motion is disturbed by the surrounding particles and which themselves perturb the particles in their vicinity. Each many-particle excited state of the interacting system may be described by listing all occupied momentum states, just as in the non-interacting system. As a consequence, quantities such as the heat capacity of the Fermi liquid behave qualitatively in the same way as in the Fermi gas (e.g. the heat capacity rises linearly with temperature). However, the following differences to the non-interacting Fermi gas arise:

•

The energy of a many-particle state is not simply a sum of the single-particle energies of all occupied states. Instead, the change in energy for a given change δnk in occupation of states k contains terms both linear and quadratic in δnk (for the Fermi gas, it would only be linear, δnk k , where k denotes the single-particle energies). The linear contribution corresponds to renormalized single-particle energies, which involve, e.g., a change in the effective mass of particles. The quadratic terms correspond to a sort of "mean-field" interaction between quasiparticles, which is parameterized by so-called Landau Fermi liquid parameters and determines the behaviour of Fermi liquid

143 density oscillations (and spin-density oscillations) in the Fermi liquid. Still, these mean-field interactions do not lead to a scattering of quasi-particles with a transfer of particles between different momentum states. Specific heat, compressibility, spin-susceptibility and other quantities show the same qualitative behaviour (e.g. dependence on temperature) as in the Fermi gas, but the magnitude is (sometimes strongly) changed.

•

In addition to the mean-field interactions, some weak interactions between quasiparticles remain, which lead to scattering of quasiparticles off each other. Therefore, quasiparticles acquire a finite lifetime. However, at low enough energies above the Fermi surface, this lifetime becomes very long, such that the product of excitation energy (expressed in frequency) and lifetime is much larger than one. In this sense, the quasiparticle energy is still well-defined (in the opposite limit, Heisenberg’s uncertainty relation would prevent an accurate definition of the energy).

•

Green’s function and momentum distribution of quasiparticles behave as for the fermions in the Fermi gas (apart from the broadening of the delta peak in the Green’s function by the finite lifetime).

DR A

FT

•

•

The structure of the "bare" particles (as opposed to quasiparticle) Green’s function is similar to that in the Fermi gas (where, for a given momentum, the Green’s function in frequency space is a delta peak at the respective single-particle energy). The delta peak in the density-of-states is broadened (with a width given by the quasiparticle lifetime). In addition (and in contrast to the quasiparticle Green’s function), its weight (integral over frequency) is suppressed by a quasiparticle weight factor 0 < Z < 1. The remainder of the total weight is in a broad "incoherent background", corresponding to the strong effects of interactions on the fermions at short time-scales.

•

The distribution of particles (as opposed to quasiparticles) over momentum states at zero temperature still shows a discontinuous jump at the Fermi surface (as in the Fermi gas), but it does not drop from 1 to 0: the step is only of size Z.

•

In a metal the resistance at low temperatures is dominated by electronelectron scattering in combination with Umklapp scattering. For a Fermi liquid, the resistance from this mechanism varies as T 2 , which is often taken as an experimental check for Fermi liquid behaviour (in addition to the linear temperature-dependence of the specific heat), although it only arises in combination with the lattice. Fermi liquid

144

Source: http://en.wikipedia.org/wiki/Fermi_liquid

FT

Principal Authors: CYD, Dschwen, RedWolf, Itai, Jofox

Fermi’s golden rule

In quantum physics, Fermi’s golden rule is a way to calculate the transition rate between two eigenstates of a quantum system using time-dependent perturbation theory, which means it’s an approximation. We consider the system to begin in an eigenstate |ii of a given Hamiltonian H0 . We consider the effect of a time-independent perturbing Hamiltonian H 0 .

The one-to-many transition probability per unit of time from the state |ii to a set of states |f i is given, to first order in the perturbation, by: 2π ~

|hf |H 0 |ii|2 ρ

DR A

Ti→f =

where ρ is the density of final states, and < f | H’ | i > is the matrix element (in bra-ket notation) of the perturbation, H’, between the final and initial states. Fermi’s golden rule is valid when H 0 is time-independent, |ii is an eigenstate of the unperturbed Hamiltonian, the states |f i form a continuum, and the initial state has not been significantly depleted (eg, by scattering into the final states). The most common way to derive the equation is to start with time-dependent perturbation theory and to take the limit for absorption under the assumption that the time of the measurement is much larger than the time needed for the transition. Although named after Fermi, most of the work leading to the Golden Rule was done by Dirac.

External links • •

More information on Fermi’s golden rule 83 Derivation using time-dependent perturbation theory 84

Source: http://en.wikipedia.org/wiki/Fermi%27s_golden_rule

83 http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/fermi.html 84 http://www.ph.utexas.edu/~schwitte/PHY362L/QMnote.pdf

Fermi’s golden rule

145

Field emission

FT

Principal Authors: Donvinzk, Laurascudder, Jheald, MikeMorley, Alfredo.correa

Also known as Fowler-Nordheim tunneling, field emission is a form of quantum tunneling in which electrons pass through a barrier in the presence of a high electric field. This phenomenon is highly dependent on both the properties of the material and the shape of the particular cathode, so that higher aspect ratios produce higher field emission currents. The current density produced by a given electric field is governed by the Fowler-Nordheim equation. Applications of field emission include its use as an electron source in flash memory, electron microscopy, MEMS systems, and field emission displays.

DR A

In the field of vacuum electronics, field emission is seen as an alternative to thermionic emission, with advantages such as dramatically higher efficiency, less scatter of emitted electrons, faster turn-on times, compactness, and, in many cases, redundancy. Some disadvantages include lower current per emission source and, often, lower overall current density. Field emission limits the maximum operating voltage for high voltage vacuum devices such as vacuum capacitors and vacuum switches. Vacuum tubes based on thermionic emission require several minutes to warm up before they can be used; by contrast, the function of field emission devices is effectively instantaneous, allowing switching times of many megahertz. The ability to modulate the electron source, rather than modifying a stream of electrons from a constant source (i.e., by velocity modulation), has allowed many vacuum devices to be greatly simplified. For instance, the Klystrode functions much like the two-chamber Klystron, without the need for a first chamber.

See also •

Cold cathode

External links •

Field emission - Fowler-Nordheim tunneling, Principles of Semiconductor Devices, Bart Van Zeghbroeck, 1997 85

85 http://ece-www.colorado.edu/~bart/book/msfield.htm

Field emission

146 Source: http://en.wikipedia.org/wiki/Field_emission

FT

Principal Authors: Ajdecon, Bert Hickman, Pjacobi, Heron, Adoarns

Finite potential well

DR A

The finite potential well (also known as the finite square well) is a simple problem from quantum mechanics. It is an extension of the infinite potential well, in which a particle is confined to a box, but one which has finite - not infinite - potential walls. This means unlike the infinite potential well, there is a probability associated with the particle being found outside of the box. The quantum mechanical interpretation is unlike the classical interpretation, where if the total energy of the particle is less than potential energy barrier of the walls it cannot be found outside the box. In the quantum interpretation, there is a probability of the particle being outside the box even when the energy of the particle is less than the potential energy barrier of the walls.

The particle in a 1-dimensional box

For the 1-dimensional case on the x-axis, the time-independent Schrödinger equation can be written as: ~2 d2 ψ + V (x)ψ = Eψ − 2m 2 dx

(1) where h ~ = 2π h is Planck’s constant m is the mass of the particle ψ is the (complex valued) wavefunction that we want to find V (x) is a function describing the potential at each point x and E is the energy, a real number.

For the case of the particle in a 1-dimensional box of length L, the potential is zero inside the box, but rises abruptly to a value Γ at x = -L/2 and x = L/2. The wavefunction is considered to be made up of different wavefuctions at different ranges of x, depending on whether x is inside or outside of the box. Therefore  the wavefunction is defined such that:

 ψ1 , if x < −L/2 (the region outside the box) ψ = ψ2 , if − L/2 < x < L/2 (the region inside the box)   ψ3 if x > L/2 (the region outside the box)

Finite potential well

147

Inside the box For the region inside the box V (x) = 0 and Equation 1 reduces to: 2

2

Letting

√

k=

FT

~ d ψ2 − 2m = Eψ2 dx2

2mE ~

the equation becomes d2 ψ2 dx2

= −k 2 ψ2

This is a well studied differential equation and eigenvalue problem with a general solution of: ψ2 = A sin(kx) + B cos(kx) Hence: k 2 ~2 2m

DR A E=

Here, A and B can be any complex numbers, and k can be any real number (k must be real because E is real).

Outside the box

For the region outside of the box, V (x) = Γ and Equation 1 becomes: 2

2

~ d ψ1 − 2m = (E − Γ)ψ1 dx2

There are two possible families of solutions, depending on whether E is less than Γ (the particle is bound in the potential) or E is greater than Γ (the particle is free). For a free particle, E > Γ, and letting √ 2m(E−Γ) κ= ~ produces

d2 ψ1 dx2

= −κ2 ψ1

with the same solution form as the inside-well case: ψ1 = C sin(κx) + D cos(κx)

Finite potential well

148 This analysis will first focus on the bound state, where Γ > E. Letting √ 2m(Γ−E) α= ~

d2 ψ1 dx2

FT

produces = α 2 ψ1

where the general solution is exponential: ψ1 = F e−αx + Geαx

Similarly, for the other region outside the box: ψ3 = He−αx + Ieαx

DR A

Now in order find the specific solution for the problem at hand, we must specify the appropriate boundary conditions and find the values for A , B , F , G , H and I that satisfy those conditions.

Finding wavefunctions for the bound state

Solutions to the Schrödinger equation must be continuous, and continuously differentiable. These requirements are boundary conditions on the differential equations previously derived. In this case, the finite potential well is symmetrical, so symmetry can be exploited to reduce the necessary calculations. Summarizing the previous section: 

 ψ1 , if x < −L/2 (the region outside the box) ψ = ψ2 , if − L/2 < x < L/2 (the region inside the box)   ψ3 if x > L/2 (the region outside the box)

where we found ψ1 , ψ2 and ψ3 to be: ψ1 = F e−αx + Geαx

ψ2 = A sin(kx) + B cos(kx)

ψ3 = He−αx + Ieαx

Finite potential well

149 We see that as x goes to −∞, the F term goes to infinity. Likewise, as x goes to +∞, the I term goes to infinity. As the wave function must be finite for all x, this means we must set F = I = 0, and we have:

FT

ψ1 = Geαx and ψ3 = He−αx

Next, we know that the overall ψ function must be continuous and differentiable. In other words the values of the functions and their derivatives must match up at the dividing points:

See also • →Potential well • Infinite potential well • →Quantum tunnelling

References

Griffiths, David J. (2005). Introduction to Quantum Mechanics, 2 nd ed., Prentice Hall. ISBN 0131118927.

DR A

•

Source: http://en.wikipedia.org/wiki/Finite_potential_well

Flux quantization

Flux quantization is a quantum phenomenon in which the magnetic field is quantized in the unit of h/2e, also known variously as flux quanta, fluxoids, vortices or fluxons. Flux quantization occurs in Type II superconductors subjected to a magnetic field. Below a critical field H c1, all magnetic flux is expulsed according to the Meissner effect and perfect diamagnetism is observed, exactly as in a Type I superconductor. Up to a second critical field value, H c2, flux penetrates in discrete units while the bulk of the material remains superconducting. Both critical fields are temperature dependent, and tabulated values are the zerotemperature extrapolation unless otherwise noted.

Flux quantization

150

• •

Flux pinning Magnetic flux quantum

FT

See also

Source: http://en.wikipedia.org/wiki/Flux_quantization

Principal Authors: Charles Matthews, Stevelihn, Eldereft, Art LaPella, Zowie

Fock matrix

In quantum mechanics, the Fock matrix is a matrix approximating the singleelectron energy operator of a given quantum system in a given set of basis vectors.

DR A

It is most often formed in computational chemistry when attempting to solve the Roothaan equations for an atomic or molecular system. The Fock matrix is actually an approximation to the true Hamiltonian operator of the quantum system. It includes the effects of electron-electron repulsion only in an average way. Importantly, because the Fock operator is a one-electron operator, it does not include the electron correlation energy.

The Fock matrix is defined by the Fock operator. For the restricted case which assumes closed-shell orbitals and single-determinantal wavefunctions, the Fock operator for the first electron is given by: ˆ core (1) + Pn [2Jˆj (1) − K ˆ j (1)] Fˆ (1) = H j=1

where: Fˆ (i)

is the Fock operator for the i -th electron in the system, ˆ core (i) H is the core Hamiltonian for the i -th electron, n

is the total number of orbitals in the system (equal to half the number of electrons), Fock matrix

151 Jˆj (i)

FT

is the Coulomb operator, defining the repulsive force between the j -th and i -th electrons in the system, ˆ j (i) K is the exchange operator, defining the effect of exchanging the two electrons.

See also • •

Roothaan equations Hartree-Fock

Source: http://en.wikipedia.org/wiki/Fock_matrix

DR A

Principal Authors: Edsanville, Agentsoo, Charles Matthews, Remuel, Njerseyguy

Fock space

The Fock space is an algebraic system (→Hilbert space) used in quantum mechanics to describe quantum states with a variable or unknown number of particles. It is named for V. A. Fock. Technically, the Fock space is the Hilbert space made from the direct sum of tensor products of single-particle Hilbert spaces: L ⊗n Fν (H) = ∞ n=0 Sν H

where S ν is the operator which symmetrizes or antisymmetrizes the space, depending on whether the Hilbert space describes particles obeying bosonic (ν = +) or fermionic (ν = -) statistics respectively. H is the single particle Hilbert space. It describes the quantum states for a single particle, and to describe the quantum states of systems with n particles, or superpositions of such states, one must use a larger Hilbert space, the Fock space, which contains states for unlimited and variable number of particles. →Fock states are the natural basis of this space. (See also the →Slater determinant.)

Fock space

152

Example An example of a state of the Fock space is

FT

|Ψiν = |φ1 , φ2 , · · · , φn iν describing n particles, one of which has wavefunction φ 1, another φ 2 and so on up to the n th particle, where each φ i is any wavefunction from the single particle Hilbert space H. When we speak of one particle in state φ i it must be borne in mind that in quantum mechanics identical particles are indistinguishable, and in a same Fock space all particles are identical (to describe many species of particles, make the tensor products of as many different Fock spaces). It is one of the most powerful features of this formalism that states are intrinsically properly symmetrized. So that for instance, if the above state |Ψ> - is fermionic, it will be 0 if two (or more) of the φ i are equal, because by the Pauli exclusion principle no two (or more) fermions can be in the same quantum state. Also, the states are properly normalized, by construction.

DR A

A useful and convenient basis for this space is the occupancy number basis. If |ψ i> is a basis of H, then we can agree to denote the state with n 0 particles in state |ψ 0>, n 1 particles in state |ψ 1>, ..., n k particles in state |ψ k> by |n0 , n1 , · · · , nk iν ,

with each n i taking the value 0 or 1 for fermionic particles and 0,1,2,... for bosonic particles. Such a state is called a →Fock state. Since |ψ i> are understood as the steady states of the free field, i.e., a definite number of particles, a Fock state describes an assembly of non-interacting particles in definite numbers. The most general pure state is the linear superposition of Fock states.

Two operators of paramount importance are the creation and annihilation operators, which upon acting on a Fock state respectively remove and add a particle, in the ascribed quantum state. They are denoted a† (φi ) and a(φi ) respectively, with φi referring to the quantum state |φi i in which the particle is removed or added. It is often convenient to work with states of the basis of H so that these operators remove and add exactly one particle in the given state. These operators also serve as a basis for more general operators acting on the Fock space, for instance the number operator giving the number of particles in a specific state |φi i is a† (φi )a(φi ).

Source: http://en.wikipedia.org/wiki/Fock_space

Principal Authors: Phys, Stupidmoron, Laussy, Maury Markowitz, Charles Matthews

Fock space

153

Fock state

FT

A Fock state, in quantum mechanics, is any state of the →Fock space with a well-defined number of particles in each state. The name is for V. A. Fock.

If we limit to a single mode for simplicity (doing so we formally describe a mere harmonic oscillator), a Fock state is of the type |n> with n an integer value. This means that there are n quanta of excitation in the mode. |0> corresponds to the ground state (no excitation). It is different from 0 which is the null vector. Fock states form the most convenient basis of the Fock space. They are defined to obey the following relations in the bosonic algebra: √ a† |ni = n + 1|n + 1i a|ni =

n|n − 1i

√1 (a† )n |0i n!

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|ni =

√

with a (resp. a †) the annihilation (resp. creation) bose operator. Similar relations hold for fermionic algebra. This allows to check that =n and Var(a †a)=0, i.e., that measuring the number of particles a †a in a Fock state returns always a definite value with no fluctuation.

Source: http://en.wikipedia.org/wiki/Fock_state

Principal Authors: Digitalme, Phys, Unyoyega, Charles Matthews, Laussy

Franck-Hertz experiment

In physics, the Franck-Hertz experiment was an early physics experiment that provided support for the Bohr model of the atom, a precursor to quantum mechanics. In 1914, physicists James Franck and Gustav Ludwig Hertz sought to experimentally probe the energy levels of the atom. The now-famous FranckHertz experiment elegantly supported Niels Bohr’s model of the atom, with electrons orbiting the nucleus with specific, discrete energies. Franck and Hertz were awarded the Nobel Prize in Physics in 1925 for this work. Franck-Hertz experiment

154

The experiment

•

• •

At low voltages–up to 7 volts when the tube contained mercury vapour– the current through the tube increased steadily with increasing potential difference. The higher voltage increased the electric field in the tube and electrons were drawn more forcefully towards and through the accelerating grid. At 7 volts the current drops sharply, almost back to zero. The current increases steadily once again if the voltage is increased further, until 11.9 volts is reached (exactly 7+4.9 volts). At 11.9 volts a similar sharp drop is observed. This series of dips in current at 4.9 volt increments will visibly continue to potentials of at least 100 volts.

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• •

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The classic experiment involved a tube containing low pressure gas and bounded at each end by an electrode, and containing a mesh accelerating grid near the ground electrode. (This ’ground’ was actually held very slightly negative, so that electrons had to have a small amount of kinetic energy to reach it.) Instruments were fitted to measure the current passing between the electrodes, and to adjust the potential difference (the voltage) between the cathode (negative electrode) and the accelerating grid.

Franck and Hertz were able to explain their experiment in terms of elastic and inelastic collisions. At low potentials, electrons acquired only a modest amount of kinetic energy. When they encountered mercury atoms in the tube, they participated in elastic collisions. The total amount of kinetic energy in the system remained the same. (Since electrons are significantly less massive than mercury atoms, this meant that the electrons held on to the vast majority of that energy, too.) Higher potentials served to drive more electrons to the ground and increase the observed current. The lowest energy electronic excitation a mercury atom can participate in requires 4.9 electron volts (eV). When the accelerating potential reached 4.9 volts, each electron possessed exactly that amount of energy when it reached the anode grid. Consequently, a collision between a mercury atom and an electron at that point could be inelastic. Its kinetic energy could be converted into potential energy, and used to excite the mercury atom. With the loss of all of its kinetic energy, the electron can no longer overcome the slight negative potential at the ground electrode, and the measured current drops sharply. As the voltage is increased, electrons will participate in one inelastic collision, lose their 4.9 eV, but then continue to be accelerated. In this manner, the current rises again after the accelerating potential exceeds 4.9 V. At 9.8 V, the Franck-Hertz experiment

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situation changes again. There, each electron now has just enough energy to participate in two inelastic collisions, excite two mercury atoms, and then be left with no kinetic energy. Once again, the observed current drops. At intervals of 4.9 volts this process will repeat; each time the electrons will undergo one additional inelastic collision. A similar pattern is observed with neon gas, but at intervals of approximately 19 volts. The process is identical, just with a much different threshold. One additional difference is that a glow will appear near the accelerating grid at 19 volts–one of the transitions of relaxing neon atoms emits red-orange light. This glow will move closer to the cathode with increasing accelerating potential, to whatever point in the tube the electrons acquire the 19 eV required to excite a neon atom. At 38 volts two distinct glows will be visible: one between the cathode and grid, and one right at the accelerating grid. Higher potentials will result in additional glowing regions in the tube, spaced at 19 volt intervals.

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The Franck-Hertz experiment confirmed Bohr’s quantized model of the atom by demonstrating that atoms could indeed only absorb (and be excited by) specific amounts of energy (quanta).

References

• •

The Franck-Hertz Experiment at Hyperphysics 86 Up-to-date literature on the Franck-Hertz Experiment 87

Source: http://en.wikipedia.org/wiki/Franck-Hertz_experiment

Principal Authors: Icairns, TenOfAllTrades, Mac Davis, Linas, Pnicolet

Free particle

In physics, a free particle is a particle that, in some sense, is not bound. In the classical case, this is represented with the particle not being influenced by any external force.

86 http://hyperphysics.phy-astr.gsu.edu/hbase/FrHz.html 87 http://users.skynet.be/P.Nicoletopoulos/references.html

Free particle

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Classical Free Particle

p = mv and the energy by E = 12 mv 2

FT

The classical free particle is characterized simply by a fixed velocity. The momentum is given by

where m is the mass of the particle and v is the vector velocity of the particle.

Non-Relativistic Quantum Free Particle The →Schrödinger equation for a free particle is: 2

~ ∂ − 2m ∇2 ψ(r, t) = i~ ∂t ψ(r, t)

The solution for a particular momentum is given by a plane wave:

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ψ(r, t) = ei(k·r−ωt)

with the constraint ~2 k 2 2m

= ~ω

where r is the position vector, t is time, k is the wave vector, and ω is the angular frequency. Since the integral of ψψ * over all space must be unity, there will be a problem normalizing this momentum eigenfunction. This will not be a problem for a general free particle which is somewhat localized in momentum and position. (See particle in a box for a further discussion.) The expectation value of the momentum p is hpi = hψ| − i~∇|ψi = ~k

The expectation value of the energy E is ∂ hEi = hψ|i~ ∂t |ψi = ~ω

Solving for k and ω and substituting into the constraint equation yields the familiar relationship between energy and momentum for non-relativistic massive particles hEi =

hpi2 2m

Free particle

157 where p=|p|. The group velocity of the wave is defined as vg = dω/dk = dE/dp = v

FT

where v is the classical velocity of the particle. The phase velocity of the wave is defined as vp = ω/k = E/p = p/2m = v/2

A general free particle need not have a specific momentum or energy. In this case, the free particle wavefunction may be represented by a superposition of free particle momentum eigenfunctions: R ψ(r, t) = A(k)ei(k·r−ωt) dk where the integral is over all k-space.

Relativistic free particle

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There are a number of equations describing relativistic particles. For a description of the free particle solutions, see the individual articles. •

The →Klein-Gordon equation describes charge-neutral, spinless, relativistic quantum particles

•

The Dirac equation describes the relativistic electron (charged, spin 1/2)

Source: http://en.wikipedia.org/wiki/Free_particle

Principal Authors: PAR, Linas, Lupin, Sverdrup, Karol Langner

Functional integration

This article’s topic is not that of functional integration (neurobiology) or functional integration (sociology).

Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations and in Feynman’s approach to the quantum mechanics of particles and fields. Functional integration

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In an ordinary integral there is a function to be integrated—the integrand—and a region of space over which to integrate the function—the domain of integration. The process of integration consists of adding the values of the integrand at each point of the domain of integration. Making this procedure rigorous requires a limiting procedure, where the domain of integration is divided into smaller and smaller regions. For each small region the value of the integrand cannot vary much so it may be replaced by a single value. In a functional integral the domain of integration is a space of functions. For each function the integrand returns a value to add up. Making this procedure rigorous poses challenges that are the topic of research in the beginning of the 21st century. Functional integration was introduced by Wiener in 1921 in his studies of Brownian motion. He developed a rigorous method —now known as the Wiener measure— for assigning a probability to a particle’s random path. Feynman developed another functional integral, the path integral, useful for computing the quantum properties of systems. In Feynman’s path integral, the classical notion of a unique trajectory for a particle is replaced by an infinite sum of classical paths, each weighed differently according to its classical properties.

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Functional integration is central to quantization techniques in theoretical physics. The algebraic properties of functional integrals are used to develop series used to calculate properties in quantum electrodynamics and the standard model.

The problem of functional integration

Integration of functions is a summation. If the domain of integration is the square [0, 1] × [0, 1], the integral is computed by breaking the region into small rectangles. Each rectangle serves as the base of a prism whose height is any value of the function within the rectangle. The integral is the sum of the volumes (base × height) of all the prisms. If the rectangles are small enough and the function smooth, the process converges. A functional is a function that associates a number to a function. This is in distinction to common functions that associate numbers to other numbers. Examples of functionals include the functional that is one for any function, or the functional the returns the integral of the function over a domain. By analogy with integration of functions, functional integration is a summation procedure where the domain of integration is a space of functions and the functional integral an addition of cylinders (just like the prisms in ordinary integration) with the functional as the height and some amount (or measure) of function space as the base. The “area” of the functions can be represented by

Functional integration

159

R

I Dω F [ω] . (1)

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Dω , the functional by F [ω] (square brackets are often used to distinguish functionals from common functions), the space of functions by I and the functional integral by

The result of integrating a functional is to be a number. Developing a definition to equation (1) that has properties similar to ordinary integration is the problem of the definition of functional integration. There is no general theory to make sense of this formal expression as there is for the conventional integration. In functional integration there are different spaces to consider: •

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The functions ω take values from a ν-dimensional space called space-time. This is how many dimensions are used to specify a point in the domain of ω. Space-time in often assumed to be a subset of R ν , the Euclidean space with ν dimensions. In applications of functional integration, the functions ω represent particle paths (in which case ν = 1) or a physical field such as the vector potential (in which case ν = 4). • The space for the range of the functions ω also varies depending on applications. This space is locally a subset of R κ. In applications it can be R 3, as in the quantum mechanics; or a more complicated space, such as a tangent bundle as in the case of quantum chromodynamics. • The domain of integration is a function space, and likely to be infinite dimensional. A definition of functional integration, by analogy with common integration, is expected to satisfy certain properties. Functional integration should be itself R R R a linear functional, such that Dω (F +α G ) = Dω F + α Dω G. Volumes in functional space should be invariant under translation. A ball in functional space centered around a function f or that same function plus a constant should result in the same value for the functional integral. Also, if the functional space happens to be finite dimensional, then the functional integral should be related to the ordinary integration. These conditions are impossible to satisfy for functional integrals.

Attempting to directly generalize the notion of volume in functional space has not led to a useful theory of functional integration. Discretization of the functional integral in equation (1) could be an approach towards its definition. For the case of one-dimensional paths (ν=1 and κ=1), the functional integration is replaced by an n-dimensional integral and the functional is computed from the value of the path ω and n points. The functional integral would then be the value of the n-dimensional integral in the limit of n going to infinity: Functional integration

160

R

Dω F (ω) = lim

n→∞

R

dω1

R

R dω2 . . . dωn F (ω1 , . . . , ωn ) (2)

FT

The “size” of a function space can be computed from this expression by using the simple functional F [ω]=1. Choosing a function space where each ω i varies over limited range of length W, the n-dimensional integral is equal to W n. This will diverge to infinity or converge to zero in the limit. Building a theory of integration when the value of the integral can only be zero or infinity is not very interesting.

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Most of the cylinders that contribute to the functional integral (2) correspond to discontinuous functions. In Brownian motion or in the path integral formulation of quantum mechanics, the paths are continuous. Both applications did not generalize the notion of volume to functional spaces, as in equation (1), but rather generalized the notion of a Gaussian integral. In applications, the functional being integrated is related to an action functional S arising from classical mechanics. Action functionals can be written as the sum of two terms, S 0 + S i , with S 0 involving the derivative of the function squared. For example, for the case of one-dimensional paths R 2 S0 [ω] = dτ ω˙2 .

Smooth paths lead to small values of the functional, and large variations of the path (as if almost discontinuous) lead to large values of the functional. Introducing a term exp(-S 0) into the functional integral should dampen the effects of discontinuous paths. This leads to Gaussian functional integrals.

Gaussian integration

Instead of generalizing the notion of volume to infinite dimensions, the Gaussian integral can be generalized. If M is a positive n×n symmetric matrix, and x and J are n-dimensional vectors, the basic Gaussian integral can be used to show that R 1 1 1 Dx e− 2 x·M ·x+x·J = √ 1 e 2 J· M ·J | det M |

The integration variables have been abbreviated with Dx=(2π ) -n/2dx 1...dx n. The determinant and the matrix operations in the result of the manydimensional Gaussian can be interpreted in terms of infinite-dimensional objects. This result then can be used as the basis for the definition of a functional integral. The action functional S of a path x(t) can be approximated by a discretization of the time domain of the path into n+1 pieces of length a. At these time values the path assumes the values x 0, ..., x n+1. The two end points

Functional integration

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x 0 and x n+1 remain fixed and are not part of the n-fold integration. The discretized action S can be put in the form x∆x + xJ. The matrix ∆ is similar to a discretized Laplace operator, the vector J will have only two non-zero entries and the vector x runs from x 1 to x n . The determinant of ∆ and the inverse matrix ∆ -1 can be evaluated as a function of the number of discretization steps n. The determinant is n+1. The inverse matrix is a matrix with entries of order one divided by the determinant. Using a limit of iterated Gaussians it is possible to define a functional integral for paths (ν=1). With the notation O (1) to indicate a matrix with entries of order one, the iterated Gaussian integral  n+1 1/2 1 R 1 O(1) Dx e− 2 x·M ·x+x·J = an+1 e 2a(n+1) will converge if the entire expression is divided by a (n+1)/2.

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In the case of a spacetime of more than one dimension, generalizations of the Gaussian integral will not converge. This is at the root of many of the difficulties in quantum field theory, as the Gaussian integral corresponds to the quantum field theory with no interactions. The difficulty in generalizing the Gaussian integral is that the growth of the determinant in the result cannot be removed by a rescaling. (The determinant of the operator in ν spacetime dimensions grows as 2nν.) In physics it is common to refer to the divergence due to the determinant as an infrared divergence and the divergence due to the limit of the discretization a going to zero as an ultraviolet divergence. Due to the behavior of Gaussian integrals, in one spacetime dimension (ν=1) it becomes possible to define a functional integral of the form R Dxe−S[x] .

In the applications of functional integration to quantum mechanics the action S[x] is pure imaginary and the exponential is an oscillating function. This leads to further difficulties relating to conditional convergence. In spacetime dimensions greater than one (ν ≥ 2), it is not known how to define a functional integral without resorting to regularization procedures that do away with many of the properties of integration.

Approaches to path integrals

Functional integrals where the space of integration are paths (ν = 1) can be defined in many different ways. The definitions fall in two different classes: the constructions derived from Wiener’s theory yield a integral based on a measure; whereas the constructions following Feynman’s path integral do not. Even Functional integration

162 within these two broad divisions, the integrals are not identical, that is, they are defined for different classes of functions.

FT

The Wiener integral In the Wiener integral a probability is assigned to a class of Brownian motion paths. The class consists of the paths w that are known to go through a small region of space at a given time. The passage through different regions of space is assumed independent of each other and the distance between any two points of the Brownian path is assumed to be Gaussian distributed with a variance that depends on the time t and on a diffusion constant D :   ||w(s+t)−w(s)||2 Prob(w(s + t), t|w(s), s) = √ 1 exp − 2Dt 2πDt

The probability for the class of paths can be found by multiplying the probabilities of starting in one region and then being at the next. The Wiener measure can be developed by considering the limit of many small regions. Ito and Stratanovich calculus

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•

The Feynman integral

• • • •

Trotter formula The Kac idea of Wick rotations. Using x-dot-dot-squared or i S[x] + x-dot-squared. The Cartier DeWitt-Morette relies on integrators rather than measures

See also • •

Feynman path integral Partition function

Reference and external links

Source: http://en.wikipedia.org/wiki/Functional_integration

Principal Authors: Michael Hardy, Miguel, XaosBits, Phys, The Anome

Functional integration

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Geiger-Marsden experiment

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The Geiger-Marsden experiment (also called the Gold foil experiment or the Rutherford experiment) was an experiment done by Hans Geiger and Ernest Marsden in 1909, under the direction of Ernest Rutherford at the Physical Laboratories of the University of Manchester which led to the downfall of the plum pudding model of the atom. They measured the deflection of alpha particles directed normally onto a sheet of very thin gold foil. Under the prevailing plum pudding model, the alpha particles should all have been deflected by, at most, a few degrees. However they observed that a very small percentage of particles were deflected through angles much larger than 90 degrees. From this observation Rutherford concluded that the atom contained a very small positive charge which could repel the alpha particles if they came close enough, subsequently developed into the →Bohr model.

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Methodology

Geiger and Marsden bombarded a number of different metal foils with alpha particles generated from a tube of radium bromide gas. A low power microscope was used to count the scattering of these particles, a procedure requiring many hours in a darkened room watching for tiny flashes of light as the scattered particles struck a zinc sulfide scintillant screen. A variety of different foils were used such as aluminium, iron, gold and lead along with different thicknesses of gold foil made by packing several pieces of very thin foil together. Given the very high mass and momentum of an alpha particle, the expectation was that the particles would pass through having been deflected by a tiny angle at most, with the number of particles penetrating falling off as the thickness of foil (and the atomic weight of its material) was increased; the remainder being absorbed. However they were astonished to find that although this was generally true, around 1 in 8000 particles were reflected through more than 90 degrees even with a single sheet of extremely thin, 6x10 -8 metre (or about 200 atoms) thick, gold foil, an observation completely at odds with the predictions of the plum pudding model.

Geiger-Marsden experiment

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Conclusions

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The result was completely unpredicted, prompting Rutherford to later comment "It was almost as incredible as if you fired a fifteen-inch shell at a piece of tissue paper and it came back and hit you".

Early in 1911 Rutherford published a revised model of the atom, known as the Rutherford atom. The observations indicated that a model of the atom with a diffuse positive charge was incorrect and that it was instead concentrated. He concluded that the atom is mostly empty space, with most of the atom’s mass concentrated in a tiny center, the nucleus, and electrons being held in orbit around it by electrostatic attraction. The nucleus was around 10 -15 meters in diameter, in the centre of a 10 -10 metre diameter atom. Those alpha particles that had come into proximity with the nucleus had been strongly deflected whereas the majority had passed at a relatively great distance to it.

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Rutherford’s model was developed by Niels Bohr into the →Bohr model proposed in 1913. The Rutherford atom had a number of problems, in particular electrons should radiate electromagnetic energy and rapidly spiral into the nucleus.

See also •

List of famous experiments

References •

•

• • •

Geiger H. & Marsden E. (1909). "On a Diffuse Reflection of the αParticles" 88. Proceedings of the Royal Society, Series A 82: 495–500. Rutherford E. (1911). "The Scattering of α and β Particles by Matter and the Structure of the Atom" 89. Philosophical Magazine, Series 6 21: 669–688. JPEG images of Rutherford’s 1911 paper 90 Description of the experiment, from the New Mexico Institute of Mining and Technology 91 Short biography of Ernest Rutherford 92

Source: http://en.wikipedia.org/wiki/Geiger-Marsden_experiment

88 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/GM-1909.html 89 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Rutherford-1911/Rutherford-1911.html 90 http://www.math.ubc.ca/~cass/rutherford/rutherford.html 91 http://www.physics.nmt.edu/~raymond/classes/ph13xbook/node193.html 92 http://www.phy.hr/~dpaar/fizicari/xrutherf.html

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Gibbs paradox

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Principal Authors: Jll, Fastfission, Sjakkalle, Raul654, Malcolm Farmer

This article is about Gibbs paradox on the extensivity of entropy; for Gibbs mixing paradox, see mixing paradox.

In statistical mechanics, a simple derivation of the entropy of an ideal gas, based on the Boltzmann distribution yields an expression for the entropy which is not an extensive variable as it must be, leading to an apparent paradox known as the Gibbs paradox. The difficulty is resolved by requiring the particles be indistinguishable which results in "correct Boltzmann counting". The resulting equation for the entropy of a classical ideal gas is extensive, and is known as the Sackur-Tetrode equation.

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If you have a fixed volume of an ideal gas, the measurement of the total entropy within the volume should not change if you were to divide the volume into two (or more) equal partitions, then remove the partitions. However, if you calculate the total entropy by measuring (and adding up) the position and momentum of each individual particle inside the volume (and keep track of each particle separately), then the entropy you calculate will change depending on whether you add and remove partitions inside the fixed volume. This discrepancy is called the Gibb’s Paradox. The paradox is resolved by concluding that every particle is indistinguishable from every other particle in the volume, thus you cannot measure entropy by measuring particles as if each were individually identifiable.

Calculating the Gibbs paradox

If we have an ideal gas of energy U, volume V and with N particles, then we can represent the state of the gas by specifying the 3D momentum vector and the 3D position vector for each of the N particles. This can be thought of as specifying the coordinates of a point in a 6N-dimensional phase space, where each of the axes corresponds to one of the momentum or position coordinates of one of the particles. The set of all possible points that the gas could ever occupy in phase space is specified by the constraint that the gas will have a particular energy: 1 PN P3 2 U = 2m i=1 j=1 pij

Gibbs paradox

166 and be contained inside of the volume V (let’s say V is a box of side X so that X 3=V ):

FT

0 ≤ xij ≤ X where [p i1, p i2, p i3] and [x i1, x i2, x i3] are the vector momentum and position of particle i. The first constraint defines the surface of a 3N-dimensional hypersphere of radius (2mU ) 1/2 and the second is a 3N-dimensional hypercube of volume V N. These combine to form a 6N-dimensional "hypercylinder". Just as the area of the wall of a cylinder is the circumference of the base times the height, so the area φ of the wall of this hypercylinder is:  3N  3N −1 2π 2 (2mU ) 2 φ(U, V, N ) = V N (1) Γ(3N/2)

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The entropy is proportional to the logarithm of the number of states that the gas could have while satisfying these constraints. Another way of stating Heisenberg’s uncertainty principle is to say that we cannot specify a volume in phase space smaller than h 3N where h is Planck’s constant. The above "area" must really be a shell of thickness h, so we therefore write the entropy as:

S = k log(φh/h3N )

where k is the constant of proportionality, Boltzmann’s constant. Using Stirling’s approximation for the Gamma function, and keeping only terms of order N the entropy becomes:   3    U 2 S(=?) kN log V N + 32 kN 53 + log 4πm 3h2 This quantity is not extensive as can be seen by considering two identical volumes with the same particle number and the same energy. Suppose the two volumes are separated by a barrier in the beginning. Removing or reinserting the wall is reversible, but the entropy difference after removing the barrier is δS = k [2N log(2V ) − N log V − N log V ] = 2kN log 2 > 0

which is in contradiction to thermodynamics. This is the Gibbs paradox. It was resolved by J.W. Gibbs himself, by postulating that the gas particles are in fact indistinguishable. This means that all states that differ only by a permutation of particles should be considered as the same point. For example, if we have a 2-particle gas and we specify AB as a state of the gas where the first particle (A ) has momentum p 1 and the second particle (B ) has momentum p 2, then this point as well as the BA point where the B particle has momentum p 1 and the A particle has momentum p 2 should be counted as the same point. It can Gibbs paradox

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be seen that for an N -particle gas, there are N! points which are identical in this sense, and so to calculate the volume of phase space occupied by the gas we must divide Equation 1 by N!. This will give for the entropy:     3    V U 2 S = kN log N + 32 kN 53 + log 4πm N 3h2 which can be easily shown to be extensive. This is the Sackur-Tetrode equation.

Source: http://en.wikipedia.org/wiki/Gibbs_paradox

Principal Authors: PAR, Marco Krohn, Charles Matthews, Kjkolb, Linas

Greenberger-Horne-Zeilinger state

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In physics, in the area of quantum information theory, a Greenberger-HorneZeilinger state is a certain type of entangled quantum state.

Definition

The GHZ state is an entangled quantum state in an M dimensions: |GHZi =

|0i⊗M√ +|1i⊗M 2

.

Most notably the 3-qubit GHZ state is: |GHZi =

|000i+|111i √ . 2

Properties

Apparently there is no standard measure of multi-partite entanglement, but many measures define the GHZ to be maximally entangled. Another important property of the GHZ state is that when we trace over one of the three systems T r3 ((|000i + |111i)(h000| + h111|)) = |00ih00| + |11ih11|

which is a mixed state.

On the other hand, if we were to measure any of subsystems we will leave behind either |00i or |11i which are not entangled. Thus, we say that the GHZ is maximally entangled. This is unlike the W state which leaves bipartite entanglements even when we measure one of its subsystems.

Source: http://en.wikipedia.org/wiki/Greenberger-Horne-Zeilinger_state

Greenberger-Horne-Zeilinger state

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Hamiltonian (quantum mechanics)

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The quantum Hamiltonian is the physical state of a system, which may be characterized as a ray in an abstract →Hilbert space (or, in the case of ensembles, as a trace class operator with trace 1). (See →Mathematical formulation of quantum mechanics) Physically observable quantities are described by self-adjoint operators acting on the Hilbert space. For example, the Hilbert space associated with the spin degrees of freedom of a spin-1/2 particle is C 2, while the Hilbert space associated to a spinless particle moving on a line is L 2(R), the space of complex-valued and square-integrable (with respect to the Lebesgue measure) functions on the real line.

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The quantum Hamiltonian H is the observable corresponding to the total energy of the system. If the state space is finite dimensional, then it is of course bounded. In the infinite dimensional case, it is almost always unbounded, therefore not defined everywhere. In introductory physics literature, the following is considered either as a definition or an axiom: The eigenkets (eigenvectors) of H, denoted |ai

(using Dirac Bra-ket notation), provide an orthonormal basis for the Hilbert space. The spectrum of allowed energy levels of the system is given by the set of eigenvalues, denoted {E a}, solving the equation: H |ai = Ea |ai.

Since H is a Hermitian operator, the energy is always a real number.

From a mathematically rigorous point of view, care must be taken with the above axiom. Operators on infinite-dimensional Hilbert spaces need not have eigenvalues (notice in the infinite-dimensional case, the set of eigenvalues need not coincide with the spectrum, which can always safely assumed to be nonempty). However, all routine quantum mechanical calculations can be done using the physical formulation. As with all observables, the spectrum of the Hamiltonian are the possible outcomes when one measures the total energy of a system. Like any other self Hamiltonian (quantum mechanics)

169

FT

adjoint operator, the spectrum of the Hamiltonian can be decomposed, via its spectral measures, into discrete, absolutely continuous, and singular parts. The discrete spectrum can be associated to eigenvectors, which in turn usually are the bound states of the system. The absolutely continuous spectrum correspond to the free states. The singular spectrum, interesting enough, are physically impossible outcomes. For example, consider the finite potential well, which admits bound states with discrete negative energies and free states with continuous positive energies.

The Hamiltonian generates the time evolution of quantum states. If |ψ(t)i is the state of the system at time t, then ∂ H |ψ(t)i = i~ ∂t |ψ(t)i.

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where ~ is h-bar. This equation is known as the →Schrödinger equation. (It takes the same form as the Hamilton-Jacobi equation, which is one of the reasons H is also called the Hamiltonian.) Given the state at some initial time (t = 0), we can integrate it to obtain the state at any subsequent time. In particular, if H is independent of time, then   |ψ(t)i = exp − iHt |ψ(0)i. ~ In physical literature, the exponential operator on the right hand side is defined by the power series. One might note that taking polynomials of unbounded and not everywhere defined operators makes no mathematical sense, much less power series. Rigorously, to take functions of unbounded operators, one requires a functional calculus. In the case of the exponential function, the continuous, or just the holomorphic functional calculus suffices. We note again, however, that for common calculations the physicist’s formulation is quite proficient. By the *-homomorphism property of the Borel functional calculus, the operator   U = exp − iHt ~ is an unitary operator, and is a common form of the time evolution operator (also called the propagator). If the Hamiltonian is time-independent, {U(t)} form a one parameter unitary group (more than a semigroup); this gives rise to the physical principle of detailed balance.

Hamiltonian (quantum mechanics)

170

Energy eigenket degeneracy, symmetry, and conservation laws

FT

In many systems, two or more energy eigenstates have the same energy. A simple example of this is a free particle, whose energy eigenstates have wavefunctions that are propagating plane waves. The energy of each of these plane waves is inversely proportional to the square of its wavelength. A wave propagating in the x direction is a different state from one propagating in the y direction, but if they have the same wavelength, then their energies will be the same. When this happens, the states are said to be degenerate.

It turns out that degeneracy occurs whenever a nontrivial unitary operator U commutes with the Hamiltonian. To see this, suppose that |a> is an energy eigenket. Then U |a> is an energy eigenket with the same eigenvalue, since U H|ai = U Ea |ai = Ea (U |ai) = H (U |ai).

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Since U is nontrivial, at least one pair of |ai and U |ai must represent distinct states. Therefore, H has at least one pair of degenerate energy eigenkets. In the case of the free particle, the unitary operator which produces the symmetry is the rotation operator, which rotates the wavefunctions by some angle while otherwise preserving their shape. The existence of a symmetry operator implies the existence of a conserved observable. Let G be the Hermitian generator of U : U = I − iG + O(2 )

It is straightforward to show that if U commutes with H, then so does G : [H, G] = 0

Therefore,

∂ ∂t hψ(t)|G|ψ(t)i

=

1 i~ hψ(t)|[G, H]|ψ(t)i

=0

In obtaining this result, we have used the Schrödinger equation, as well as its dual, ∂ hψ(t)|H = −i~ ∂t hψ(t)|.

Thus, the expected value of the observable G is conserved for any state of the system. In the case of the free particle, the conserved quantity is the angular momentum.

Hamiltonian (quantum mechanics)

171

Hamilton’s equations

hn0 |ni = δnn0

FT

Hamilton’s equations in classical Hamiltonian mechanics have a direct analogy in quantum mechanics. Suppose we have a set of basis states {|ni}, which need not necessarily be eigenstates of the energy. For simplicity, we assume that they are discrete, and that they are orthonormal, i.e.,

Note that these basis states are assumed to be independent of time. We will assume that the Hamiltonian is also independent of time.

The instantaneous state of the system at time t, |ψ (t)i, can be expanded in terms of these basis states: P |ψ(t)i = n an (t)|ni where

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an (t) = hn|ψ(t)i

The coefficients a n(t) are complex variables. We can treat them as coordinates which specify the state of the system, like the position and momentum coordinates which specify a classical system. Like classical coordinates, they are generally not constant in time, and their time dependence gives rise to the time dependence of the system as a whole. The expectation value of the Hamiltonian of this state, which is also the mean energy, is P hH(t)i ≡ hψ(t)|H|ψ(t)i = nn0 a∗n0 an hn0 |H|ni

where the last step was obtained by expanding |ψ (t)i in terms of the basis states.

Each of the a n(t) s actually corresponds to two independent degrees of freedom, since the variable has a real part and an imaginary part. We now perform the following trick: instead of using the real and imaginary parts as the independent variables, we use a n(t) and its complex conjugate a n*(t). With this choice of independent variables, we can calculate the partial derivative P ∂hHi 0 0 n an hn |H|ni = hn |H|ψi ∂a∗ = n0

By applying Schrödinger’s equation and using the orthonormality of the basis states, this further reduces to Hamiltonian (quantum mechanics)

172 ∂hHi ∂a∗n0

= i~

∂an0 ∂t

∂hHi ∂an

= −i~

∂a∗n ∂t

FT

Similarly, one can show that

If we define "conjugate momentum" variables π n by πn (t) = i~a∗n (t) then the above equations become ∂hHi ∂πn

=

∂an ∂t

,

∂hHi ∂an

n = − ∂π ∂t

which is precisely the form of Hamilton’s equations, with the an s as the generalized coordinates, the πn s as the conjugate momenta, and hHi taking the place of the classical Hamiltonian.

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See also •

Hamiltonian mechanics

Source: http://en.wikipedia.org/wiki/Hamiltonian_%28quantum_mechanics%29 Principal Authors: CYD, Lethe, Mct mht, Enormousdude, Oleg Alexandrov, Looxix, SeventyThree

Heisenberg picture

In physics, the Heisenberg picture is that formulation of quantum mechanics where the operators (observables and others) are time-dependent and the state vectors are time-independent. It stands in contrast to the →Schrödinger picture in which operators are constant and the states evolve in time.

The "Heisenberg Picture" is not to be confused with matrix mechanics which is sometimes called Heisenberg quantum mechanics.

Mathematical details

In quantum mechanics in the Heisenberg picture the state vector, |ψ> does not change with time, and an observable A satisfies   d ∂A −1 . dt A = (i~) [A, H] + ∂t classical

Heisenberg picture

173 In some sense, the Heisenberg picture is more natural and fundamental than the →Schrödinger picture, especially for relativistic theories. Lorentz invariance is manifest in the Heisenberg picture.

FT

Moreover, the similarity to classical physics is easily seen: by replacing the commutator above by the Poisson bracket, the Heisenberg equation becomes an equation in Hamiltonian mechanics. By the Stone-von Neumann theorem, the Heisenberg picture and the →Schrödinger picture are unitarily equivalent. See also →Schrödinger picture.

Deriving Heisenberg’s equation

Suppose we have an observable A (which is a Hermitian linear operator). The expectation value of A for a given state |ψ(t)> is given by: hAit = hψ(t)|A|ψ(t)i

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or if we write following the →Schrödinger equation |ψ(t)i = e−iHt/~ |ψ(0)i

(where H is the Hamiltonian and hbar is Planck’s constant divided by 2·π) we get hAit = hψ(0)|eiHt/~ Ae−iHt/~ |ψ(0)i,

and so we define

A(t) := eiHt/~ Ae−iHt/~ .

Now,

d dt A(t)

= ~i HeiHt/~ Ae−iHt/~ +





∂A ∂t classical

+ ~i eiHt/~ A · (−H)e−iHt/~

(differentiating according to the product rule),   = ~i eiHt/~ (HA − AH) e−iHt/~ + ∂A ∂t classical =  

i ~

(HA(t) − A(t)H) +

∂A ∂t classical

(the last passage is valid since exp(-iHt/hbar) commutes with H )   = ~i [H, A(t)] + ∂A ∂t classical

Heisenberg picture

174 (where [X, Y ] is the commutator of two operators and defined as [X, Y ] := XY - YX ) So we get = ~i [H, A(t)] +





∂A ∂t classical .

FT

d dt A(t)

for a Time independent observable A and using Jacobi identity and integration by parts respect to time we get: 2

A = C + i~t [A, H] + ~t 2 [A, [A, H]] where C is a constant operator, due to the relationship between Poisson Bracket and Commutators this relation also holds for classical mechanics

See also →Interaction picture →Quantum mechanics →Schrödinger equation →Bra-ket notation →Matrix mechanics

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• • • • •

Source: http://en.wikipedia.org/wiki/Heisenberg_picture

Principal Authors: Lethe, Pjacobi, -Ril-, Karl-H, MathKnight, MarSch, GangofOne, GregorB, Naar

Hilbert space

In mathematics, a Hilbert space is a generalization of Euclidean space that is not restricted to finite dimensions. Thus it is an inner product space, which means that it has notions of distance and of angle (especially the notion of orthogonality or perpendicularity). Moreover, it satisfies a more technical completeness requirement which ensures that limits exist when expected, which facilitates various definitions from calculus. Hilbert spaces allow geometric intuition to be applied to certain infinite dimensional functional spaces. They provide a context with which to formalize and generalize the concepts of the Fourier series in terms of arbitrary orthogonal polynomials and of the Fourier transform, which are central concepts from functional analysis. Hilbert spaces are of crucial importance in the mathematical formulation of quantum mechanics. Hilbert space

175

Introduction

FT

Hilbert spaces were named after David Hilbert, who studied them in the context of integral equations. John von Neumann originated the designation "der abstrakte Hilbertsche Raum" in his famous work on unbounded Hermitian operators published in 1929. Von Neumann was perhaps the mathematician who most clearly recognized their importance as a result of his seminal work on the foundations of quantum mechanics begun with Hilbert and Lothar (Wolfgang) Nordheim and continued with Eugene Wigner. The name "Hilbert space" was soon adopted by others, for example by Hermann Weyl in his book The Theory of Groups and Quantum Mechanics published in 1931 (English language paperback ISBN 0486602699).

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The elements of an abstract Hilbert space are sometimes called "vectors". In applications, they are typically sequences of complex numbers or functions. In quantum mechanics for example, a physical system is described by a complex Hilbert space which contains the "wavefunctions" that stand for the possible states of the system. See mathematical formulation of quantum mechanics for details. The Hilbert space of plane waves and bound states commonly used in quantum mechanics is known more formally as the rigged Hilbert space.

Definition

Every inner product <·,·> on a real or complex vector space H gives rise to a norm ||·|| as follows: p ||x|| = hx, xi. In any normed space, the open balls constitute a compatible topology; any normed vector space is a topological vector space (and even a uniform structure) and therefore so is any inner product space.

The Cauchy criterion may be defined for sequences in this space (as it can in any uniform space): a sequence {x n } n is a Cauchy sequence if for every natural number N there is a real number  such that for all m, n > N, ||x n – x m || < . We call H a Hilbert space if it is complete with respect to this norm, that is if every Cauchy sequence All finite-dimensional inner product spaces (such as Euclidean space with the ordinary dot product) are Hilbert spaces. However, the infinite-dimensional examples are much more important in applications. These applications include:

• • •

The theory of unitary group representations The theory of square integrable stochastic processes The Hilbert space theory of partial differential equations, in particular formulations of the Dirichlet problem Hilbert space

176 • •

Spectral analysis of functions, including theories of wavelets Mathematical formulations of quantum mechanics

FT

The inner product allows one to adopt a "geometrical" view and use geometrical language familiar from finite dimensional spaces. Of all the infinitedimensional topological vector spaces, the Hilbert spaces are the most "wellbehaved" and the closest to the finite-dimensional spaces.

One goal of Fourier analysis is to write a given function as a (possibly infinite) sum of multiples of given base functions. This problem can be studied abstractly in Hilbert spaces: every Hilbert space has an orthonormal basis, and every element of the Hilbert space can be written in a unique way as a sum of multiples of these base elements.

Examples

In these examples, we will assume the underlying field of scalars is C, although the definitions apply to the case in which the underlying field of scalars is R.

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Euclidean spaces

C n with the inner product definition P hx, yi = nk=1 xk yk

where the bar over a complex number denotes its complex conjugate.

Sequence spaces

Much more typical are the infinite dimensional Hilbert spaces however. If B is any set, we define the sequence space little l 2 over B, denoted by

`2 (B) =

  P x : B → C b∈B |x (b)|2 < ∞

This space becomes a Hilbert space with the inner product P hx, yi = b∈B x(b)y(b) for all x and y in l 2(B ). B does not have to be a countable set in this definition, although if B is not countable, the resulting Hilbert space is not separable. In a sense made more precise below, every Hilbert space is isomorphic to one the form l 2(B ) for a suitable set B. If B =N, we write simply l 2.

Hilbert space

177

Lebesgue spaces

FT

These are function spaces associated to measure spaces (X, M, µ), where M is a σ-algebra of subsets of X and µ is a countably additive measure on M. Let L 2 µ(X ) be the space of complex-valued square-integrable measurable functions on X, modulo equality almost everywhere. Square integrable means the integral of the square of its absolute value is finite. Modulo equality almost everywhere means functions are identified if and only if they are equal outside of a set of measure 0. The inner product of functions f and g is here given by R hf, gi = X f (t)g(t) dµ(t) One needs to show: • •

That this integral indeed makes sense; The resulting space is complete.

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These are technically easy facts, and the interested reader should consult the Halmos reference below, Section 42. Note that the use of the Lebesgue integral ensures that the space will be complete. See L p space for further discussion of this example.

Sobolev spaces

Sobolev spaces, denoted by H s or W s,2 , are another example of Hilbert spaces, and are used very often in the field of Partial differential equations.

Operations on Hilbert spaces

Given two (or more) Hilbert spaces, we can combine them into a single Hilbert space by taking their direct sum or their tensor product.

Bases

An important concept is that of an orthonormal basis of a Hilbert space H : this is a family {e k } k ∈ B of H satisfying: •

Elements are normalized: Every element of the family has norm 1: ||e k || = 1 for all k in B • Elements are orthogonal: Every two different elements of B are orthogonal: <e k , e j > = 0 for all k, j in B with k 6= j. • Dense span: The linear span of B is dense in H. We also use the expressions orthonormal sequence and orthonormal set. Hilbert space

178 Examples of orthonormal bases include: the set {(1,0,0),(0,1,0),(0,0,1)} forms an orthonormal basis of R 3 the sequence {f n : n ∈ Z} with f n (x) = exp(2πinx) forms an orthonormal basis of the complex space L 2([0,1]) • the family {e b : b ∈ B } with e b (c) = 1 if b =c and 0 otherwise forms an orthonormal basis of l 2(B ).

FT

• •

Note that in the infinite-dimensional case, an orthonormal basis will not be a basis in the sense of linear algebra; to distinguish the two, the latter basis is also called a Hamel basis. That the span of the basis vectors is dense means that every vector in the space can be written as the limit of an infinite series and the orthogonality implies that this decomposition is unique.

Using Zorn’s lemma, one can show that every Hilbert space admits an orthonormal basis; furthermore, any two orthonormal bases of the same space have the same cardinality. A Hilbert space is separable if and only if it admits a countable orthonormal basis.

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Since all infinite-dimensional separable Hilbert spaces are isomorphic, and since almost all Hilbert spaces used in physics are separable, when physicists talk about the Hilbert space they mean any separable one.

If {e k } k ∈ B is an orthonormal basis of H, then every element x of H may be written as P x = k∈B hek , xiek Even if B is uncountable, only countably many terms in this sum will be nonzero, and the expression is therefore well-defined. This sum is also called the Fourier expansion of x. If {e k } k ∈ B is an orthonormal basis of H, then H is isomorphic to l 2(B ) in the following sense: there exists a bijective linear map Φ : H → l 2(B ) such that hΦ (x) , Φ (y)i = hx, yi

for all x and y in H.

Orthogonal complements and projections If S is a subset of a Hilbert space H, we define the set of vectors orthogonal to S S perp = {x ∈ H : hx, si = 0 ∀s ∈ S}

Hilbert space

179

FT

S perp is a closed subspace of H and so forms itself a Hilbert space. If V is a closed subspace of H, then V perp is called the orthogonal complement of V. In fact, every x in H can then be written uniquely as x = v + w, with v in V and w in V perp. Therefore, H is the internal Hilbert direct sum of V and V perp. The linear operator P V : H → H which maps x to v is called the orthogonal projection onto V. Theorem. The orthogonal projection P V is a self-adjoint linear operator on H of norm ≤ 1 with the property P V 2 = P V . Moreover, any self-adjoint linear operator E such that E 2 = E is of the form P V , where V is the range of E. For every x in H, P V (x) is the unique element v of V which minimizes the distance ||x - v||. This provides the geometrical interpretation of P V (x): it is the best approximation to x by elements of V.

Reflexivity

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An important property of any Hilbert space is its reflexivity. In fact, more is true: one has a complete and convenient description of its dual space (the space of all continuous linear functions from the space H into the base field), which is itself a Hilbert space. Indeed, the Riesz representation theorem states that to every element φ of the dual H there exists one and only one u in H such that φ (x) = hu, xi

for all x in H and the association φ ↔ u provides an antilinear isomorphism between H and H . This correspondence is exploited by the bra-ket notation popular in physics but frowned upon by mathematicians.

Bounded operators

For a Hilbert space H, the continuous linear operators A : H → H are of particular interest. Such a continuous operator is bounded in the sense that it maps bounded sets to bounded sets. This allows to define its norm as

kAk = sup { kAxk : kxk ≤ 1 } .

The sum and the composition of two continuous linear operators is again continuous and linear. For y in H, the map that sends x to is linear and continuous, and according to the Riesz representation theorem can therefore be represented in the form hA∗ y, xi = hy, Axi.

Hilbert space

180 This defines another continuous linear operator A * : H → H, the adjoint of A.

FT

The set L(H ) of all continuous linear operators on H, together with the addition and composition operations, the norm and the adjoint operation, forms a C *algebra; in fact, this is the motivating prototype and most important example of a C *-algebra. An element A of L(H ) is called self-adjoint or Hermitian if A * = A. These operators share many features of the real numbers and are sometimes seen as generalizations of them.

An element U of L(H ) is called unitary if U is invertible and its inverse is given by U *. This can also be expressed by requiring that = <x, y> for all x and y in H. The unitary operators form a group under composition, which can be viewed as the automorphism group of H.

Unbounded operators

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If a linear operator has a closed graph and is defined on all of a Hilbert space, then, by the closed graph theorem in Banach space theory, it is necessarily bounded. However, if we allow ourselves to define a linear map that is defined on a proper subspace of the Hilbert space, then we can obtain unbounded operators.

In quantum mechanics, several interesting unbounded operators are defined on a dense subspace of Hilbert space. It is possible to define self-adjoint unbounded operators, and these play the role of the observables in the mathematical formulation of quantum mechanics. Examples of self-adjoint unbounded operator on the Hilbert space L 2(R) are: •

A suitable extension of the differential operator d [Af ](x) = i dx f (x),

where i is the imaginary unit and f is a differentiable function of compact support.

•

The multiplication by x operator: [Bf ](x) = xf (x).

These correspond to the momentum and position observables, respectively. Note that neither A nor B is defined on all of H, since in the case of A the derivative need not exist, and in the case of B the product function need not Hilbert space

181

See also

FT

be square integrable. In both cases, the set of possible arguments form dense subspaces of L 2(R).

• • • •

Topologies on the set of operators on a Hilbert space Operator algebra Reproducing kernel Hilbert space Rigged Hilbert space

• • •

Mathematical analysis Functional analysis Harmonic analysis

References

Jean Dieudonné, Foundations of Modern Analysis, Academic Press, 1960. Paul Halmos, Measure Theory, D. van Nostrand Co, 1950. David Hilbert, Lothar Nordheim, and John von Neumann, "Über die Grundlagen der Quantenmechanik," Mathematische Annalen, volume 98, pages 1-30, 1927. John von Neumann, "Allgemeine Eigenwerttheorie Hermitescher Funktionaloperatoren," Mathematische Annalen, volume 102, pages 49-131, 1929. Hermann Weyl, The Theory of Groups and Quantum Mechanics, Dover Press, 1950. This book was originally published in German in 1931.

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• • •

• •

Source: http://en.wikipedia.org/wiki/Hilbert_space

Principal Authors: CSTAR, Jitse Niesen, Michael Hardy, Vilemiasma, Lupin, Lethe, Paul August, Zundark

Hilbert space

182

Hydrogen atom

FT

An hydrogen atom is an atom of the chemical element hydrogen. It is composed of a single negatively-charged electron circling a single positivelycharged proton which is the nucleus of the hydrogen atom. The electron is bound to the proton by the Coulomb force. The hydrogen atom has special significance in quantum mechanics and quantum field theory as a simple physical system for which the solution to the →Schrödinger equation is analytical, from which the positions of energy levels (thus, the frequencies of the hydrogen spectral lines) can be calculated.

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In 1913, Niels Bohr had deduced the spectral frequencies of the hydrogen atom making several assumptions (see The Bohr Model). The results of Bohr for the frequencies and underlying energy values are confirmed by the full quantummechanical analysis which uses the Schrödinger equation, as was shown in 1925/26. The solution of the Schrödinger equation goes much further, because it also yields the shape of the electron’s wave function ("orbital") for the various possible quantum-mechanical states - thus explaining the anisotropic character of atomic bonds. The Schrödinger equation also applies to more complicated atoms and molecules, however, in most cases the solution is not analytical and either computer calculations are necessary or some simplifying assumptions must be made.

Solution of Schrödinger equation: Overview of results

The solution of the Schrödinger equation for the hydrogen atom uses the fact that the Coulomb potential produced by the nucleus is isotropic (it only depends on the distance to the nucleus). Although the resulting energy eigenfunctions (the "orbitals") are not necessarily isotropic themselves, their dependence on the angular coordinates follows completely generally from this isotropy of the underlying potential: The states are not only eigenstates of the Hamiltonian, but also eigenstates of the angular momentum operator (so called spherical harmonics). This corresponds to the fact that angular momentum is conserved in the orbital motion of the electron around the nucleus. Therefore, the energy eigenstates may be classified by two angular momentum quantum numbers, l and m (integer numbers). The "angular momentum" quantum number l = 0, 1, 2, ... determines the magnitude of the angular momentum. The "magnetic" quantum number m = -l, .., +l determines the projection of the angular momentum on the (arbitrarily chosen) z-axis. Hydrogen atom

183

FT

In addition, the radial dependence of the wave functions has to be found. It is only here that the details of the 1/r Coulomb potential enter (leading to Laguerre polynomials in r). This leads to a third quantum number, the principal quantum number n = 1, 2, 3, ... Note that the angular momentum quantum number can run only up to n - 1, i.e. l = 0, 1, ..., n - 1.

Due to angular momentum conservation, states of the same l but different m have the same energy (this holds for all problems with rotational symmetry). In addition, for the hydrogen atom, the states of the same n are also degenerate (i.e. they have the same energy); but this is a specialty and it is no longer true for more complicated atoms which have a (effective) potential differing from the form 1/r (due to the presence of the inner electrons shielding the nucleus potential).

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Taking into account the spin of the electron adds a last quantum number, the projection of the electrons spin along the z axis, which can take on two values. Therefore, any eigenstate of the electron in the hydrogen atom is described fully by four quantum numbers. According to the usual rules of quantum mechanics, the actual state of the electron may be any superposition of these states. This explains also why the choice of z-axis for the quantization of angular momentum is immaterial: An orbital of given l and m’ obtained for another preferred axis z’ can always be represented as a suitable superposition of the various states of different m (but same l) that have been obtained for z.

Mathematical summary of eigenstates of hydrogen atom Main article: hydrogen-like atom

The normalized position wavefunctions, given in spherical coordinates are: r  3 (n−l−1)! 2 ψnlm (θ, φ, r) = e−ρ/2 ρl L2l+1 na0 n−l−1 (ρ) · Yl,m (θ, φ) 2n[(n+l)!]

where:

•

2r na0 and a 0 is the Bohr radius. L2l+1 n−l−1 (ρ) are the Generalized Laguerre

•

Yl,m (θ, φ) is a spherical harmonic.

•

ρ=

polynomials of degree n-l-1.

The eigenvalues are: •

For Angular momentum operator:

Hydrogen atom

184 L2 |n, l, mi = ~2 l(l + 1)|n, l, mi

•

For the Hamiltonian: H|n, l, mi = En |n, l, mi where 2

2

2

Z α m En = − mc2·n = − 2~ 2 2



FT

Lz |n, l, mi = ~m|n, l, mi

Ze2 4π0

2

1 n2

and α is the fine structure constant

and in hydrogen atom Z =1.

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Visualizing the hydrogen electron orbitals

The image to the right shows the first few hydrogen atom orbitals (energy eigenfunctions). These are cross-sections of the probability density that are

Hydrogen atom

185

FT

color-coded (black=zero density, white=highest density). The angular momentum quantum number l is denoted in each column, using the usual spectroscopic letter code ("s" means l = 0; "p": l = 1; "d ": l = 2). The main quantum number n (= 1, 2, 3, ...) is marked to the right of each row. For all pictures the magnetic quantum number m has been set to 0, and the crosssectional plane is the xz-plane (z is the vertical axis). The probability density in three-dimensional space is obtained by rotating the one shown here around the z-axis. The "ground state", i.e. the state of lowest energy, in which the electron is usually found, is the first one, the "1s" state (n = 1, l = 0). An image with more orbitals is also available (up to higher numbers n and l).

Note the number of black lines that occur in each but the first orbital. These are "nodal lines" (which are actually nodal surfaces in three dimensions). Their total number is always equal to n - 1, which is the sum of the number of radial nodes (equal to n - l - 1) and the number of angular nodes (equal to l ).

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Features going beyond the Schrödinger solution

There are several important effects that are neglected by the Schrödinger equation and which are responsible for certain small but measurable deviations of the real spectral lines from the predicted ones: •

Although the maximum effective speed of the electron in hydrogen is only 1/137th of the speed of light there is an increase in the electron’s mass, as predicted by special relativity. For heavier elements, this is more significant (see 93). • The spin of the electron has a magnetic moment attached to it. Even when there is no external magnetic field, within the inertial frame of the moving electron the electric field of the nucleus partly acts like a magnetic field. This effect is also explained by special relativity, and it leads to the so-called spin-orbit coupling, i.e. an influence of the electron’s orbital motion around the nucleus onto its spin.

Both of these features (and more) are incorporated in the relativistic Dirac equation, whose predictions come still closer to experiment. It can still be solved exactly for the hydrogen atom. The resulting states now must be classified by the total angular momentum number j (arising through the coupling between electron spin and orbital angular momentum). States of the same j and the same n are still degenerate.

93 http://www.chem1.com/acad/webtut/atomic/qprimer/#Q26

Hydrogen atom

186 There are always vacuum fluctuations of the electromagnetic field, according to quantum mechanics. This means in particular that the electron undergoes a kind of "jitter" motion. As a consequence, the degeneracy between states of the same j but different l is lifted. This has been demonstrated in the famous Lamb-Rutherford experiment and was the starting point for the development of the theory of Quantum electrodynamics (which is able to deal with these vacuum fluctuations and employs the famous Feynman diagrams for approximations using perturbation theory). This effect is now called Lamb shift.

FT

•

For these developments, it was essential that the solution of the Dirac equation for the hydrogen atom could be worked out exactly, such that any experimentally observed deviation had to be taken seriously as a signal of failure of the theory.

See also Hydrogen quantum mechanics quantum chemistry quantum field theory quantum state

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• • • • •

References •

Griffiths, David (1995). Introduction to Quantum Mechanics. Upper Saddle River, NJ: Prentice Hall. ISBN 0-13-111892-7.

Section 4.2 deals with the hydrogen atom specifically, but all of Chapter 4 is relevant. •

Bransden, B.H.; C.J. Joachain (1983). Physics of Atoms and Molecules. London: Longman. ISBN 0-582-44401-2.

External links • • •

Physics of hydrogen atom on Scienceworld 94 Interactive graphical representation of orbitals 95 Applet which allows viewing of all sorts of hydrogenic orbitals 96

94 http://scienceworld.wolfram.com/physics/HydrogenAtom.html 95 http://webphysics.davidson.edu/faculty/dmb/hydrogen/ 96 http://www.falstad.com/qmatom/

Hydrogen atom

187

Source: http://en.wikipedia.org/wiki/Hydrogen_atom

WAS 4.250, Jsalazar, Karol Langner

FT

Principal Authors: Oo64eva, DÅ‚ugosz, Edsanville, The Anome, Anville, Michael Hardy, John C PI,

Hydrogen-like atom

Hydrogen-like atoms (or hydrogenic atoms) are atoms with one single electron. Like the hydrogen atom, hydrogen-like atoms are one of the few quantum mechanical problems which can be exactly solved. Atoms or ions whose valence shell is made of one single electron (e.g. alkali metals) have similar chemical bonding or spectroscopic propreties to hydrogen-like atoms.

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The simplest atomic orbitals are those that occur in an atom with a single electron, such as the hydrogen atom. In this case the atomic orbitals are the eigenstates of the hydrogen Hamiltonian. They can be obtained analytically (see →Hydrogen atom). An atom of any other element ionized down to a single electron is very similar to hydrogen, and the orbitals take the same form.

For atoms with two or more electrons, the governing equations can only be solved with the use of methods of iterative approximation. Orbitals of multielectron atoms are qualitatively similar to those of hydrogen, and in the simplest models, they are taken to have the same form. For more rigorous and precise analysis, numerical approximations must be used. Atomic orbitals are often expanded in a basis set of Slater-type orbitals which are orbitals of hydrogen-like atoms with arbitrary nuclear charge Z. A given (hydrogen-like) atomic orbital is identified by unique values of three quantum numbers: n, l, and m l. The rules restricting the values of the quantum numbers, and their energies (see below), explain the electron configuration of the atoms and the periodic table. The stationary states (quantum states) of the hydrogen-like atoms are its atomic orbital. However, in general, an electron’s behavior is not fully described by a single orbital. Electron states are best represented by time-depending "mixtures" (linear combinations) of multiple orbitals. See Linear combination of atomic orbitals molecular orbital method. The quantum number n first appeared in the →Bohr model. It determines, among other things, the distance of the electron from the nucleus; all electrons with the same value of n lay at the same distance. Modern quantum mechanics confirms that these orbitals are closely related. For this reason, orbitals with Hydrogen-like atom

188 the same value of n are said to comprise a "shell". Orbitals with the same value of n and also the same value of l are even more closely related, and are said to comprise a "subshell".

FT

Mathematical characterization Derivation

Atomic orbitals are solutions to the →Schrödinger equation. In this case, the potential term is the potential given by Coulomb’s law: 1 V = − 4π 0

Ze2 r

where

The first term is a constant, usually abbreviated by the letter k, Z is the atomic number, e is the elementary charge, r is the magnitude of the distance from the nucleus.

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• • • •

The wavefunction is a function of three spatial variables, so that after removing the time-dependence, the Schrödinger equation is a partial differential equation in three variables (see separation of variables). However, since the potential is spherically symmetric, it is profitable to write the equation in spherical coordinates. In this form, any individual eigenfunction ψ can be written as a product of three single-variable functions, often denoted as follows: ψ(r, θ, φ) = R(r)f (θ)g(φ)

(where θ represents the polar angle (colatitude) and φ the azimuthal angle.) It can further be reduced to three separate equations, each in one variable.

Two separations are required, resulting in two separation constants. A third arbitrary constant results from the application of boundary conditions to R. The equations given below use a form of the separation constants that seems arbitrary, but it simplifies matters later on.   1 d 2 dR + 2µr2 (E − V (r)) = l(l + 1) r dr dr R(r) ~2 2 1 d g(φ) g(φ) dφ2

= −m2

l(l + 1) sin2 (θ) +

sin(θ) d f (θ) dθ

h

i df sin(θ) dθ = m2

where:

Hydrogen-like atom

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h h is the reduced Planck constant ( 2π ), and µ is the reduced mass of the electron vis-à-vis the nucleus.

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Results In addition to ` and m, a third arbitrary integer, called n, emerges from the boundary conditions placed on R. The functions R, f and g that solve the equations above depend on the values of these integers, called quantum numbers. As a result, it is customary to subscript the functions with the values of the quantum numbers they depend on. The forms of the functions are: ψ = Cnlm Rnl (r) flm (θ) gm (φ) Zr − na µ

Rnl (r) = e flm (θ) =



(sin θ)|m| 2l l!

2Zr naµ

h

functions)

2l+1 Ln−l−1

d d(cos θ)



il+|m|

2Zr naµ



(cos2 (θ) − 1)l

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gm (φ) = eimφ

l

(the associated Legendre

n, l, m are integer numbers, with the restrictions: n = 1, 2, 3, ...

l = 0, 1, 2, ..., n − 1

m = −l, −(l − 1), ..., 0, ..., l − 1, l

where: •

L2l+1 n−l−1 are the generalized Laguerre polynomials.

•

aµ is defined by: aµ =

4πε0 ~2 µe2

Note that aµ is approximately equal to a0 (the Bohr radius)

•

•

i is the imaginary number.

Cnlm is a normalization constant. Since the wavefunction must be normal-

ized (

R

R3

|Ψ|2 d3~r = 1 ), by calculating this integral we get: Hydrogen-like atom

190 Cnlm =



2Z naµ

3

(n−l−1)! 2n(n+l)!

1/2

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The functions f and g are usually consolidated into the function Y (θ, φ) = f (θ)g(φ). This function is a spherical harmonic.

Thus, the complete expression for the normalized wavefunctions is: r 3     (n−l−1)! −Zr/naµ 2Zr l 2l+1 2Z ψnlm (θ, φ, r) = e Ln−l−1 2Zr · naµ naµ naµ 2n[(n+l)!] Yl,m (θ, φ)

Angular momentum

Each atomic orbital is associated with an angular momentum L. It is a vector, and its magnitude is given by: p |L| = ~ l(l + 1)

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The projection of this vector onto any arbitrary direction is quantized. If the arbitrary direction is called z, the quantization is given by: Lz = ml ~

where ml is restricted as described above. This value is always less than the total angular momentum. Thus, if the L-vector is measured in some direction, it will not lie entirely in that direction; part of it will lie in perpendicular directions. This allows the uncertainty principle to stand. It mandates that no two components of L may be known at once. If one component were known to be equal to the total L, the other two would necessarily be zero. These two relations do not give the total angular momentum of the electron. For that, electron spin must be included. This quantization of angular momentum closely parallels that proposed by Niels Bohr (see →Bohr model) in 1913, with no knowledge of wavefunctions.

See also • •

Positronium Exotic atom

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References Tipler, Paul & Ralph Llewellyn (2003). Modern Physics (4th ed.). New York: W. H. Freeman and Company. ISBN 0-7167-4345-0

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Source: http://en.wikipedia.org/wiki/Hydrogen-like_atom

Principal Authors: Pfalstad, John C PI, Eequor, JimR, Achoo5000

Hyperfine structure

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In atomic physics, hyperfine structure is a small perturbation in the energy levels (or spectra) of atoms or molecules due to the magnetic dipole-dipole interaction, arising from the interaction of the nuclear magnetic dipole with the magnetic field of the electron.

Theory

According to classical thinking, the electron moving around the nucleus has a magnetic dipole moment, because it is charged. The interaction of this magnetic dipole moment with the magnetic moment of the nucleus (due to its spin) leads to hyperfine splitting. However, due to the electron’s spin, there is also hyperfine splitting for s-shell electrons, which have zero orbital angular momentum. In this case, the magnetic dipole interaction is even stronger, as the electron probability density does not vanish inside the nucleus (r = 0). The amount of correction to the Bohr energy levels due to hyperfine splitting of the hydrogen atom is on the order of: m 4 2 mp α mc

where

m is the mass of an electron m p is the mass of a proton

α is the fine structure constant (1/137.036 )

Hyperfine structure

192 c is the speed of light.

2

where ~

a = √gI µ~ N BJ , J(J+1)

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For atoms other than hydrogen, the nuclear spin I~ and the total electron angu~ +S ~ get coupled, giving rise to the total angular momenlar momentum J~ = L ~ ~ ~ tum F = J + I. The hyperfine splitting is then ~ J = a [F (F + 1) − I(I + 1) − J(J + 1)], ∆Ehf s = −~ µI B

with µ ~ N the magnetic dipole moment of the nucleus.

This interaction obeys the Lande interval rule: The energy level is split into (J + I) − |J − I| + 1 energy levels, where J denotes the total electron angular momentum and I denotes the nuclear spin.

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With ∆Ehf s ≈ ~, the hyperfine splitting is a much smaller perturbation than the fine structure. In a more advanced treatment, one also has to take the nuclear magnetic quadrupole moment into account. This is sometimes (?) referred to as "hyperfine structure anomaly".

History

The optical hyperfine structure was already observed in 1881 by Albert Abraham Michelson. It could, however, only be explained in terms of quantum mechanics in the 1920s. Wolfgang Pauli proposed the existence of a small nuclear magnetic moment in 1924. In 1935, M. Schiiler and T. Schmidt proposed the existence of a nuclear quadrupole moment in order to explain anomalies in the hyperfine structure.

Applications Astrophysics

As the hyperfine splitting is very small, the transition frequencies usually are not optical, but in the range of radio- or microwave frequencies. Hyperfine structure gives the 21 cm line observed in HI region in interstellar medium. Carl Sagan and Frank Drake considered the hyperfine transition of hydrogen to be a sufficiently universal phenomenon so as to be used as a base unit of time and length on the Pioneer plaque and later Voyager Golden Record. Hyperfine structure

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Nuclear technology

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The AVLIS and MLIS processes use hyperfine splitting caused by differences between the mass of atomic nucleus for uranium-235 and uranium-238 to selectively photoionize only the uranium-235 atoms and then separate the ionized particles from the non-ionized ones. Precisely tuned dye lasers are used as the sources of the necessary exact wavelength radiation.

Use in defining the SI second and meter

The hyperfine structure transition can be used to make a microwave notch filter with very high stability, repeatability and Q factor, which can thus be used as a basis for very precise atomic clocks. Typically, the hyperfine structure transition frequency of a particular isotope of caesium or rubidium atoms is used as a basis for these clocks.

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Due to the accuracy of hyperfine structure transition-based atomic clocks, they are now used as the basis for the definition of the second. One second is now defined to be exactly 9,192,631,770 cycles of the hyperfine structure transition frequency of caesium-133 atoms. Since 1983, the meter is defined by declaring the speed of light in a vacuum to be exactly 299,792,458 metres per second. Thus: The metre is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

Qubit in ion-trap quantum computing

The hyperfine states of a trapped ion are commonly used for storing qubits in ion-trap quantum computing. They have the advantage of having a very long lifetimes, experimentally exceeding 10 min (compared to 1 s for metastable electronic levels). The frequency associated with the states’ energy separation is in the microwave region, making it possible to drive hyperfine transitions using microwave radiation. However, at present no emitter is available that can be focused to address a particular ion from a sequence. Instead, a pair of laser pulses can be used to drive the transition, by having their frequency difference (detuning) equal to the required transition’s frequency. This is essentially a stimulated Raman transition.

See also • •

Energy levels Quantum numbers

Hyperfine structure

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Source: http://en.wikipedia.org/wiki/Hyperfine_structure

Imaginary time

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Principal Authors: The Anome, LAk loho, ArnoldReinhold, Icairns, Linas

Imaginary time is a concept derived from quantum mechanics. It is used to describe models of the universe in physical cosmology. Stephen Hawking popularized the concept of imaginary time in his book A Brief History of Time.

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Imaginary time is difficult to visualize. If we imagine "regular time" as a horizontal line with "past" on one side and "future" on the other, then imaginary time would run perpendicular to this line as the imaginary numbers run perpendicular to the real numbers in the complex plane. However, imaginary time is not imaginary in the sense that it is unreal or made-up—it simply runs in a direction different from the type of time we experience. In essence, imaginary time is a way of looking at the time dimension as if it were a dimension of space: you can move forward and backward along imaginary time, just like you can move right and left in space. The concept is useful in cosmology because it can help smooth out gravitational singularities in models of the universe (see Hartle-Hawking state. Singularities pose a problem for physicists because they are areas where known physical laws do not apply. The Big Bang, for example, appears as a singularity in "regular time." But when visualized with imaginary time, the singularity is removed and the Big Bang functions like any other point in spacetime.

The No-Boundary Universe and imaginary time To further illustrate this concept, imagine spacetime as the surface of the Earth, with all three space dimensions combined into the "east-west" axis, and the imaginary time dimension running along the "north-south" axis. In this model, spacetime is both finite and boundless—like the surface of the Earth, it has a finite area, but lacks any edge or boundary. The "North Pole" in this model would be analogous to the Big Bang. It is the "northernmost" point on the surface of the earth, just as it is the "earliest" point of time in the universe. But in the imaginary time/space cosmology, the Big Bang/north pole is not a singularity; it is a point of spacetime just like any other. If you travel "north" or "before" the Big Bang, physics would not break down, you would simply encounter more of the universe.

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See also • •

Wick rotation Euclidean quantum gravity

External links •

The Beginning of Time 97 — Lecture by Stephen Hawking which discusses imaginary time. Stephen Hawking’s Universe: Strange Stuff Explained 98 — PBS site on imaginary time

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It must be noted that this globe visualization of the No-Boundary Universe is not without faults. The visualization suggests that space will expand until it reaches an "equator" and then contract, due to gravity, until it becomes the South Pole, or the Big Crunch. However, current models of the universe suggest that space will continue to expand faster and faster. Imaginary time, as a quantum mechanics concept, is perhaps more accurately integrated with cosmology via the wavefunction model of the universe.

Source: http://en.wikipedia.org/wiki/Imaginary_time

Implicate and Explicate Order according to David Bohm

The physicist David Bohm proposed a conception of order which is radically different from most conceptions of order, and in doing so made a distinction between the Implicate and Explicate Order, which he characterised as follows: In the enfolded [or Implicate] order, space and time are no longer the dominant factors determining the relationships of dependence or independence of different elements. Rather, an entirely different sort of basic connection of elements is possible, from which our ordinary notions of space and time, along with those of separately existent material particles, are abstracted as forms derived from the deeper order. These ordinary notions in fact appear

97 http://www.hawking.org.uk/text/public/bot.html 98 http://www.pbs.org/wnet/hawking/strange/html/imaginary.html

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in what is called the "explicate" or "unfolded" order, which is a special and distinguished form contained within the general totality of all the Implicate Orders (Bohm, 1980, p. xv).

David Bohm’s challenges to some generally prevailing views In proposing this new notion of order, Bohm explicitly challenged a number of tenets that are fundamental to much scientific work. The tenets challenged by Bohm include: •

•

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That phenomena are reducible to fundamental particles and laws describing the behaviour of particles, or more generally to any static (i.e. unchanging) entities, whether separate events in space-time, quantum states, or static entities of some other nature. Related to (1), that human knowledge is most fundamentally concerned with mathematical prediction of statistical aggregates of particles. That an analysis or description of any aspect of reality (e.g. quantum theory, the speed of light) can be unlimited in its domain of relevance. That the Cartesian coordinate system, or its extension to a curvilinear system, is the deepest conception of underlying order as a basis for analysis and description of the world. That there is ultimately a sustainable distinction between reality and thought, and that there is a corresponding distinction between the observer and observed in an experiment or any other situation (other than a distinction between relatively separate entities valid in the sense of Explicate Order). That it is, in principle, possible to formulate a final notion concerning the nature of reality; e.g. a Theory of Everything.

•

•

•

Bohm’s proposals have at times been ’dismissed’ largely on the basis of such tenets, without due consideration necessarily given to the fact that they had been challenged by Bohm.

Bohm’s paradigm is inherently antithetical to reductionism, in most forms, and accordingly can be regarded as a form of ontological holism. On this, Bohm noted of prevailing views among physicists: "the world is assumed to be constituted of a set of separately existent, indivisible and unchangeable ’elementary particles’, which are the fundamental ’building blocks’ of the entire universe · · · there seems to be an unshakable faith among physicists that either such particles, or some other kind yet to be discovered, will eventually make possible a complete and coherent explanation of everything" (Bohm, 1980, p. 173). Implicate and Explicate Order according to David Bohm

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Figure 16 A Helium Atom and its constituent particles: an example of a small collection of the posited building blocks of the universe

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In Bohm’s conception of order, then, primacy is given to the undivided whole, and the Implicate Order inherent within the whole, rather than to parts of the whole, such as particles, quantum states, and continua. For Bohm, the whole encompasses all things, structures, abstractions and processes, including processes that result in (relatively) stable structures as well as those that involve metamorphosis of structures or things. In this view, parts may be entities normally regarded as physical, such as atoms or sub-atomic particles, but they may also be abstract entities, such as quantum states. Whatever their nature and character, according to Bohm, these parts are considered in terms of the whole, and in such terms, they constitute relatively autonomous and independent "sub-totalities". The implication of the view is, therefore, that nothing is entirely separate or autonomous. Bohm (1980, p. 11) said: "The new form of insight can perhaps best be called Undivided Wholeness in Flowing Movement. This view implies that flow is, in some sense, prior to that of the ‘things’ that can be seen to form and dissolve in this flow". According to Bohm, a vivid image of this sense of analysis of the whole is afforded by vortex structures in a flowing stream. Such vortices can be relatively stable patterns within a continuous flow, but such an analysis does not imply that the flow patterns have any sharp division, or that they are literally separate and independently existent entities; rather, they are most fundamentally undivided. Thus, according to Bohm’s view, the whole is in continuous flux, and hence is referred to as the holomovement (movement of the whole). Implicate and Explicate Order according to David Bohm

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Figure 17 A vortex in a stream - a relatively stable pattern which occurs within a continuous flow of liquid

Quantum theory and relativity theory

A key motivation for Bohm in proposing a new notion of order was what he saw as the incompatibility of quantum theory with relativity theory, in terms of certain features of the theoris as observed in relevant experimental contexts. Bohm (1980, p. xv) summarised the state of affairs he perceived to exist in the following terms: · · ·in relativity, movement is continuous, causally determinate and well defined, while in quantum mechanics it is discontinuous, not causally determinate and not well-defined. Each theory is committed to its own notions of essentially static and fragmentary modes of existence (relativity to that of separate events connectible by signals, and quantum mechanics to a well-defined quantum state). One thus sees that a new kind of theory is needed which drops these basic commitments and at most recovers some essential features of the older theories as abstract forms derived from a deeper reality in which what prevails is unbroken wholeness.

Bohm maintained that relativity and quantum theory are in basic contradiction in these essential respects, and that a new notion of order should begin with that which both point toward: undivided wholeness. This should not be Implicate and Explicate Order according to David Bohm

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taken, however, to imply that he considered such powerful theories should be discarded. Nevertheless, he argued that each was relevant in a certain context - i.e. a set of interrelated conditions within the Explicate Order - rather than having unlimited relevance, and that apparent contradictions stem from attempts to overgeneralize by superposing the theories on one another, implying greater generality or broader relevance than is ultimately warranted. Thus, Bohm (1980, pp. 156-167) argued: "... in sufficiently broad contexts such analytic descriptions cease to be adequate ... ’the law of the whole’ will generally include the possibility of describing the ’loosening’ of aspects from each other, so that they will be relatively autonomous in limited contexts ... however, any form of relative autonomy (and heteronomy) is ultimately limited by holonomy, so that in a broad enough context such forms are seen to be merely aspects, relevated in the holomovement, rather than disjoint and separately existent things in interaction".

Hidden variable quantum theory

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Bohm proposed a hidden variable theory of quantum physics (see Bohm interpretation). According to Bohm, a key motivation for doing so was purely to show the possibility of such theories. On this, Bohm (1980, p. 81) said "... it should be kept in mind that before this proposal was made there had existed the widespread impression that no conceptions of hidden variables at all, not even if they were abstract, and hypothetical, could possibly be consistent with the quantum theory". Bohm (1980, p. 110) also claimed that "the demonstration of the possibility of theories of hidden variables may serve in a more general philosophical sense to remind us of the unreliability of conclusions based on the assumption of the complete universality of certain features of a given theory, however general their domain of validity seems to be". Another aspect of Bohm’s motivation was to point out a confusion he perceived to exist in quantum theory. On the dominant approaches in quantum theory, he said: "...we wish merely to point out that this whole line of approach re-establishes at the abstract level of statistical potentialities the same kind of analysis into separate and autonomous components in interaction that is denied at the more concrete level of individual objects".

Quantum entanglement

Central to Bohm’s schema are correlations between observables of entities which seem separated by great distances in the Explicate Order (such as a particular electron here on earth and an alpha particle in one of the stars in the Abell 1835 galaxy, the farthest galaxy from Earth known to humans), manifestations of the Implicate Order. Within quantum theory there is entanglement of such objects. Implicate and Explicate Order according to David Bohm

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This view of order necessarily departs from any notion which entails signalling, and therefore causality. The correlation of observables does not imply a causal influence, and in Bohm’s schema the latter represents ’relatively’ independent events in space-time; and therefore Explicate Order. He also used the term unfoldment to characterise processes in which the Explicate Order becomes relevant (or "relevated"). Bohm likens unfoldment also to the decoding of a television signal to produce a sensible image on a screen. The signal, screen, and television electronics in this analogy represent the Implicate Order whilst the image produced represents the Explicate Order. He also uses an interesting example in which an ink droplet can be introduced into a highly viscous substance (such as glycerine), and the substance rotated very slowly such that there is negligible diffusion of the substance. In this example, the droplet becomes a thread which, in turn, eventually becomes invisible. However, by rotating the substance in the reverse direction, the droplet can essentially reform. When it is invisible, according to Bohm, the order of the ink droplet as a pattern can be said to be implicate within the substance.

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In another analogy, Bohm asks us to consider a pattern produced by making small cuts in a folded piece of paper and then, literally, unfolding it. Widely separated elements of the pattern are, in actuality, produced by the same original cut in the folded piece of paper. Here the cuts in the folded paper represent the Implicate Order and the unfolded pattern represents the Explicate Order.

The hologram as analogy for the Implicate Order Bohm employed the hologram as a means of characterising Implicate Order, noting that each region of a photographic plate in which a hologram is observable contains within it the whole three-dimensional image, which can be viewed from a range of perspectives. That is, each region contains a whole and undivided image. In Bohm’s words: "There is the germ of a new notion of order here. This order is not to be understood solely in terms of a regular arrangement of objects (eg., in rows) or as a regular arrangement of events (e.g. in a series). Rather, a total order is contained, in some implicit sense, in each region of space and time. Now, the word ’implicit’ is based on the verb ’to implicate’. This means ’to fold inward’ ... so we may be led to explore the notion that in some sense each region contains a total structure ’enfolded’ within it". (Bohm, 1980, p. 149). Bohm noted that although the hologram conveys undivided wholeness, it is nevertheless static. In this view of order, laws represent invariant relationships between explicate entities and structures, and thus Bohm maintained that in physics, the Explicate Order generally reveals itself within well-constructed experimental contexts as, for example, in the sensibly observable results of instruments. With Implicate and Explicate Order according to David Bohm

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Figure 18 In a holographic reconstruction, each region of a photographic plate contains the whole image

respect to Implicate Order, however, Bohm (1980, p. 147) asked us to consider the possibility instead "that physical law should refer primarily to an order of undivided wholeness of the content of description similar to that indicated by the hologram rather than to an order of analysis of such content into separate parts · · ·".

A common grounding for consciousness and matter The Implicate Order represents the proposal of a general metaphysical concept in terms of which it is claimed that matter and consciousness might both be understood, in the sense that it is proposed that both matter and consciousness: (i) enfold the structure of the whole within each region, and (ii) involve continuous processes of enfoldment and unfoldment. For example, in the case of matter, entities such as atoms may represent continuous enfoldment and unfoldment which manifests as a relatively stable and autonomous entity that can be observed to follow a relatively well-defined path in space-time. In the case of consciousness, Bohm pointed toward evidence presented by Karl Pribram

Implicate and Explicate Order according to David Bohm

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Figure 19 Karl Pribram and colleagues have presented evidence which indicates that memories do not in general appear to be localized in specific regions of brains

that memories may be enfolded within every region of the brain rather than being localized (for example in particular regions of the brain, cells, or atoms).

Bohm (1980, p. 205) went on to say: "As in our discussion of matter in general, it is now necessary to go into the question of how in consciousness the Explicate Order is what is manifest ... the manifest content of consciousness is based essentially on memory, which is what allows such content to be held in a fairly constant form. Of course, to make possible such constancy it is also necessary that this content be organized, not only through relatively fixed association but also with the aid of the rules of logic, and of our basic categories of space, time causality, universality, etc. ... there will be a strong background of recurrent stable, and separable features, against which the transitory and changing aspects of the unbroken flow of experience will be seen as fleeting impressions that tend to be arranged and ordered mainly in terms of the vast totality of the relatively static and fragmented content of [memories]". Bohm also claimed that "as with consciousness, each moment has a certain Explicate Order, and in addition it enfolds all the others, though in its own way. So the relationship of each moment in the whole to all the others is implied by its total content: the way in which it ’holds’ all the others enfolded within it". Bohm characterises consciousness as a process in which at each moment, content that was previously implicate is presently explicate, and content which was previously explicate has become implicate. He said: "One may indeed say that our Implicate and Explicate Order according to David Bohm

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memory is a special case of the process described above, for all that is recorded is held enfolded within the brain cells and these are part of matter in general. The recurrence and stability of our own memory as a relatively independent sub-totality is thus brought about as part of the very same process that sustains the recurrence and stability in the manifest order of matter in general. It follows, then, that the explicate and manifest order of consciousness is not ultimately distinct from that of matter in general" (Bohm, 1980, p. 208).

Connections with other works

Many, along with Bohm himself, have seen strong connections between his ideas and ideas from the East. There are particularly strong connections to Buddhism, for which Einstein also shared sympathy. Some proponents of alternative religions (such as shamanism) claim a connection with their belief systems as well.

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Bohm may have known that his idea is a striking analogy to "intensional and extensional aboutness" to which R. A. Fairthorne (1969) insightfully referred information scientists (although a Google search reveals that few paid attention to this suggestion). John Searle treated aboutness and network in his Intentionality (1983), contemporarily with Bohm’s Wholeness (1983)! Searle’s concept of aboutness is in sharp contrast to, and is as odd as Bohm’s idea of wholeness. As the former is to the content, so the latter is to the context as the ultimate determiner of meaning. The holistic view of context, hence another striking analogy of wholeness, was first put forward in The Meaning of Meaning by C. K. Ogden & I. A. Richards (1923), including the literary, psychological, and external. These are respectively analogous to Karl Popper’s world 3, 2, and 1 appearing in his Objective Knowledge (1972 and later ed.). Bohm’s worldview of "undivided wholeness" is contrasted with Popper’s three divided worlds. The direct causality among these and other authorships may be actually evident in the Implicate Order, though apparently not in the Explicate Order in spite of a great deal of reasonable doubt in terms of locality, ethnicity, ideology, academic tendency, and so on. Bohm and Popper favored Einstein above all. Suppose that someone intends to convey a definite thought or story with the following word string: woman, street, crowd, traffic, noise, haste, thief, bag, loss, scream, police, .....

which looks almost non-sensical as a whole. Then, what will happen to us listeners? We have a dictionary, but we cannot simply sum up the meanings of individual words. That "a whole is more than the sum of the parts" is too plain Implicate and Explicate Order according to David Bohm

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a saying. There seems to be no grammar to which the speaker might have conformed. He merely suggests rather than tells the story, which in other words is implied or implicit in the word string. From this awkward symbology we can guess the story with varying accuracies, if we are ready to take risks. In this case, the meaning of such symbology may be said to be connotative, implicit, implicate or intensional, in contrast to denotative, explicit, explicate or extensional. Consult a dictionary for these words. Note that the more context that unfolds, the less uncertainty remains folded. Most importantly, note that interpretation or making sense of Explicate in Implicate Order, that is, aboutness in wholeness or in context is an outstanding analogy as well as the very principle of subject indexing as a prerequisite of information retrieval that has now become an everyday concern. This principle’s actual implication for and impact on a number of other disciplines should be unfolded if any. Why not unfold who on earth played an inspiring or leading role in shaping contextualism in the spotlight?

Figure 20 Cells stained for keratin and DNA: such parts of life exist because of the whole, but also in order to sustain it

Bohm’s views also connect with those of Immanuel Kant in some key respects. For example, Kant held that the parts of an organism, such as cells, simultaneously exist in order to sustain the whole, and depend upon the whole for their own existence and functioning. Also, as noted by Bohm, Kant proposed that the process of thought plays an active role in organizing knowledge, which implies theoretical insights are instrumental to the process of acquiring factual

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knowledge. This perspective is also congruent with an analysis of the function of measurement in physical science by Thomas Kuhn in 1961.

Figure 21 For Bohm, life is a continuous flowing process of enfoldment and unfoldment involving relatively autonomous entities. DNA ’directs’ the environment to form a living thing. Life can be said to be implicate in ensembles of atoms that ultimately form life.

There are also connections to views expressed by Stuart Kauffman, who noted Kant’s perspective on organisms in his book At Home in the Universe in a section given the evocative title An Unrepentant Holism. Kauffman’s concept of an autocatalytic set, as it was originally conceived in terms of molecules, elaborates on Kant’s perspective in terms of modern scientific concepts. In his later book Investigations, Kauffman attempts to define, or at least characterize, the notion of an autonomous agent. If viewed as "relatively autonomous", this concept is potentially congruous with Bohm’s view. Bohm’s views are also echoed in Kauffman’s (2000, p. 137) statement: "... our incapacity to prestate the configuration space of the biosphere is not a failure to prestate the consequences of the primitives, it appears to be a failure to prestate the primitives themselves". Kauffman suggests that such a failure may stem from more generally applicable foundations applicable also within physics. Consistent with Bohm, this potentially calls into question whether we should presuppose that it is possible (even in principle) to formulate a final and complete theory of everything.

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• • • • • • • • • • • •

Brahman Buddhism Holographic principle Immanuel Kant Laminar flow Mind’s eye Noumenon Parable of the cave Plato Samsara Taoism Unobservables

References

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See also

Bohm, D. (1980). Wholeness and the Implicate Order. London: Routledge. ISBN 0-710-00971-2

•

Kauffman, S. (1995). At Home in the Universe. New York: Oxford University Press. hardcover: ISBN 0-19-509599-5, paperback ISBN 0-19-511130-3

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Kauffman, S. (2000). Investigations. New York: Oxford University Press.

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Kuhn, T.S. (1961). The function of measurement in modern physical science. ISIS, 52, 161-193.

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Further reading •

Talbot, M. (1991). The Holographic Universe. Harpercollins

External links

Interview with David Bohm 99 – An interview with Bohm concerning this particular subject matter conducted by F. David Peat. • Excerpt from The Holographic Universe 100 – Parallels some of the experiences of 18th century Swedish mystic, Emanuel Swedenborg, with David Bohm’s ideas. •

99 http://www.fdavidpeat.com/interviews/bohm.htm

100 http://www.soultravel.nu/2004/040907-swedenborg/index.asp

Implicate and Explicate Order according to David Bohm

207 Mohamad Latiff’s Paradigm of Everything 101 - An Essay by a young Bohmenthusiast concerning the Interconnectedness of Everything using the Bohm model using a practical analogy.

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Source: http://en.wikipedia.org/wiki/Implicate_and_Explicate_Order_according_to_David_Bohm Principal Authors: Goethean, Holon, Floorsheim, Togo, Ciprianman

Incompleteness of quantum physics

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Incompleteness of quantum physics is the assertion that the state of a physical system, as formulated by quantum mechanics, does not give a complete description for the system. A complete description is one which uniquely determines the values of all its measurable properties. The existence of indeterminacy for some measurements is a characteristic of quantum mechanics; moreover, bounds for indeterminacy can be expressed in a quantitative form by the Heisenberg uncertainty principle. Incompleteness can be understood in two fundamentally different ways: •

•

QM is incomplete because it is not the "right" theory; the right theory would provide descriptive categories to account for all observable behavior and not leave "anything to chance". QM is incomplete, but it accurately reflects the way nature is.

Incompleteness understood as 1) is now considered highly controversial, since it contradicts the impossibility of a hidden variables theory which is shown by Bell test experiments. There are many variants of 2) which is widely considered to be the more orthodox view of quantum mechanics.

Einstein’s argument for the incompleteness of quantum physics Albert Einstein may have been the first person to carefully point out the radical effect the new quantum physics would have on our notion of physical state. For a historical background of Einstein’s thinking in regard to QM, see Jankiw and Kleppner [2000], although his best known critique was formulated in the EPR thought experiment. See Bell [1964].

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208 According to Fuchs [2002], Einstein developed a very good argument for incompleteness:

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The best [argument of Einstein] was in essence this. Take two spatially separated systems A and B prepared in some entangled quantum state |ψ AB >. By performing the measurement of one or another of two observables on system A alone, one can immediately write down a new state for system B. Either the state will be drawn from one set of states

Incompleteness of quantum physics

209

Reality of incompleteness

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’s argument shows that quantum state is not a complete description of a physical system, according to Fuchs [2002]: Thus one must take it seriously that the new state (either a |φ i B > or |η i B >) represents information about system B. In making a measurement on A, one learns something about B, but that is where the story ends. The state change cannot be construed to be something more physical than that. More particularly, the final state itself for B cannot be viewed as more than a reflection of some tricky combination of one’s initial information and the knowledge gained through the measurement. Expressed in the language of Einstein, the quantum state cannot be a “complete” description of the quantum system.

Although Einstein was one of the first to formulate the necessary incompleteness of quantum physics, he never fully accepted it. In a 1926 letter to Max Born, he made a remark that is now famous: Quantum mechanics is certainly imposing. But an inner voice tells me it is not yet the real thing. The theory says a lot, but does not really bring us any closer to the secret of the Old One. I, at any rate, am convinced that He does not throw dice.

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Einstein was mistaken according to Steven Hawking in Does God Play Dice 102, Einstein’s view was what would now be called, a hidden variable theory. Hidden variable theories might seem to be the most obvious way to incorporate the Uncertainty Principle into physics. They form the basis of the mental picture of the universe, held by many scientists, and almost all philosophers of science. But these hidden variable theories are wrong. The British physicist, John Bell, who died recently, devised an experimental test that would distinguish hidden variable theories. When the experiment was carried out carefully, the results were inconsistent with hidden variables. Thus it seems that even God is bound by the Uncertainty Principle, and can not know both the position, and the speed, of a particle. So God does play dice with the universe. All the evidence points to him being an inveterate gambler, who throws the dice on every possible occasion.

Chris Fuchs [2002] summed up the reality of the necessary incompleteness of information in quantum physics as follows, attributing this idea to Einstein "He [Einstein] was the first person to say in absolutely unambiguous terms why the quantum state should be viewed as information (or, to say the same thing, as a representation of one’s beliefs and gambling commitments, credible or otherwise). Fuchs adds: Incompleteness, it seems, is here to stay: The theory prescribes that no matter how much we know about a quantum system—even when we have maximal information about it—there will always be a statistical residue. There will always be questions that we can ask of a system for which we cannot predict the outcomes. In quantum theory, maximal information is simply not complete information [Caves and Fuchs 1996]. But neither can it be completed.

The kind of information about the physical world that is available to us according to Fuchs [2002] is “the potential consequences of our experimental interventions into nature” which is the subject matter of quantum physics.

Relational Quantum Physics

According to Relational Quantum Physics [Laudisa and Rovelli 2005], the way distinct physical systems affect each other when they interact (and not of the way physical systems "are") exhausts all that can be said about the physical world. The physical world is thus seen as a net of interacting components, where there is no meaning to the state of an isolated system. A physical system (or, more precisely, its contingent state) is described by the net of relations it entertains Incompleteness quantum with the surrounding systems, and the physicalof structure of thephysics world is identified as this net of relationships. In other words, “Quantum physics is the theoretical formalization of the experimental discovery that the descriptions that different observers give of the same events are not universal.”

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Source: http://en.wikipedia.org/wiki/Incompleteness_of_quantum_physics

Interaction picture

In quantum mechanics, the Interaction picture (or Dirac picture) is an intermediate between the →Schrödinger picture and the →Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of observables.

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Warning: Operator equations which hold in the interaction picture don’t necessarily hold in the Schrödinger or the Heisenberg picture. This is because the operators A I, A S and A H are not the same but are related by unitary transformations. Unfortunately, most textbooks and articles omit the subscript which can often lead to confusion and mistakes when an unwary student applies an equation for one picture to another picture.

Switching pictures

To switch into the interaction picture, we divide the Schrödinger picture Hamiltonian into two parts, HS = H0,S + H1,S . Then the state vector is defined as |ψI (t)i = eiH0,S t/~ |ψS (t)i

Operators transform between the pictures as AI = eiH0 t/~ AS e−iH0 t/~ .

The →Schrödinger equation then becomes in this picture: d i~ dt |ψI (t)i = H1,I |ψI (t)i.

This equation is referred to as the Schwinger- Tomonaga equation.

The purpose of this picture is to shunt all the time dependence due to H 0 onto the operators, leaving only H 1, I affecting the time-dependence of the state vectors. The interaction picture is convenient when considering the effect of a small interaction term, H 1, S, being added to the Hamiltonian of a solved system, H 0, S. By switching into the interaction picture, you can use time-dependent perturbation theory to find the effect of H 1, I. Interaction picture

211

References Townsend, John S. (2000). A Modern Approach to Quantum Mechanics, 2nd ed.. Sausalito, CA: University Science Books. ISBN 1891389130.

See also • • •

→Bra-ket notation →Schrödinger equation Haag’s theorem

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Principal Authors: Laurascudder, Phys, Elroch, Charles Matthews, Pjacobi

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Internal conversion

This article is about the nuclear process. For the chemical process, see Internal conversion (chemistry).

Internal conversion is a radioactive decay process where an excited nucleus interacts with an electron in one of the lower electron shells causing the electron to be emitted. This is not to be confused with a photoelectric effect where a photon emitted from the nucleus interacts with the electron. The internal conversion process is not actually the photoelectric ejection of an atomic electron, as the nucleus does not actually emit a gamma ray in the first place in this process. What happens is that the wavefunction of an inner shell electron penetrates the nucleus (ie there is a finite probability of the electron being found in the nucleus) and when this is the case the electron takes the energy of the nuclear transition without an intermediary gamma ray being produced. The energy of the emitted electron is equal to the transition energy minus the binding energy of the electron. Most internal conversion electrons come from the K shell as this electron has the highest probability of being found inside the nucleus. After the electron has been emitted, the atom is left with a vacancy in one of the inner electron shells. This hole will be filled with an electron from one of the higher shells and subsequently a characteristic x-ray or →Auger electron will be emitted. Internal conversion

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Internal conversion is favoured when the energy gap between nuclear levels is small, and is also the only mode of de-excitation for 0+ -> 0+ (i.e. E0) transitions. It is the predominant mode of de-excitation whenever the initial and final spin states are the same, but the multi-polarity rules for nonzero initial and final spin states do not necessarily forbid the emission of a gamma ray in such a case.

The tendency towards internal conversion can be determined by the internal conversion coefficient, which is empirically determined by the ratio of deexcitations that go by the emission of electrons to those that go by gamma emission.

References

Krane, Kenneth S. (1988). Introductory Nuclear Physics. J. Wiley & Sons. ISBN 0-471-80553-X.

External links HyperPhysics 107

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Internal conversion coefficient

Source: http://en.wikipedia.org/wiki/Internal_conversion

Principal Authors: Sunborn, GangofOne, Jpau, AjAldous, Bensaccount

Interpretation of quantum mechanics

An interpretation of quantum mechanics is an attempt to answer the question, What exactly is quantum mechanics talking about? The question has its historical roots in the nature of quantum mechanics itself which was considered as a radical departure from previous physical theories. However, quantum mechanics has been described as "the most precisely tested and most successful theory in the history of science" (c.f. Jackiw and Kleppner, 2000.)

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Historical background

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The operational meaning of the technical terms used by researchers in quantum theory (such as wavefunctions and matrix mechanics) progressed through various intermediate stages. For instance Schrödinger originally viewed the wavefunction associated to the electron as the charge density of an object smeared out over an extended, possibly infinite, volume of space. Max Born later proposed its interpretation as the probability distribution in the space of the electron’s position. Other leading scientists, such as Albert Einstein, had great difficulty in accepting some of the more radical consequences of the theory, such as quantum indeterminacy. Even if these matters could be treated as ’teething troubles’, they have lent importance to the activity of interpretation.

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It should not, however, be assumed that most physicists consider quantum mechanics as requiring interpretation, other than very minimal instrumentalist interpretations, which are discussed below. The Copenhagen interpretation, as of 2006, appears to be the most popular one among scientists, followed by the many worlds and consistent histories interpretations. But it is also true that most physicists consider non-instrumental questions (in particular ontological questions) to be irrelevant to physics. They fall back on Paul Dirac’s point of view, later expressed in the famous dictum: "Shut up and calculate" often (perhaps erroneously) attributed to Richard Feynman (see 108).

Obstructions to direct interpretation The perceived difficulties of interpretation reflect a number of points about the orthodox description of quantum mechanics, including: • • • •

The abstract, mathematical nature of the description of quantum mechanics. The existence of what appear to be non-deterministic and irreversible processes in quantum mechanics. The phenomenon of entanglement, and in particular, the higher correlations between remote events than would be expected in classical theory. The complementarity of possible descriptions of reality.

First, the accepted mathematical structure of quantum mechanics is based on fairly abstract mathematics, such as →Hilbert spaces and operators on those Hilbert spaces. In classical mechanics and electromagnetism, on the other hand, properties of a point mass or properties of a field are described by real numbers or functions defined on two or three dimensional sets. These have

108 http://www.physicstoday.org/vol-57/iss-5/p10.html

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214 direct, spatial meaning, and in these theories there seems to be less need to provide a special interpretation for those numbers or functions.

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Further, the process of measurement plays an apparently essential role in the theory. It relates the abstract elements of the theory, such as the wavefunction, to operationally definable values, such as probabilities. Measurement interacts with the system state, in somewhat peculiar ways, as is illustrated by the double-slit experiment. The mathematical formalism used to describe the time evolution of a nonrelativistic system proposes two somewhat different kinds of transformations:

Reversible transformations described by unitary operators on the state space. These transformations are determined by solutions to the →Schrödinger equation.

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Non-reversible and unpredictable transformations described by mathematically more complicated transformations (see quantum operations). Examples of these transformations are those that are undergone by a system as a result of measurement.

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A restricted version of the problem of interpretation in quantum mechanics consists in providing some sort of plausible picture, just for the second kind of transformation. This problem may be addressed by purely mathematical reductions, for example by the many-worlds or the consistent histories interpretations. In addition to the unpredictable and irreversible character of measurement processes, there are other elements of quantum physics that distinguish it sharply from classical physics and which cannot be represented by any classical picture. One of these is the phenomenon of entanglement, as illustrated in the EPR paradox, which seemingly violates principles of local causality.

Another obstruction to direct interpretation is the phenomenon of complementarity, which seems to violate basic principles of propositional logic. Complementarity says there is no logical picture (obeying classical propositional logic) that can simultaneously describe and be used to reason about all properties of a quantum system S. This is often phrased by saying that there are "complementary" sets A and B of propositions that can describe S, but not at the same time. Examples of A and B are propositions involving a wave description of S and a corpuscular description of S. The latter statement is one part of Niels Bohr’s original formulation, which is often equated to the principle of complementarity itself.

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Complementarity is not usually taken to mean that classical logic fails, although Hilary Putnam did take that view in his paper Is logic empirical?. Instead complementarity means that composition of physical properties for S (such as position and momentum both having values in certain ranges) using propositional connectives does not obey rules of classical propositional logic. As is now well-known (Omnès, 1999) the "origin of complementarity lies in the noncommutativity of operators" describing observables in quantum mechanics.

Problematic status of pictures and interpretations

The precise ontological status, of each one of the interpreting pictures, remains a matter of philosophical argument.

In other words, if we interpret the formal structure X of quantum mechanics by means of a structure Y (via a mathematical equivalence of the two structures), what is the status of Y ? This is the old question of saving the phenomena, in a new guise.

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Some physicists, for example Asher Peres and Chris Fuchs, seem to argue that an interpretation is nothing more than a formal equivalence between sets of rules for operating on experimental data. This would suggest that the whole exercise of interpretation is unnecessary.

Instrumentalist interpretation

Any modern scientific theory requires at the very least an instrumentalist description which relates the mathematical formalism to experimental practice and prediction. In the case of quantum mechanics, the most common instrumentalist description is an assertion of statistical regularity between state preparation processes and measurement processes. That is, if a measurement of a real-valued quantity is performed many times, each time starting with the same initial conditions, the outcome is a well-defined probability distribution over the real numbers; moreover, quantum mechanics provides a computational instrument to determine statistical properties of this distribution, such as its expectation value. Calculations for measurements performed on a system S postulate a →Hilbert space H over the complex numbers. When the system S is prepared in a pure state, it is associated with a vector in H. Measurable quantities are associated with Hermitian matrices acting on H : these are referred to as observables. Repeated measurement of an observable A for S prepared in state ψ yields a distribution of values. The expectation value of this distribution is given by the expression

Interpretation of quantum mechanics

216 hψ|A|ψi.

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This mathematical machinery gives a simple, direct way to compute a statistical property of the outcome of an experiment, once it is understood how to associate the initial state with a vector, and the measured quantity with an observable (that is, a specific Hermitian matrix).

As an example of such a computation, the probability of finding the system in a given state |φi is given by computing the expectation value of a (rank-1) projection operator Π = |φihφ|

The probability is then the non-negative real number given by P = hψ|Π|ψi = |hψ|φi|2 .

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By abuse of language, the bare instrumentalist description can be referred to as an interpretation, although this usage is somewhat misleading since instrumentalism explicitly avoids any explanatory role; that is, it does not attempt to answer the question of what quantum mechanics is talking about.

Summary of common interpretations of QM Properties of interpretations

An interpretation can be characterized by whether it satisfies certain properties, such as: • • • •

Realism Completeness Local realism Determinism

To explain these properties, we need to be more explicit about the kind of picture an interpretation provides. To that end we will regard an interpretation as a correspondence between the elements of the mathematical formalism M and the elements of an interpreting structure I, where: •

The mathematical formalism consists of the Hilbert space machinery of ketvectors, self-adjoint operators acting on the space of ket-vectors, unitary time dependence of ket-vectors and measurement operations. In this context a measurement operation can be regarded as a transformation which

Interpretation of quantum mechanics

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•

carries a ket-vector into a probability distribution on ket-vectors. See also quantum operations for a formalization of this concept. The interpreting structure includes states, transitions between states, measurement operations and possibly information about spatial extension of these elements. A measurement operation here refers to an operation which returns a value and results in a possible system state change. Spatial information, for instance would be exhibited by states represented as functions on configuration space. The transitions may be non-deterministic or probabilistic or there may be infinitely many states. However, the critical assumption of an interpretation is that the elements of I are regarded as physically real.

In this sense, an interpretation can be regarded as a semantics for the mathematical formalism. In particular, the bare instrumentalist view of quantum mechanics outlined in the previous section is not an interpretation at all since it makes no claims about elements of physical reality.

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The current use in physics of "completeness" and "realism" is often considered to have originated in the paper (Einstein et al., 1935) which proposed the EPR paradox. In that paper the authors proposed the concept "element of reality" and "completeness" of a physical theory. Though they did not define "element of reality", they did provide a sufficient characterization for it, namely a quantity whose value can be predicted with certainty before measuring it or disturbing it in any way. EPR define a "complete physical theory" as one in which every element of physical reality is accounted for by the theory. In the semantic view of interpretation, an interpretation of a theory is complete if every element of the interpreting structure is accounted for by the mathematical formalism. Realism is a property of each one of the elements of the mathematical formalism; any such element is real if it corresponds to something in the interpreting structure. For instance, in some interpretations of quantum mechanics (such as the many-worlds interpretation) the ket vector associated to the system state is assumed to correspond to an element of physical reality, while in others it does not. Determinism is a property characterizing state changes due to the passage of time, namely that the state at an instant of time in the future is a function of the state at the present (see time evolution). It may not always be clear whether a particular interpreting structure is deterministic or not, precisely because there may not be a clear choice for a time parameter. Moreover, a given theory may have two interpretations, one of which is deterministic, and the other not. Local realism has two parts:

Interpretation of quantum mechanics

218 •

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The value returned by a measurement corresponds to the value of some function on the state space. Stated in another way, this value is an element of reality; • The effects of measurement have a propagation speed not exceeding some universal bound (e.g., the speed of light). In order for this to make sense, measurement operations must be spatially localized in the interpreting structure. A precise formulation of local realism in terms of a local hidden variable theory was proposed by John Bell. Bell’s theorem and its experimental verification restrict the kinds of properties a quantum theory can have. For instance, Bell’s theorem implies quantum mechanics cannot satisfy local realism.

Consistent histories

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The consistent histories generalizes the conventional Copenhagen interpretation and attempts to provide a natural interpretation of quantum cosmology. The theory is based on a consistency criterion that then allows the history of a system to be described so that the probabilities for each history obey the additive rules of classical probability while being consistent with the →Schrödinger equation. According to this interpretation, the purpose of a quantum-mechanical theory is to predict probabilities of various alternative histories.

Many worlds

The many-worlds interpretation (or MWI) is an interpretation of quantum mechanics that rejects the non-deterministic and irreversible wavefunction collapse associated with measurement in the Copenhagen interpretation in favor of a description in terms of quantum entanglement and reversible time evolution of states. The phenomena associated with measurement are explained by decoherence which occurs when states interact with the environment. As result of the decoherence the world-lines of macroscopic objects repeatedly split into mutally unobservable, branching histories – distinct universes within a greater multiverse.

The Copenhagen Interpretation

The Copenhagen interpretation is an interpretation of quantum mechanics formulated by Niels Bohr and Werner Heisenberg while collaborating in Copenhagen around 1927. Bohr and Heisenberg extended the probabilistic interpretation of the wavefunction, proposed by Max Born. The Copenhagen interpretation rejects questions like "where was the particle before I measured Interpretation of quantum mechanics

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Quantum Logic

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its position" as meaningless. The act of measurement causes an instantaneous "collapse of the wave function". This means that the measurement process randomly picks out exactly one of the many possibilities allowed for by the state’s wave function, and the wave function instantaneously changes to reflect that pick.

Quantum logic can be regarded as a kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. This research area and its name originated in the 1936 paper by Garrett Birkhoff and John von Neumann, who attempted to reconcile some of the apparent inconsistencies of classical boolean logic with the facts related to measurement and observation in quantum mechanics.

The Bohm interpretation

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The Bohm interpretation of quantum mechanics is an interpretation postulated by David Bohm in which the existence of a non-local universal wavefunction allows distant particles to interact instantaneously. The interpretation generalizes Louis de Broglie’s pilot wave theory from 1927, which posits that both wave and particle are real. The wave function ’guides’ the motion of the particle, and evolves according to the Schrödinger equation. The interpretation assumes a single, nonsplitting universe (unlike the Everett many-worlds interpretation) and is deterministic (unlike the Copenhagen interpretation). It says the state of the universe evolves smoothly through time, without the collapsing of wavefunctions when a measurement occurs, as in the Copenhagen interpretation. However, it does this by assuming a number of hidden variables, namely the positions of all the particles in the universe, which, like probability amplitudes in other interpretations, can never be measured directly.

Transactional interpretation

The transactional interpretation of quantum mechanics (TIQM) by John Cramer is an unusual interpretation of quantum mechanics that describes quantum interactions in terms of a standing wave formed by retarded (forward-in-time) and advanced (backward-in-time) waves. The author argues that it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and resolves various quantum paradoxes.

Consciousness causes collapse

Consciousness causes collapse is the speculative theory that observation by a conscious observer is responsible for the wavefunction collapse. It is an attempt Interpretation of quantum mechanics

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to solve the Wigner’s friend paradox by simply stating that collapse occurs at the first "conscious" observer. Supporters claim this is not a revival of substance dualism, since (in a ramification of this view) consciousness and objects are entangled and cannot be considered as distinct. The consciousness causes collapse theory can be considered as a speculative appendage to almost any interpretation of quantum mechanics and most physicists reject it as unverifiable and introducing unnecessary elements into physics.

Relational Quantum Mechanics

According to the Stanford Encyclopedia of Philosophy, Relational Quantum Mechanics is an interpretation of Quantum mechanics which:

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discards the notions of absolute state of a system, absolute value of its physical quantities, or absolute event. The theory describes only the way systems affect each other in the course of physical interactions. State and physical quantities refer always to the interaction, or the relation, between two systems. Nevertheless, the theory is assumed to be complete. The physical content of quantum theory is understood as expressing the net of relations connecting all different physical systems. (Stanford Encyclopedia of Philosophy, Federico Laudisa, Università degli Studi di Milano-Bicocca)

Modal Interpretations of Quantum Theory

Modal interpretations of Quantum mechanics were first conceived of in 1972 by B. van Fraassen, in his paper “A formal approach to the philosophy of science.” However, this term now is used to describe a larger set of models that grew out of this approach. The Stanford Encyclopedia of Philosophy describes several versions: • • •

The Copenhagen Variant Kochen-Dieks-Healey Interpretations Motivating Early Modal Interpretations, based on the work of R. Clifton, M. Dickson and J. Bub.

Comparison

At the moment, there is no experimental evidence that would allow us to distinguish between the various interpretations listed below. To that extent, the physical theory stands, and is consistent with, itself and with reality; troubles come only when one attempts to "interpret" it. Nevertheless, there is active Interpretation of quantum mechanics

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research in attempting to come up with experimental tests which would allow differences between the interpretations to be experimentally tested. Some of the most common interpretations are summarized here (however, the assignment of values in this table is not without controversy, for the precise meanings of some of the concepts involved are unclear and, in fact, the subject of the very controversy itself): Deterministic? Waveform Real? Unique History? Avoids Hidden Variables? No

Copenhagen interpretation (Waveform real)

No

Yes

Consistent histories (Decoherent approach)

Agnostic 1

Agnostic 1

Yes

Yes

No

Yes

Yes

No

No

Yes

Yes

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Copenhagen No interpretation (Waveform not real)

Avoids Collapsing Wavefunctions?

Many-worlds interpretation (Decoherent approach)

Yes

Yes

No

Yes

Yes

Bohm-de Broglie interpretation ("Pilot-wave" approach)

Yes

Yes 2

Yes 3

No

Yes

Transactional interpretation

No

Yes

Yes

Yes

No

Consciousness No causes collapse

Yes

Yes

Yes

No

1

If wavefunction is real then this becomes the Many-Worlds Interpretation. If wavefunction less than real, but more than just information, then Zurek calls this the Existential Interpretation. 2 Both particle AND guiding wavefunction are real. 3 Unique particle history, but multiple wave histories. Each interpretation has many variants. It is difficult to get a precise definition of the Copenhagen Interpretation — in the table above, two variants are shown — one that regards the waveform as being a tool for calculating probabilities only, and the other regards the waveform as an "element of reality".

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See also

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Afshar experiment Bell’s theorem Bohm interpretation Bohr-Einstein debates Consistent Histories Copenhagen interpretation Many-minds interpretation Many-worlds interpretation Measurement problem →Penrose Interpretation Philosophical interpretation of classical physics Quantum computation →Quantum indeterminacy →Quantum mechanics Quantum metaphysics Transactional interpretation Wavefunction collapse

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• • • • • • • • • • • • • • • • •

Related lists • •

List of physics topics Unsolved problems in physics

References •

Bub, J. and Clifton, R. 1996. “A uniqueness theorem for interpretations of quantum mechanics,” Studies in History and Philosophy of Modern Physics, 27B, 181-219 • R. Carnap, The interpretation of physics, Foundations of Logic and Mathematics of the International Encyclopedia of Unified Science, University of Chicago Press, 1939. • D. Deutsch, The Fabric of Reality, Allen Lane, 1997. Though written for general audiences, in this book Deutsch argues forcefully against instrumentalism. • Dickson, M. 1994. Wavefunction tails in the modal interpretation, Proceedings of the PSA 1994, Hull, D., Forbes, M., and Burian, R. (eds), Vol. 1, pp. 366-376. East Lansing, Michigan: Philosophy of Science Association.

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• • • • • • •

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Dickson, M. and Clifton, R. 1998. Lorentz-invariance in modal interpretations The Modal Interpretation of Quantum Mechanics, Dieks, D. and Vermaas, P. (eds), pp. 9-48. Dordrecht: Kluwer Academic Publishers A. Einstein, B. Podolsky and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? Phys. Rev. 47 777, 1935. C. Fuchs and A. Peres, Quantum theory needs no ‘interpretation’ , Physics Today, March 2000. Christopher Fuchs, Quantum Mechanics as Quantum Information (and only a little more), arXiv:quant-ph/0205039 v1, (2002) N. Herbert. Quantum Reality: Beyond the New Physics, New York: Doubleday, ISBN 0385235690, LoC QC174.12.H47 1985. R. Jackiw and D. Kleppner, One Hundred Years of Quantum Physics, Science, Vol. 289 Issue 5481, p893, August 2000. M. Jammer, The Conceptual Development of Quantum Mechanics. New York: McGraw-Hill, 1966. M. Jammer, The Philosophy of Quantum Mechanics. New York: Wiley, 1974. W. M. de Muynck, Foundations of quantum mechanics, an empiricist approach, Dordrecht: Kluwer Academic Publishers, 2002, ISBN 1-4020-09321 R. Omnès, Understanding Quantum Mechanics, Princeton, 1999. K. Popper, Conjectures and Refutations, Routledge and Kegan Paul, 1963. The chapter "Three views Concerning Human Knowledge", addresses, among other things, the instrumentalist view in the physical sciences. H. Reichenbach, Philosophic Foundations of Quantum Mechanics, Berkeley: University of California Press, 1944. M. Tegmark and J. A. Wheeler, 100 Years of Quantum Mysteries", Scientific American 284, 68, 2001. van Fraassen, B. 1972. A formal approach to the philosophy of science, in Paradigms and Paradoxes: The Philosophical Challenge of the Quantum Domain, Colodny, R. (ed.), pp. 303-366. Pittsburgh: University of Pittsburgh Press. J. A. Wheeler and H. Z. Wojciech (eds), Quantum Theory and Measurement, Princeton: Princeton University Press, ISBN 0691083169, LoC QC174.125.Q38 1983.

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Willem M. de Muynck 109 Broad overview, realist vs. empiricist interpretations, against oversimplified view of the measurement process Comparative interpretations 110 Skeptical View of "New Age" Interpretations of QM 111 The many worlds of quantum mechanics 112 Erich Joos’ Decoherence Website 113 Basic and indepth information on decoherence Quantum Mechanics for Philosophers 114 An argument for the superiority of the Bohm interpretation. Many-Worlds Interpretation of Quantum Mechanics 115 Numerous Many Worlds-related Topics and Articles 116 Relational Quantum Mechanics 117

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Introduction to quantum mechanics This article is intended as a general, non-technical introduction. For the proper encyclopedia article, please see →Quantum mechanics.

109 http://www.phys.tue.nl/ktn/Wim/muynck.htm#quantum 110 http://members.aol.com/jmtsgibbs/Interpretation.htm 111 http://www.csicop.org/si/9701/quantum-quackery.html 112 http://www.sankey.ws/qm.html 113 http://www.decoherence.de 114 http://home.sprynet.com/~owl1/qm.htm

115 http://plato.stanford.edu/entries/qm-manyworlds/ 116 http://www.station1.net/DouglasJones/many.htm 117 http://plato.stanford.edu/entries/qm-relational/

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Quantum mechanics is a physical science dealing with the behaviour of matter and waves on the scale of atoms and subatomic particles. It also forms the basis for the contemporary understanding of how large objects such as stars and galaxies, and cosmological events such as the Big Bang, can be analyzed and explained. Its acceptance by the general physics community is due to its accurate prediction of the physical behaviour of systems, including systems where Newtonian mechanics fails. This difference between the success of classical and quantum mechanics is most often observed in systems at the atomic scale or smaller, or at very low or very high energies, or at extremely low temperatures. Quantum mechanics is the basis of modern developments in chemistry, molecular biology, and electronics, and the foundation for the technology that has transformed the world in the last fifty years.

Background

Through a century of experimentation and applied science, quantum mechanical theory has proven to be very successful and practical. The term "quantum mechanics" was first coined by Max Born in 1924. Quantum mechanics is the

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Despite the success of quantum mechanics, it does have some controversial elements. For example, the behaviour of microscopic objects described in quantum mechanics is very different from our everyday experience, which may provoke an incredulous reaction. Moreover, some of the consequences of quantum mechanics appear to be inconsistent with the consequences of other successful theories, such as Einstein’s Theory of Relativity, especially general relativity. Some of the background of quantum mechanics dates back to the early 1800’s, but the real beginnings of quantum mechanics date from the work of Max Planck in 1900. Albert Einstein, Niels Bohr, and Louis de Broglie soon made important contributions. However, it was not until the mid-1920’s that a more complete picture emerged, and the true importance of quantum mechanics became clear. Some of the most prominent scientists to contribute were Max Born, Paul Dirac, Werner Heisenberg, Wolfgang Pauli, and Erwin Schrödinger [#endnote_Schrödinger1].

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Later, the field was further expanded with work by Julian Schwinger, Murray Gell-Mann, and Richard Feynman, in particular, with the development of Quantum Electrodynamics in 1947.

Early researchers differed in their explanations of the fundamental nature of what we now call electromagnetic radiation. In 1690, Christian Huygens explained the laws of reflection and refraction on the basis of a wave theory. Sir Isaac Newton believed that light consisted of particles which he designated corpuscles. In 1827 Thomas Young and Augustin Fresnel made experiments on interference that showed that a corpuscular theory of light was inadequate. Then in 1873 James Clerk Maxwell showed that by making an electrical circuit oscillate it should be possible to produce electromagnetic waves. His theory made it possible to compute the speed of electromagnetic radiation purely on the basis of electrical and magnetic measurements, and the computed value corresponded very closely to the empirically measured speed of light. In 1888, Heinrich Hertz made an electrical device that actually produced what we would now call microwaves — essentially radiation at a lower frequency than visible light. Everything up to that point suggested that Newton had been entirely wrong to regard light as corpuscular. Then it was discovered that when light strikes an electrical conductor it causes electrons to move away from their original positions, and, furthermore, the phenomenon observed could only be explained if the light delivered energy in definite packets. In a photoelectric device such as the light meter in a camera, when light hits the metallic detector electrons are caused to move. Greater intensities of light at one frequency can cause more electrons to move, but they will not move any faster. In contrast, Introduction to quantum mechanics

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Figure 22 The interference that produces colored bands on bubbles cannot be explained by a model that depicts light as a particle. It can be explained by a model that depicts it as a wave. The drawing shows sine waves that resemble waves on the surface of water being reflected from two surfaces of a film of varying width, but that depiction of the wave nature of light is only a crude analogy.

higher frequencies of light can cause electrons to move faster. So intensity of light controls the amperes of current produced, but frequency of light controls the voltage produced. This appeared to raise a contradiction when compared to sound waves and ocean waves, where only intensity was needed to predict the energy of the wave. In the case of light, frequency appeared to predict energy. Something was needed to explain this phenomenon and also to reconcile experiments that had shown light to have a particle nature with experiments that had shown it to have a wave nature.

Spectroscopy and Onward

It is fairly easy to see a spectrum produced by white light when it passes through a prism, the beveled edge of a mirror or a special pane of glass, or through drops of rain to form a rainbow. When samples of single elements are caused to emit light they may emit light at several characteristic frequencies. The frequency profile produced is characteristic of that element. Instead of there being a wide band filled with colors from violet to red, there will be isolated bands of single colors separated by darkness. Such a display is called Introduction to quantum mechanics

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Figure 23

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a line spectrum. Some lines go beyond the visible frequencies and can only be detected by special photographic film or other such devices. Scientists hypothesized that an atom could radiate light the way the string on a fine violin radiates sound – not only with a fundamental frequency (in which the entire string moves the same way at once) but with several higher harmonics (formed when the string divides itself into halves and other divisions that vibrate in coordination with each other as when one half of the string is going one way as the other half of the string is going the opposite way). For a long time nobody could find a mathematical way to relate the frequencies of the line spectrum of any element.

NASA photo of the bright-line spectrum of hydrogen

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In 1885, Johann Jakob Balmer (1825-1898) figured out how the frequencies of atomic are related to each other. The formula is a simple one:  hydrogen  1 1 1 L = R 4 − n2 where L is wavelength, teger (2, 3,· · ·n).

R is Rydberg’s constant and n is an in-

This formula can be generalized to apply to atoms that are more complicated than hydrogen, but we will stay with hydrogen for this general exposition. (That is the reason that the denominator in the first fraction is expressed as a square.)

The next development was the discovery of the →Zeeman effect, named after Pieter Zeeman (1865-1943). The physical explanation of the Zeeman effect was worked out by Hendrik Anton Lorentz (1853-1928). Lorentz hypothesized that the light emitted by hydrogen was produced by vibrating electrons. It was possible to get feedback on what goes on within the atom because moving electrons create a magnetic field and so can be influenced by the imposition of an external magnetic field in a manner analogous to the way that one iron magnet will attract or repel another magnet. The Zeeman effect could be interpreted to mean that light waves are originated by electrons vibrating in their orbits, but classical physics could not explain why electrons should not fall out of their orbits and into the nucleus of their atoms, nor could classical physics explain why their orbits would be such as to produce the series of frequencies derived by Balmer’s formula and displayed in the line spectra. Why did the electrons not produce a continuous spectrum? Introduction to quantum mechanics

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Old quantum theory

Planck’s constant

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Quantum mechanics developed from the study of electromagnetic waves through spectroscopy which includes visible light seen in the colors of the rainbow, but also other waves including the more energetic waves like ultraviolet light, x-rays, and gamma rays plus the waves with longer wavelengths including infrared waves, microwaves and radio waves. We are not, however, speaking of sound waves, but only of those waves that travel at the speed of light. Also, when the word "particle" is used below, it always refers to elementary or subatomic particles.

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Classical physics predicted that a black-body radiator would produce infinite energy, but that result was not observed in the laboratory. If black-body radiation was dispersed into a spectrum, then the amount of energy radiated at various frequencies rose from zero at one end, peaked at a frequency related to the temperature of the radiating object, and then fell back to zero. In 1900, Max Planck developed an empirical equation that could account for the observed energy curves, but he could not harmonize it with classical theory. He concluded that the classical laws of physics do not apply on the atomic scale as had been assumed. In this theoretical account, Planck allowed all possible frequencies, all possible wavelengths. However, he restricted the energy that is delivered. "In classical physics,... the energy of a given oscillator depends merely on its amplitude, and this amplitude is subject to no restriction." But, according to Planck’s theory, the energy emitted by an oscillator is strictly proportional to its frequency. The higher the frequency, the greater the energy. To reach this theoretical conclusion, he made an assumption about the inner structure of black-body radiators:

He postulated that a radiating body consisted of an enormous number of elementary oscillators, some vibrating at one frequency and some at another, with all frequencies from zero to infinity being represented....The energy E of any one oscillator was not permitted to take on any arbitrary value, but was proportional to some integral multiple of the frequency f of the oscillator. That is, E = nhf,

where n = 1, 2, 3,... etc. The proportionality constant h is called Planck’s constant.One of the most direct applications is finding the energy of photons. If you know h (Planck’s constant), and you know the frequency of the photon, Introduction to quantum mechanics

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then you can calculate the energy of the photons. For instance, if a beam of light illuminated a target, and the light frequency was 540 × 10 12 hertz, then the energy of each photon would be h × 540 × 10 12 hertz. The value of h itself is exceedingly small, about 6.6260693 x 10 -34 joule seconds in scientific notation). This means that the photons in the beam of light have a energy of about 3.58 × 10−19 Joules which is approximately 2.23 eV. When you describe the energy of a wave in this manner, it seems that the wave is carrying its energy in a certain number of little packets per second. This discovery then seemed to remake the wave into a particle. These packets of energy carried along with the wave were called quanta by Planck. Quantum mechanics began with the discovery that energy is delivered in packets whose size is related to the frequencies of all electromagnetic waves (and to the color of visible light since in that case frequency determines color). Be aware, however, the descriptions in terms of wave and particle import macro world concepts into the quantum world, where they have only provisional relevance or appropriateness.

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In early research on light, there were two competing ways to describe light, either as a wave propagated through empty space, or as small particles traveling in straight lines. Because Planck showed that the energy of the wave is made up of packets, the particle analogy became favored to help understand how light delivers energy in multiples of certain set values designated as quanta of energy. Nevertheless, the wave analogy is also indispensable for helping to understand other light phenomena. In 1905, Albert Einstein used Planck’s constant to postulate that the energy in a beam of light occurs in concentrations that he called photons. According to that account, a single photon of a given frequency delivers an invariant amount of energy. In other words, individual photons can deliver more or less energy, but only depending on their frequencies. Although the description that stemmed from Planck’s research sounds like Newton’s corpuscular account, Einstein’s photon was still said to have a frequency, and the energy of the photon was accounted proportional to that frequency. The particle account had been compromised once again. Both the idea of a wave and the idea of a particle are models derived from our everyday experience. We cannot see photons. We can only investigate their properties indirectly. We look at some phenomena, such as the rainbow of colors that we see when a thin film of oil rests on the surface of a puddle of water, and we can explain that phenomenon to ourselves by comparing light to waves. We look at other phenomena, such as the way a photoelectric meter in our camera works, and we explain it by analogy to particles colliding with the detection screen in the meter. In both cases we take concepts from our everyday experience and apply them to a world we have never seen. Introduction to quantum mechanics

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Neither form of explanation is entirely satisfactory. In general any model can only approximate that which it models. A model is useful only within the range of conditions where it is able to predict the real thing with accuracy. Newtonian physics is still a good predictor of many of the phenomena in our everyday life. To remind us that both "wave" and "particle" are concepts imported from our macro world to explain the world of atomic-scale phenomena, some physicists such as George Gamow have used the term "wavicle" to refer to whatever it is that is really there. In the following discussion, "wave" and "particle" may both be used depending on which aspect of quantum mechanical phenomena is under discussion.

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Reduced Planck’s constant

Figure 24 Relation between a cycle and a wave; half of a circle describes half of the cycle of a wave

Planck’s constant originally represented the energy that a light wave carries as a function of its frequency. A step in the development of this concept appeared in Bohr’s work. Bohr was using a "planetary" or particle model of the electron, and could not understand why a 2π factor was essential to his experimentally derived formulae. Later, de Broglie postulated that electrons have frequencies, just as do photons, and that the frequency of an electron must conform to the conditions for a standing wave that can exist in a certain orbit. That is to say, the beginning of one cycle of a wave at some point on the circumference of a circle (since that is what an orbit is) must coincide with the end of some cycle. There can be no gap, no length along the circumference that is not participating Introduction to quantum mechanics

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in the vibration, and there can be no overlap of cycles. So the circumference of the orbit, C, must equal the wavelength, , of the electron multiplied by some positive integer (n = 1, 2, 3...). Knowing the circumference one can calculate wavelengths that fit that orbit, and knowing the radius, r, of the orbit one can calculate its circumference. To put all that in mathematical form, C = 2π*r = n* and so = 2πr/n and the appearance of the 2π factor is seen to occur simply because we need it to calculate possible wavelengths (and therefore possible frequencies) when we already know the radius of an orbit.

Again in 1925 when Werner Heisenberg developed his full quantum theory, calculations involving wave analysis called Fourier series were fundamental, and so the "reduced" version of Planck’s constant (h/2π) became invaluable because it includes a conversion factor to facilitate calculations involving wave analysis. Finally, when this reduced Planck’s constant appeared naturally in Dirac’s equation it was then given an alternate designation, "Dirac’s constant." Therefore, it is appropriate to begin with an explanation of what this constant is, even though we haven’t yet touched on the theories that made its use convenient.

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As noted above, the energy of any wave is given by its frequency multiplied by Planck’s constant. A wave is made up of crests and troughs. In a wave, a cycle is defined by the return from a certain position to the same position such as from the top of one crest to the next crest. A cycle actually is mathematically related to a circle, and both have 360 degrees. A degree is a unit of measure for the amount of turn needed to produce an arc of a certain length at a given distance. A sine curve is generated by a point on the circumference of a circle as that circle rotates. (See a demonstration at: Rotation Applet 118) There are 2 π radians per cycle in a wave, which is mathematically related to the way a circle has 360◦ (which are equal to two π radians). (A radian is simply the angle you would get if you measured a distance along the circumference of the circle equal to the radius of the circle, and then drew lines to the center of the circle and looked at the angle thus formed.) Since one cycle is 2 π radians, when h is divided by 2 π the two "2 π" factors will cancel out leaving just the radian to contend with. So, dividing h by 2 π describes a constant that, when multiplied by the frequency of a wave, gives the energy in joules per radian. h And h/2 π is h-bar or ~ = 2π . The reduced Planck’s constant is written in mathematical formulas as ~, and is read as "h-bar". The reduced Planck’s constant allows computation of the energy of a wave in units per radian instead of in units per cycle. These two constants h and h-bar are merely conversion factors between energy units and

118 http://www.math.utah.edu/~cherk/ccli/bob/Rotation/sin12.swf

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Bohr atom

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frequency units. The reduced Planck’s constant is used more often than h (Planck’s constant) alone in quantum mechanical mathematical formulas for many reasons, one of which is that angular velocity or angular frequency is ordinarily measured in radians per second so using h-bar that works in radians too will save a computation to put radians into degrees or vice-versa. Also, when equations relevant to those problems are written in terms of ~, the frequently occurring 2 π factors in numerator and denominator can cancel out, saving a computation. However, in other cases, as in the orbits of the Bohr atom, h/π was obtained naturally for the angular momentum of the orbits. Another expression for the relation between energy and wave length is given in electron volts for energy and angstroms for wavelength: E photon (eV) = 12,400/(Å) – it appears not to involve h at all, but that is only because a different system of units has been used and now, numerically, the appropriate conversion factor is 12,400.

Figure 25 The Bohr model of the atom, showing electron quantum jumping to ground state n=1

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In 1897 the particle called the electron was discovered. By means of the gold foil experiment physicists discovered that matter is, volume for volume, largely space. Once that was clear, it was hypothesized that negative charge entities called electrons surround positively charged nuclei. So at first all scientists believed that the atom must be like a miniature solar system. But that simple analogy predicted that electrons would, within about one hundredth of a microsecond, crash into the nucleus of the atom. The great question of the early 20th century was, "Why do electrons normally maintain a stable orbit around the nucleus?" In 1913, Niels Bohr removed this substantial problem by applying the idea of discrete (non-continuous) quanta to the orbits of electrons. This account became known as the Bohr model of the atom. Bohr basically theorized that electrons can only inhabit certain orbits around the atom. These orbits could be derived by looking at the spectral lines produced by atoms.

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Bohr explained the orbits that electrons can take by relating the angular momentum of electrons in each "permitted" orbit to the value of h, Planck’s constant. He held that an electron in the lowest orbital has an angular momentum equal to h/2π. Each orbit after the initial orbit must provide for an electron’s angular momentum being an integer multiple of that lowest value. He depicted electrons in atoms as being analogous to planets in a solar orbit. However, he took Planck’s constant to be a fundamental quantity that introduces special requirements at this subatomic level and that explains the spacing of those "planetary" orbits. Bohr’s analysis of electron orbits as circular A little math on circular orbits. Bohr was very familiar with the dynamics of simple circular orbits in an inverse square field as described in classical mechanics. Simply explained: To find the acceleration of a circle, place it inside the shape of a square where tangents meet, then find the linear speed along one side of the square, then square the speed of one side to complete the speed of the entire square, then divide by the radius of the circle placed in the square to get the speed around the circle. Therefore, circular (centripetal) acceleration is v squared over r where v is speed and r is radius. The equation for the centripetal acceleration is a = v 2 /r. That is, acceleration is inversely proportional to the

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radius of the circle. If the radius is doubled, then the acceleration is halved. Also, Kepler’s Third Law is that the radius cubed equals the circumference of the orbit squared. It immediately follows that the radius of any n orbit is proportional to the orbit n squared, and the speed in that orbit is proportional to 1/n. Speed times radius gives angular momentum. That leaves n-squared over n. It then follows that the angular momentum for any orbit n is just proportional to n. Bohr argued then that the angular momentum in any orbit n was nKh, where h is Planck’s constant and K is some multiplying factor, the same for all the orbits, which was later determined to be 1/2π.

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Bohr considered one revolution in orbit to be equivalent to one cycle in an oscillator (as in Planck’s initial measurements to define the constant h) which is in turn similar to one cycle in a wave. The number of revolutions per second is (or defines) what we call the frequency of that electron or that orbital. Specifying that the frequency of each orbit must be an integer multiple of Planck’s constant h would only permit certain orbits, and would also fix their size. Bohr generalized Balmer’s formula for hydrogen by replacing denominator in the term 1/4 with an explicit squared variable:   1 1 1 λ = RH m2 − n2 , m=1,2,3,4,5,..., and n > m

where is the wavelength of the light and R H is the Rydberg constant for hydrogen. This generalization predicted many more line spectra than had been previously detected, and experimental confirmation of this prediction followed. It follows almost immediately that if λ is quantized as the formula above indicates, then the momentum of any photon must be quantized. The frequency of light, v, at a given wavelength λ is given by the relationship v=

c λ

λ=

hc hv ,

and :λ =

c v

and multiplying by h/h = 1,

and we know that

E = hv so λ =

hc E

which we can rewrite as:

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h , E/c

λ=

h p

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or p =

Beginning with line spectra, physicists were able to deduce empirically the rules according to which the orbits of electrons are determined and to discover something vital about the momentums involved–that they are quantized.

Bohr next realized how the angular momentum of an electron in its orbit, L, is quantized, i.e., he determined that there is some constant value K such that when it is multiplied by Planck’s constant, h, it will yield the angular momentum that pertains to the lowest orbital. When it is multiplied by successive integers it will then give the values of other possible orbitals. He later determined that K = 1/2π . (See the detailed argument at 119.)

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Bohr’s theory represented electrons as orbiting the nucleus of an atom in a way that was amazingly different from what we see in the world of our everyday experience. He showed that when an electron changed orbits it did not move in a continuous trajectory from one orbit around the nucleus to another. Instead, it suddenly disappeared from its original orbit and reappeared in another orbit. Each distance at which an electron can orbit is a function of a quantized amount of energy. The closer to the nucleus an electron orbits, the less energy it takes to remain in that orbital. Electrons that absorb a photon gain a quantum of energy, so they jump to an orbit that is farther from the nucleus, while electrons that emit a photon lose a quantum of energy and so jump to an inner orbital. Electrons cannot gain or lose a fractional quantum of energy, and so, it is argued, they cannot have a position that is at a fractional distance between allowed orbitals. Allowed orbitals were designated as whole numbers using the letter n with the innermost orbital being designated n = 1, the next out being n = 2, and so on. Any orbital with the same value of n is called an electron shell. Bohr’s model of the atom was essentially two-dimensional because it depicts electrons as particles in circular orbits. In this context, two-dimensional means something that can be described on the surface of a plane. One-dimensional means something that can be described by a line. Because circles can be described by their radius, which is a line segment, sometimes Bohr’s model of the atom is described as one-dimensional.

119 http://galileo.phys.virginia.edu/classes/252/Bohr_Atom/Bohr_Atom.html

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Wave-particle duality

Figure 26

Probability distribution of the Bohr atom

Niels Bohr determined that it is impossible to describe light adequately by the sole use of either the wave analogy or of the particle analogy. Therefore he enunciated the principle of complementarity, which is a theory of pairs, such as the pairing of wave and particle or the pairing of position and momentum. Louis de Broglie worked out the mathematical consequences of these findings. In quantum mechanics, it was found that electromagnetic waves could react in certain experiments as though they were particles and in other experiments as though they were waves. It was also discovered that subatomic particles could sometimes be described as particles and sometimes as waves. This discovery led to the theory of wave-particle duality by Louis-Victor de Broglie in 1924, which states that subatomic entities have properties of both waves and particles at the same time. The Bohr atom model was enlarged upon with the discovery by de Broglie that the electron has wave-like properties. In accord with de Broglie’s conclusions, electrons can only appear under conditions that permit a standing wave. A standing wave can be made if a string is fixed on both ends and made to vibrate Introduction to quantum mechanics

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(as it would in a stringed instrument). That illustration shows that the only standing waves that can occur are those with zero amplitude at the two fixed ends. The waves created by a stringed instrument appear to oscillate in place, simply changing crest for trough in an up-and-down motion. A standing wave can only be formed when the wave’s length fits the available vibrating entity. In other words, no partial fragments of wave crests or troughs are allowed. In a round vibrating medium, the wave must be a continuous formation of crests and troughs all around the circle. Each electron must be its own standing wave in its own discrete orbital.

Development of modern quantum mechanics Full quantum mechanical theory

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Werner Heisenberg developed the full quantum mechanical theory in 1925 at the young age of 23. Following his mentor, Niels Bohr, Werner Heisenberg began to work out a theory for the quantum behavior of electron orbitals. Because electrons could not be observed in their orbits, Heisenberg went about creating a mathematical description of quantum mechanics built on what could be observed, that is, the light emitted from atoms in their characteristic atomic spectra. Heisenberg studied the electron orbital on the model of a charged ball on a spring, an oscillator, whose motion is anharmonic (not quite regular). For a picture of the behavior of a charged ball on a spring see: Vibrating Charges 120. Heisenberg first explained this kind of observed motion in terms of the laws of classical mechanics known to apply in the macro world, and then applied quantum restrictions, discrete (non-continuous) properties, to the picture. Doing so causes gaps to appear between the orbitals so that the mathematical description he formulated would then represent only the electron orbitals predicted on the basis of the atomic spectra. In 1925 Heisenberg published a paper (in Z. Phys. vol. 33, p. 879-893) entitled "Quantum-mechanical re-interpretation of kinematic and mechanical relations." So ended the old quantum theory and began the age of quantum mechanics. Heisenberg’s paper gave few details that might aid readers in determining how he actually contrived to get his results for the one-dimensional models he used to form the hypothesis that proved so useful. In his paper, Heisenberg proposed to "discard all hope of observing hitherto unobservable quantities, such as the position and period of the electron," and restrict himself strictly to actually observable quantities. He needed mathematical rules for predicting the relations actually observed in nature, and the rules he produced worked differently depending on the sequence in which they were applied. "It

120 http://www.colorado.edu/physics/2000/waves_particles/wavpart4.html

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quickly became clear that the non-commutativity (in general) of kinematical quantities in quantum theory was the really essential new technical idea in the paper." (Aitchison, p. 5) But it was unclear why this non-commutativity was essential. Could it have a physical interpretation? At least the matter was made more palatable when Max Born discovered that the Heisenberg computational scheme could be put in a more familiar form present in elementary mathematics.

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The special type of multiplication that turned out to be required in his formula was most elegantly described by using special arrays of numbers called matrices. In ordinary situations it does not matter in which order the operations involved in multiplication are performed, but matrix multiplication does not commute. Essentially that means that it matters which order given operations are performed in. Multiplying matrix A by matrix B is not the same as multiplying matrix B by matrix A. In symbols, AxB is in general not equal to BxA. (The important thing in quantum theory is that it turned out to matter whether one measures velocity first and then measures position, or vice-versa.) The matrix convention turned out to be a convenient way of organizing information and making clear the exact sequence in which calculations must be made. In matrix mathematics sets of numbers are given in rows and columns, and there are conventions for the way multiplication of matrices is performed. If everybody arranged their matrices entirely as they pleased, then understanding every new matrix calculation would involve learning the personal plan of the person who made the matrix, so certain conventions have evolved. Reverse the order of the multiplication of the matrices and the numerical results will go wrong. In other words, Matrix multiplication is noncommutative (to be precise, reversing the order will in general cause the multiplication to become undefined except for the special case where the matrices are square). Because these complex operations are, by analogy, called "multiplication," it is tempting to imagine that it ought to be irrelevant whether one multiplies matrix A by matrix B, or one multiplies matrix B by matrix A. Because of the complications involved in the rules of matrix multiplication, in almost every case the ordinary mathematical expectation of commutation does not hold. (Sometimes matrices even anticommute.) In Heisenberg’s matrix mechanics, the sets of numbers are infinite, representing all possible positions of the electron, and those matrices cannot be multiplied in reverse order and still produce the correct results. The essential point is that Heisenberg first learned what ways he had to operate on measured quantities to be able to account for the line spectra that had been observed. The operations to be performed seemed complicated and arbitrary, but when specialists realized that what he was doing could be represented in a

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Heisenberg approached quantum mechanics from the historical perspective that treated an electron as an oscillating charged particle. Bohr’s use of this analogy had already allowed him to explain why the radii of the orbits of electrons could only take on certain values. It followed from this interpretation of the experimental results available and the quantum theory that Heisenberg subsequently created that an electron could not be at any intermediate position between two "permitted" orbits. Therefore electrons were described as "jumping" from orbit to orbit. The idea that an electron might now be in one place and an instant later be in some other place without having traveled between the two points was one of the earliest indications of the "spookiness" of quantum phenomena. Although the scale is smaller, the "jump" from orbit to orbit is as strange and unexpected as would be a case in which someone stepped out of a doorway in London onto the streets of Los Angeles.

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Amplitudes of position and momentum that have a period of 2 π like a cycle in a wave are called Fourier series variables. Heisenberg described the particlelike properties of the electron in a wave as having position and momentum in his matrix mechanics. When these amplitudes of position and momentum are measured and multiplied together, they give intensity. However, he found that when the position and momentum were multiplied together in that respective order, and then the momentum and position were multiplied together in that respective order, there was a difference or deviation in intensity between them of h/2π. Heisenberg would not understand the reason for this deviation until two more years had passed, but for the time being he satisfied himself with the idea that the math worked and provided an exact description of the quantum behavior of the electron. Matrix mechanics was the first complete definition of quantum mechanics, its laws, and properties that described fully the behavior of the electron. It was later extended to apply to all subatomic particles. Schrödinger subsequently produced a quantum wave theory that was computationally easier and avoided some of the odd-sounding ideas like "quantum leaps" of an electron from one orbit to another, and finally Dirac made the idea of non-commutativity central to his own theory that proved the formulations of Heisenberg and of Schrödinger to be special cases of his own.

Schrödinger wave equation

Because particles could be described as waves, later in 1925 Erwin Schrödinger analyzed what an electron would look like as a wave around the nucleus of the Introduction to quantum mechanics

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atom. Using this model, he formulated his equation for particle waves. Rather than explaining the atom by analogy to satellites in planetary orbits, he treated everything as waves whereby each electron has its own unique wavefunction. A wavefunction is described in Schrödinger’s equation by three properties (later Paul Dirac added a fourth). The three properties were (1) an "orbital" designation, indicating whether the particle wave is one that is closer to the nucleus with less energy or one that is further from the nucleus with more energy, (2) the shape of the orbital, i.e. an indication that orbitals were not just spherical but other shapes, and (3) the magnetic moment of the orbital, which is a manifestation of force exerted by the charge of the electron as it rotates around the nucleus.

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These three properties were called collectively the wavefunction of the electron and are said to describe the quantum state of the electron. "Quantum state" means the collective properties of the electron describing what we can say about its condition at a given time. For the electron, the quantum state is described by its wavefunction which is designated in physics by the Greek letter ψ (psi, pronounced "sigh"). The three properties of Schrödinger’s equation that describe the wavefunction of the electron and therefore also describe the quantum state of the electron as described in the previous paragraph are each called quantum numbers. The first property which describes the orbital was numbered according to Bohr’s model where n is the letter used to describe the energy of each orbital. This is called the principal quantum number. The next quantum number that describes the shape of the orbital is called the azimuthal quantum number and it is represented by the letter l (lower case L). The shape is caused by the angular momentum of the orbital. The rate of change of the angular momentum of any system is equal to the resultant external torque acting on that system. In other words, angular momentum represents the resistance of a spinning object to speed up or slow down under the influence of external force. The azimuthal quantum number "l" represents the orbital angular momentum of an electron around its nucleus. However, the shape of each orbital has its own letter as well. So for the letter "l" there are other letters to describe the shapes of "l". The first shape is spherical and is described by the letter s. The next shape is like a dumbbell and is described by the letter p. The other shapes of orbitals become more complicated (see Atomic Orbitals 121) and are described by the letters d, f, and g. To see the shape of a carbon atom, see Carbon atom 122. The third quantum number of Schrödinger’s equation describes the magnetic moment of the electron and is designated by the letter m

121 http://orbitals.com/orb/ 122 http://library.thinkquest.org/C0110925/carbon.htm

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242 and sometimes as the letter m with a subscript l because the magnetic moment depends upon the second quantum number l.

Uncertainty Principle

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In May 1926 Schrödinger published a proof that Heisenberg’s matrix mechanics and Schrödinger’s wave mechanics gave equivalent results: mathematically they were the same theory. Both men claimed to have the superior theory. Heisenberg insisted on the existence of discontinuous quantum jumps in his particle-like examination of the oscillation of a charged electron giving more precise definitions and Schrödinger insisted that a theory based on continuous wave-like properties which he called "matter-waves" was better.

Main article: →Uncertainty principle

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In 1927, Heisenberg made a new discovery on the basis of his quantum theory that had further practical consequences of this new way of looking at matter and energy on the atomic scale. In Heisenberg’s matrix mechanics formula, Heisenberg had encountered an error or difference of h/2π between position and momentum. This represented a deviation of one radian of a cycle when the particle-like aspects of the wave were examined. Heisenberg analyzed this difference of one radian of a cycle and divided the difference or deviation of one radian equally between the measurement of position and momentum. This had the consequence of being able to describe the electron as a point particle in the center of one cycle of a wave so that its position would have a standard deviation of plus or minus one-half of one radian of the cycle (1/2 of h-bar). A standard deviation can be either plus or minus the measurement i.e. it can add to the measurement or subtract from it. In three-dimensions a standard deviation is a displacement in any direction. What this means is that when a moving particle is viewed as a wave it is less certain where the particle is. In fact, the more certain the position of a particle is known, the less certain the momentum is known. This conclusion came to be called "Heisenberg’s Indeterminacy Principle," or Heisenberg’s Uncertainty Principle. To understand the real idea behind the uncertainty principle imagine a wave with its undulations, its crests and troughs, moving along. A wave is also a moving stream of particles, so you have to superimpose a stream of particles moving in a straight line along the middle of the wave. An oscillating ball of charge creates a wave larger than its size depending upon the length of its oscillation. Therefore, the energy of a moving particle is as large as the cycle of the wave, but the particle itself has a location. Because the particle and the wave are the same thing, then the particle is really located somewhere in the width of the wave. Its position could be anywhere from the crest to the trough. The math for the uncertainty principle Introduction to quantum mechanics

243 says that the measurement of uncertainty as to the position of a moving particle is one-half the width from the crest to the trough or one-half of one radian of a cycle in a wave.

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For moving particles in quantum mechanics, there is simply a certain degree of exactness and precision that is missing. You can be precise when you take a measurement of position and you can be precise when you take a measurement of momentum, but there is an inverse imprecision when you try to measure both at the same time as in the case of a moving particle like the electron. In the most extreme case, absolute precision of one variable would entail absolute imprecision regarding the other. Heisenberg voice recording in an early lecture on the uncertainty principle pointing to a Bohr model of the atom: "You can say, well, this orbit is really not a complete orbit. Actually at every moment the electron has only an inactual position and an inactual velocity and between these two inaccuracies there is an inverse correlation."

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The consequences of the uncertainty principle were that the electron could no longer be considered as in an exact location in its orbital. Rather the electron had to be described by every point where the electron could possibly inhabit. By creating points of probable location for the electron in its known orbital, this created a cloud of points in a spherical shape for the orbital of a hydrogen atom which points gradually faded out nearer to the nucleus and farther from the nucleus. This is called a probability distribution. Therefore, the Bohr atom number n for each orbital became known as an n-sphere in the three dimensional atom and was pictured as a probability cloud where the electron surrounded the atom all at once. This led to the further description by Heisenberg that if you were not making measurements of the electron that it could not be described in one particular location but was everywhere in the electron cloud at once. In other words, quantum mechanics cannot give exact results, but only the probabilities for the occurrence of a variety of possible results. Heisenberg went further and said that the path of a moving particle only comes into existence once we observe it. However strange and counter-intuitive this assertion may seem, quantum mechanics does still tell us the location of the electron’s orbital, its probability cloud. Heisenberg was speaking of the particle itself, not its orbital which is in a known probability distribution. It is important to note that although Heisenberg used infinite sets of positions for the electron in his matrices, this does not mean that the electron could be anywhere in the universe. Rather there are several laws that show the electron must be in one localized probability distribution. An electron is described by its energy in Bohr’s atom which was carried over to matrix mechanics. Therefore, an electron in a certain n-sphere had to be within a certain range from the nucleus depending upon its energy. This restricts its location. Also, the number Introduction to quantum mechanics

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of places an electron can be is also called "the number of cells in its phase space". The uncertainty principle set a lower limit to how finely one can chop up classical phase space. Therefore, the number of places that an electron can be in its orbital becomes finite due to the Uncertainty Principle. Therefore, an electron’s location in an atom is defined to be in its orbital and its orbital although being a probability distribution does not extend out into the entire universe, but stops at the nucleus and before the next n-sphere orbital begins and the points of the distribution are finite due to the Uncertainty Principle creating a lower limit.

Classical physics had shown since Newton that if you know the position of stars and planets and details about their motions that you can predict where they will be in the future. For subatomic particles, Heisenberg denied this notion showing that due to the uncertainty principle one cannot know the precise position and momentum of a particle at a given instant, so its future motion cannot be determined, but only a range of possibilities for the future motion of the particle can be described.

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These notions arising from the uncertainty principle only arise at the subatomic level and were a consequence of wave-particle duality. As counter-intuitive as they may seem, quantum mechanical theory with its uncertainty principle has been responsible for major improvements in the world’s technology from computer components to fluorescent lights to brain scanning techniques.

Wavefunction collapse

Schrödinger’s wave equation with its unique wavefunction for a single electron is also spread out in a probability distribution like Heisenberg’s quantized particle-like electron. This is because a wave is naturally a widespread disturbance and not a point particle. Therefore, Schrödinger’s wave equation has the same predictions made by the uncertainty principle because uncertainty of location is built into the definition of a widespread disturbance like a wave. Uncertainty only needed to be defined from Heisenberg’s matrix mechanics because the treatment was from the particle-like aspects of the electron. Schrödinger’s wave equation shows that the electron is in the probability cloud at all times in its probability distribution as a wave that is spread out. Max Born discovered in 1928 that when you compute the square of Schrödinger’s wavefunction (psi-squared), you get the electron’s location as a probability distribution. Therefore, if a measurement of the position of an electron is made as an exact location in space instead of as a probability distribution, it ceases to have wave-like properties. Without wave-like properties, none of Schrödinger’s definitions of the electron being wave-like make sense

Introduction to quantum mechanics

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anymore. The measurement of the position of the particle nullifies the wavelike properties and Schrödinger’s equation then fails. Because the electron can no longer be described by its wavefunction when measured due to it becoming particle-like, this is called wavefunction collapse.

Eigenstates and eigenvalues

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The term eigenstate is derived from the German/Dutch word "eigen," which means "inherent" or "characteristic." The word eigenstate is descriptive of the measured state of some object that possesses quantifiable characteristics such as position, momentum, etc. The state being measured and described must be an "observable" (i.e. something that can be experimentally measured either directly or indirectly like position or momentum), and must have a definite value. In the everyday world, it is natural and intuitive to think of everything being in its own eigenstate. Everything appears to have a definite position, a definite momentum, a definite value of measure, and a definite time of occurrence. However, quantum mechanics affirms that it is impossible to pinpoint exact values for the momentum of a certain particle like an electron in a given location at a particular moment in time, or, alternatively, that it is impossible to give an exact location for such an object when the momentum has been measured. Due to the uncertainty principle, statements regarding both the position and momentum of particles can only be given in terms of a range of probabilities, a "probability distribution". Eliminating uncertainty in one term maximizes uncertainty in regard to the second parameter.

Therefore it became necessary to have a way to clearly formulate the difference between the state of something that is uncertain in the way just described, such as an electron in a probability cloud, and effectively contrast it to the state of something that is not uncertain, something that has a definite value. When something is in the condition of being definitely "pinned-down" in some regard, it is said to possess an eigenstate. For example, if the position of an electron has been made definite, it is said to have an eigenstate of position. A definite value, such as the position of an electron that has been successfully located, is called the eigenvalue of the eigenstate of position. The German word "eigen" was first used in this context by the mathematician David Hilbert in 1904. Schrödinger’s wave equation gives wavefunction solutions, meaning the possibilities where the electron might be, just as does Heisenberg’s probability distribution. As stated above, when a wavefunction collapse occurs because something has been done to locate the position of an electron, the electron’s state becomes an eigenstate of position, meaning that the position has a known value.

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The Pauli Exclusion Principle

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There was a doublet, meaning a pair of lines, in the spectrum of a hydrogen atom that was unaccounted for. This meant that there was more energy in the electron orbital from magnetic moment than had previously been described. Wolfgang Pauli when studying alkali metals had introduced what he called a "two-valued quantum degree of freedom" associated with the electron in the outermost shell. "Degrees of freedom" simply means the number of possible independent ways a particle may move. This led to the Pauli Exclusion Principle that predicted that no more than two electrons can inhabit the same orbital. It also predicted that any neutron, electron, or proton (types of fermions) could not exist in the same quantum state. We learned in Schrödinger’s equation that there were three quantum states of the electron, but if two electrons could be in the same orbital, there had to be another quantum number to distinguish those two electrons from each other and to describe the extra magnetic moment shown in the atomic spectrum. In early 1925, the young physicist Ralph Kronig had introduced a theory to Pauli that the electron rotates in space in the same way that the earth rotates on its axis. This would account for the missing magnetic moment and allow for two electrons in the same orbital to be different if their spin was in opposite directions to each other thus satisfying the Exclusion Principle. The Pauli Exclusion Principle states that no electron (or other fermion) can be in the same quantum state as another. This has an effect on the probability distribution of the electron further defining the number of cells in its phase space. The minimum limit is the limit of the Uncertainty Principle and the Exclusion Principle states that no two electrons can be within this same minimum space defined by the Uncertainty Principle.

Therefore, a single electron in its orbital when defined by its quantum state which is its wavefunction which is defined by its four quantum numbers cannot have the same four quantum numbers of another electron in that atom. Where two electrons are in the same n-sphere and therefore share the same principal quantum number, they must then have some other unique quantum number of shape l, magnetic moment m or spin s. Even in the formation of degenerate gases where the electrons are not in an orbital around the nucleus of an atom, they must still follow the Pauli Exclusion Principle when in a confined space.

Dirac wave equation

In 1928, Paul Dirac worked out a variation of Schrödinger’s equation that accounted for a fourth property of the electron in its orbital. Paul Dirac introduced the fourth quantum number called the spin quantum number designated by the letter s to the new Dirac equation of the wavefunction of the electron. In Introduction to quantum mechanics

247 1930, Dirac combined Heisenberg’s matrix mechanics with Schrödinger’s wave mechanics into a single quantum mechanical representation in his Principles of Quantum Mechanics. The quantum picture of the electron was now complete.

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All of the above development of quantum theory was based mainly on the atomic spectrum of the hydrogen atom. This is due to the fact that each atom of each element produces a unique pattern of spectral lines when light from each different kind of element is passed through a prism. Scientists could not study the electron and nucleus of the atom itself because they cannot be seen. Even today with High-resolution Scanning Tunneling Electron Microscopes we can only get images of the atom as a blurry fuzzball. However, the spectral lines of the atom reveal the orbits of electrons and the energy that can be expected. It was basically this study of the spectroscopic analysis of first the hydrogen atom and then the helium atom that led to quantum theory. Therefore, the mathematical formula were made to fit the picture of the atomic spectrum. That is why quantum mechanics is sometimes referred to as a form of mathematical physics.

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Quantum entanglement

Albert Einstein rejected Heisenberg’s Uncertainty Principle insofar as it seemed to imply more than a necessary limitation on human ability to actually know what occurs in the quantum realm. In a letter to Max Born in 1926, Einstein claimed that God "does not play dice." Heisenberg’s quantum mechanics, based on Bohr’s initial explanation, became known as the Copenhagen Interpretation of quantum mechanics. Both Bohr and Einstein spent many years defending and attacking this interpretation. After the 1930 Solvay conference, Einstein never again challenged the Copenhagen interpretation on technical points, but did not cease a philosophical attack on the interpretation, on the grounds of realism and locality. Einstein, in trying to show that it was not a complete theory, recognized that the theory predicted that two or more particles which have interacted in the past exhibit surprisingly strong correlations when various measurements are made on them. Einstein called this "spooky action at a distance". In 1935, Schrödinger published a paper explaining the argument which had been denoted the EPR paradox (Einstein-Podolsky-Rosen, 1935). Einstein showed that the Copenhagen Interpretation predicted quantum entanglement which he was trying to prove was incorrect in that it would defy the law of physics that nothing could travel faster than the speed of light. Quantum entanglement means that when there is a change in one particle at a distance from another particle then the other particle automatically changes to counter-balance the system. In quantum entanglement, the act of measuring one entangled particle defines its properties and seems to influence the properties of its partner or partners instantaneously, no matter how far apart they are. Introduction to quantum mechanics

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Because the two particles are in an entangled state, changes to the one cause instantaneous effects on the other. Einstein had calculated that quantum theory would predict this, he saw it as a flaw and therefore challenged it. However, instead of showing a weakness in quantum mechanics, this forced quantum mechanics to acknowledge that quantum entanglement did in fact exist and it became another foundation theory of quantum mechanics. The 1935 paper is currently Einstein’s most cited publication in physics journals.

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Bohr’s original response to Einstein was that the particles were in a system. However, Einstein’s challenge led to decades of substantial research into this quantum mechanical phenomenon of quantum entanglement. This research clarified by Yanhua Shih points out that the two entangled particles can be viewed as somehow not separate, which removes the locality objection. This means that no matter the distance between the entangled particles, they remain in the same quantum state so that one particle is not sending information to another particle faster than the speed of light, but rather a change to one particle is a change to the entire system or quantum state of the entangled particles and therefore changes the state of the system without information transference.

Interpretations

As a system becomes larger or more massive (action » h ) the classical picture tends to emerge, with some exceptions, such as superfluidity. The emergence of behaviour as we scale up that matches our classical intuition is called the correspondence principle and is based on Ehrenfest’s theorem. This why we can usually ignore quantum mechanics when dealing with everyday objects. Even so, trying to make sense of quantum theory is an ongoing process which has spawned a number of interpretations of quantum theory, ranging from the conventional Copenhagen Interpretation to hidden variables & many worlds. There seems to be no end in sight to the philosophical musings on the subject; however the empirical or technical success of the theory is unrivalled; all modern fundamental physical theories are quantum theories.

See also • • • • •

Atom →Quantum mechanics →Uncertainty principle Correspondence principle Quantum number • Principal quantum number • Azimuthal quantum number

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References • •

Mara Beller, Quantum Dialogue: The Making of a Revolution. University of Chicago Press, Chicago, 2001. Bohr, Niels (1958). Atomic Physics and Human Knowledge. John Wiley and Sons. ISBN 000000000X. De Broglie, Louis. The Revolution in Physics, Noonday Press, 1953. Feynman, Richard P., QED: The Strange Theory of Light and Matter, Princeton University Press, 1985. ISBN 0-691-08388-6 Feigl, Herbert and May Brodbeck, Readings in the Philosophy of Science, Appleton-Century-Crofts, 1953. Einstein, Albert. Essays in Science, Philosophical Library, 1934. Prof. Michael Fowler, The Bohr Atom, A series of lectures, 1999, University of Virginia. Heisenberg, Werner. Physics and Philosophy, Harper and Brothers, 1958. S Lakshmibala, "Heisenberg, Matrix Mechanics and the Uncertainty Principle", Resonance, Journal of Science Education, Volume 9, Number 8, August 2004. Richard L. Liboff, Introductory Quantum Mechanics, 2nd ed. 1992. Lindsay, Robert Bruce and Henry Margenau, Foundations of Physics, Dover, 1936. McEnvoy, J.P., and Zarate, Oscar. Introducing Quantum Theory, ISBN 184046-577-9 Carl Rod Nave, Hyperphysics-Quantum Physics, Department of Physics and Astronomy, Georgia State University, CD 2005. F. David Peat, "From Certainty to Uncertainty: The Story of Science and Ideas in the Twenty-First Century", Joseph Henry Press, 2002. Reichenbach, Hans, Philosophic Foundations of Quantum Mechanics, University of California Press, 1944. Schilpp, Paul Arthur, Albert Einstein: Philosopher-Scientist, Tudor Publishing Commpany, 1949. Scientific American Reader, 1953.

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• Magnetic quantum number • Spin quantum number Planck’s constant Standard Model →Matrix mechanics Schrödinger’s equation →Quantum mechanics, philosophy and controversy →Interpretation of quantum mechanics

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Introduction to quantum mechanics

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Notes

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Sears, Francis Weston, Optics, Addison-Wesley, 1949. Dr. Kenjiro Takada, Emeritus professor of Kyushu University, Microscopic World-Introduction to Quantum Mechanics, Internet seminar, http: //www2.kutl.kyushu-u.ac.jp/seminar/MicroWorld1_E/MicroWorld_1_E. html. • "Uncertainty Prirnciple" Werner Heisenberg actual voice recording, http: //www.thebigview.com/spacetime/index.html. • J.H. Van Vleck, The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics, Proc. Nat. Acad. Sci., Vol. 14, p.179, 1928. • Carl Wieman and Katherine Perkins, "Transforming Physics Education", Physics Today, November 2005,

↑ Events leading up to the December 1900 publication of Planck’s quantum hypothesis are related by Werner Heisenberg in his Physics and Philosophy, pp. 30f. Heisenberg believes that Planck was clearly aware that his ideas would have very far-reaching consequences.

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↑ It is remarkable that Einstein made significant contributions to quantum mechanics at the same time he was revolutionizing physics with his relativity theories. The far-reaching nature of his contributions to quantum mechanics is noted by Richard Feynman in QED: The Strange Theory of Light and Matter, p. 112: "[The] phenomenon of ’stimulated emission’ was discovered by Einstein when he launched the quantum theory proposing the photon model of light. Lasers work on the basis of this phenomenon." ↑ Albert Einstein characterized Niels Bohr’s contributions to the quantum revolution by saying that history "will have to connect one of the most important advances ever made in our knowledge of the nature of the atom with the name of Niels Bohr." He added, "The boldly selected hypothetical basis of his speculations soon became a mainstay for the physics of the atom....The theory of the Röntgen spectra of the visible spectra, and the periodic system of the elements is primarily based on the ideas of Bohr." See Einstein’s Essays in Science, p. 46f.

↑ Niels Bohr records the contributions of Louis de Broglie toward "a more comprehensive quantum theory" that would take into account that "the wavecorpuscle duality was not confined to the properties of radiation, but was equally unavoidable in accounting for the behaviour of material particles." See his Atomic Physics and Human Knowledge, p. 37 et passim. ↑ See Max Born, Atomic Physics, especially p. 90, where he says of quantum mechanics that it is "in the nature of the case indeterministic, and therefore the affair of statistics." Introduction to quantum mechanics

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↑ A two-page account of the highlights of Dirac’s work, including his speculation that a positron would be found, appears in an article on "The Ultimate Particles" by George W. Gray, in The Scientific American Reader, p. 100f. (Simon and Schuster, 1953). ↑ Werner Heisenberg is well known for his "indeterminancy principle" or "uncertainty principle." ↑ Wolfgang Pauli’s name is most closely associated with what is known as the "Pauli Exclusion Principle," according to which, it is impossible, in the words of Louis de Broglie, "for two electrons to have rigorously identical quantized states, i.e., defined by the same quantum numbers.... Translated into wave mechanics, Pauli’s principle is expressed as follows: ’for electrons, the only states realized in nature are the antisymmetric states.’" (See de Broglie’s The Revolution in Physics, p. 267)

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↑ Schrödinger’s cat was originally a character in an example intended to be critical of an apparent difficulty in Heisenberg’s exposition of his principle of indeterminancy. The story has been taken somewhat out of context and the cat has assumed a minor literary life of its own. Schrödinger’s other contributions to understanding quantum mechanics and to making the mathematics easier to handle were, of course, much more important. The translation of his 1935 essay that includes the story is to be found at http://www.qedcorp.com/pcr /pcr/qcat.html. Schrödinger describes a situation in which a cat will live or die depending on whether a quantum mechanically probabalistic radioactive emission event occurs within the hour the cat is confined to a box. To Heisenberg’s interpretation of quantum mechanics he objects: "If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The psi-function of the entire system would express this by having in it the living and dead cat (pardon the expression) mixed or smeared out in equal parts."

↑ Huygens’ principle is explained in Optics, by Francis Weston Sears, AddisonWesley, 1949, pp. 5f. ↑ Sears, Optics, p. 2.

↑ Sears, Optics, p. 2f.

↑ Lindsay and Margenau, Foundations of Physics, p. 388}} ↑ Sears, Mechanics, Wave Motion, and Heat, p. 537.

↑ See Sears, Optics, pp. 282-293.

↑ Planck is quoted by Louis de Broglie, The Revolution in Physics, p. 106. The material in this paragraph summarizes de Broglie’s account given on pages 105 to 108. (Noonday Press, New York, 1953) Introduction to quantum mechanics

252 ↑ A. Einstein, Ann. d. Phys., 17, 132, (1905). ↑ J. P. McEvoy and Oscar Zarate, Introducing Quantum Theory, p. 114 and p. 118.

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↑ A. P. French and Edwin F. Taylor, An Introduction to Quantum Physics,, p. 18.

↑ For the length of time involved, see George Gamow’s One, Two, Three...Infinity, p. 140. ↑ A. P. French and Edwin F. Taylor, An Introduction to Quantum Physics,, p. 23.

↑ A very clear explanation of interference in thin films may be found in Sears, op. cit., p. 203ff.

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↑ See Linus Pauling, The Nature of the Chemical Bond, p. 35f. He explains that a circular electron orbit in the case of the first orbital of the hydrogen atom would predict orbital angular momentum for the atom, which it does not in fact have. He therefore depicts the electron as moving from the nucleus and back in a straight line.

↑ The German and English forms of this quotation appear in slightly different versions from place to place, probably because Einstein repeated his original remark several time. The earliest German version can be found at http://www.goettingen.de/kultur/gott_wuerfelt_nicht.htm. In it, Einstein first speaks of God and then says, "And this one does not (dice =) play dice."

↑ Hans Reichenback works out the mathematics in one sample case of quantum entanglement. See his Philosophic Foundations of Quantum Mechanics, p. 170 ff. ↑ Yanhua H. Shih (2001), "Quantum Entanglement and Quantum Teleportation" Annals of Physics 10 (2001) 1-2 pp.45-61 as referenced by Amir Aczel (2003), Entanglement p.252 ISBN 0-452-28457-0

External links • • •

Development of Current Atomic Theory 123 Quantum Mechanics 124 Planck’s original paper on Planck’s constant 125

123 http://chemed.chem.purdue.edu/genchem/topicreview/bp/ch6/bohr.html 124 http://www.aip.org/history/heisenberg/p07.htm 125 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html

Introduction to quantum mechanics

253 Everything you wanted to know about the quantum world 126 — Provided by New Scientist. • Quantum Articles 127

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Source: http://en.wikipedia.org/wiki/Introduction_to_quantum_mechanics

Principal Authors: Voyajer, Patrick0Moran, Paulc1001, Ancheta Wis, David R. Ingham

Josephson effect

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The Josephson effect is a term given to the phenomenon of current flow across two superconductors separated by a very thin insulating barrier. This arrangement—two superconductors linked by a non-conducting oxide barrier—is known as a Josephson junction; the current that crosses the barrier is the Josephson current. The terms are named eponymously after British physicist Brian David Josephson, who predicted the existence of the effect in 1962 128. It has important applications in quantum-mechanical circuits.

The effect

The basic equations 129 governing the dynamics of the Josephson effect are U (t) =

~ ∂φ 2e ∂t ,

I(t) = Ic sin(φ(t))

where U (t) and I(t) are the voltage and current across the Josephson junction, φ(t) is the phase difference between the wave functions in the two superconductors comprising the junction, and Ic is a constant, the critical current of the junction. The critical current is an important phenomenological parameter of the device that can be affected by temperature as well as by an applied magh

netic field. The physical constant, 2e is the magnetic flux quantum, the inverse of which is the Josephson constant. The three main effects predicted by Josephson follow from these relations:

126 http://www.newscientist.com/channel/fundamentals/quantum-world 127 http://www.thequantumsite.com/ 128 B. D. Josephson. The discovery of tunnelling supercurrents(http://prola.aps.org/abstract/RMP/v46

/i2/p251_1). Rev. Mod. Phys. 1974; 46(2): 251-254.

129 Barone A, Paterno G. Physics and Applications of the Josephson Effect. New York: John Wiley & Sons;

1982.

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The DC Josephson effect. This refers to the phenomenon of a direct current crossing the insulator in the absence of any external electromagnetic field, owing to tunneling. This DC Josephson current is proportional to the sine of the phase difference across the insulator, and may take values between −Ic and Ic . The AC Josephson effect. With a fixed voltage UDC across the junctions, the phase will vary linearly with time and the current will be an AC current 2e with amplitude Ic and frequency h UDC . This means a Josephson junction can act as a perfect voltage-to-frequency converter. The inverse AC Josephson effect. If the phase takes the form φ(t) = φ0 + nωt + a sin(ωt), the voltage and current will be P ~ :U (t) = 2e ω(n + a cos(ωt)), I(t) = Ic ∞ m=−∞ Jn (a) sin(φ0 + (n + m)ωt) The DC components will then be ~ ω, I(t) = Ic J−n (a) sin φ0 Hence, for distinct DC voltages, :UDC = n 2e the junction may carry a DC current and the junction acts like a perfect frequency-to-voltage converter.

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The device

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The Josephson junction finds numerous important applications. Its properties are exploited in →SQUIDs used to measure magnetic flux at the quantum level. This finds application in medicine for measurement of small currents in the brain and the heart. A version using different superfluids can be used as a quantum gyroscope. Josephson junctions are also used in Rapid Single Flux Quantum integrated circuits, and some other of their properties can be exploited to build photon or particle detectors. Josephson junctions are used as microwave detectors in the giga- and terahertz range. When assembled in two dimensional arrays, "testboards" for the physical realization of mathematical model systems are created. When assembled in linear arrays (connected in series) the inverse Josephson effect is used as a representation of the SI unit volt. It is also speculated that Josephson junctions may allow the realisation of qubits, the key elements of a future quantum computer. There are two general types of Josephson junctions: overdamped and underdamped. In overdamped junctions, the barrier is conducting (ie. it is a normal metal or superconductor bridge). The effects of the junction’s internal electrical resistance will be large compared to its small capacitance. An overdamped junction will quickly reach a unique equilibrium state for any given set of conditions. The barrier of an underdamped junction is an insulator. The effects of the junction’s internal resistance will be minimal. Underdamped junctions do not have unique equilibrium states, but are hysteretic. Josephson effect

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Figure 27 The current - voltage curve of a Josephson junction at low temperature. The vertical portions (zero voltage) of the curve represent Cooper pair tunnelling. There is a small magnetic field applied, so that the maximum Josephson current is severely reduced. Hysteresis is clearly visible around 100 microvolts. The portion of the curve between 100 and 300 microvolts is current independent, and is the regime where the device can be used as a detector.

A Josephson junction can be transformed into the so-called Giaever tunneling junction by the application of a small, well defined magnetic field. In such a situation, the new device is called a superconducting tunneling junction (STJ) 130 and is used as a very sensitive photon detector throughout a wide range of the spectrum, from infrared to hard x-ray. Each photon breaks up a number of Cooper pairs. This number depends on the ratio of the photon energy to approximately twice the value of the gap parameter of the material of the junction. The detector can be operated as a photon-counting spectrometer, with a spectral resolution limited by the statistical fluctuations in the number of released charges. The detector has to be cooled to extremely low temperature, typically below 1 kelvin, to distinguish the signals generated by the detector from the thermal noise. Small arrays of STJs have demonstrated their potential

130 European Space Agency.

Payload and Advanced Concepts: Superconducting Tunnel Junction (STJ)(http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=33525). Last updated: February 17 2005.

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Quantum gyroscope Quantum computer

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as spectro-photometers and could further be used in astronomy. They are also used to perform energy dispersive X-ray spectroscopy and in principle they could be used as elements in infrared imaging devices as well. 131

Source: http://en.wikipedia.org/wiki/Josephson_effect

Principal Authors: Filou, Encephalon, Danko Georgiev MD, Rorro, Marudubshinki

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Klein-Gordon equation

The Klein-Gordon equation (Klein-Fock-Gordon equation or sometimes Klein-Gordon-Fock equation) is the relativistic version of the →Schrödinger equation. It was named after Oskar Klein and Walter Gordon.

Details

The Schrödinger equation for a free particle is p2 2m ψ

∂ = i~ ∂t ψ

where

p = −i~∇ is the momentum operator (∇ being the del operator).

The Schrödinger equation suffers from not being relativistically covariant, meaning it does not take into account Einstein’s special relativity. It is natural to try to use the identity from special relativity p E = p2 c2 + m2 c4 for the energy; then, just inserting the quantum mechanical momentum operator, yields the equation

131 Enss C (ed). Cryogenic Particle Detection. Topics in Applied Physics Vol. 99. Springer; 2005. ISBN

3-540-20113-0

Klein-Gordon equation

257 p

∂ ψ. (−i~∇)2 c2 + m2 c4 ψ = i~ ∂t

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This, however, is a cumbersome expression to work with because of the square root. In addition, this equation, as it stands, is nonlocal.

Klein and Gordon instead worked with the more general square of this equation (the Klein-Gordon equation for a free particle), which in covariant notation reads (2 + µ2 )ψ = 0, where µ=

mc ~

and 2 =

1 ∂2 c2 ∂t2

− ∇2 .

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This operator is called the d’Alembert operator. Today this form is interpreted as the relativistic field equation for a scalar (i.e. spin-0) particle.

The Klein-Gordon equation was allegedly first found by Schrödinger, before he made the discovery of the equation that now bears his name. He rejected it because he couldn’t make it include the spin of the electron. The way Schrödinger found his equation was by making simplifications in the KleinGordon equation. In 1926, soon after the Schrödinger equation was introduced, Fock wrote an article about its generalization for the case of magnetic fields, where forces were dependent on velocity, and independently derived this equation. Both Klein and Fock used Kaluza and Klein’s method. Fock also determined the gauge theory for the wave equation. The Klein-Gordon equation for a free particle has a simple plane wave solution.

Relativistic free particle solution

The Klein-Gordon equation for a free particle can be written as ∇2 ψ −

1 ∂2 ψ c2 ∂t2

=

m2 c2 ψ ~2

with the same solution as in the non-relativistic case: ψ(r, t) = ei(k·r−ωt)

except with the constraint

Klein-Gordon equation

258 −k 2 +

ω2 c2

=

m2 c2 . ~2

hpi = hψ| − i~∇|ψi = ~k, ∂ |ψi = ~ω. hEi = hψ|i~ ∂t

FT

Just as with the non-relativistic particle, we have for energy and momentum:

Except that now when we solve for k and ω and substitute into the constraint equation, we recover the relationship between energy and momentum for relativistic massive particles: hEi2 = m2 c4 + hpi2 c2 .

For massless particles, we may set m = 0 in the above equations. We then recover the relationship between energy and momentum for massless particles:

DR A

hEi = h|p|ic.

See also

• Dirac equation • Oskar Klein • →Quantum field theory

References •

Sakurai, J. J. (1967). Advanced Quantum Mechanics. Addison Wesley. ISBN 0201067102.

External links •

Linear Klein-Gordon Equation 132 at EqWorld: The World of Mathematical Equations.

•

Nonlinear Klein-Gordon Equation 133 at EqWorld: The World of Mathematical Equations. generalizing the Klein-Gordon equation 134 to include a generalized space

•

132 http://eqworld.ipmnet.ru/en/solutions/lpde/lpde203.pdf 133 http://eqworld.ipmnet.ru/en/solutions/npde/npde2107.pdf

134 http://www.public.asu.edu/~kevinlg/i256/grand_paper.pdf

Klein-Gordon equation

259

Source: http://en.wikipedia.org/wiki/Klein-Gordon_equation

FT

Principal Authors: PAR, Charles Matthews, Kurtan, Light current, MarSch, Cyp, Phys

Ladder operators

In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum. Suppose that two operators X and N have the commutation relation

DR A

[N, X] = cX

for some scalar c. Then the operator X will act in such a way as to shift the eigenvalue of an eigenstate of N by c:

In other words, if |ni is an eigenstate of N with eigenvalue n then X|ni is an eigenstate of N with eigenvalue n + c. A raising operator for N is an operator X for which c is real and positive and a lowering operator is one for which c is real and negative. If N is a Hermitian operator then c must be real and the Hermitian adjoint of X obeys the commutation relation: [N, X † ] = −cX † .

In particular, if X is a lowering operator for N then X † is a raising operator for N (and vice-versa).

See also •

→Creation and annihilation operators

Source: http://en.wikipedia.org/wiki/Ladder_operators

Principal Authors: Fropuff, Aza, Yevgeny Kats, RedWordSmith

Ladder operators

260

Laplace-Runge-Lenz vector

FT

In the classical mechanics of a particle moving under a central force, the Laplace-Runge-Lenz vector is defined as: A = p × L − mk rr where: • • •

r is the position vector of the particle of mass m, L is the angular momentum, k is a parameter that describes strength of the central force

The Laplace-Runge-Lenz vector is constant when only inverse-square central forces such as gravity or electrostatics act on the particle. In quantum mechanics, the Laplace-Runge-Lenz vector can be used to derive the spectrum of the hydrogen atom (without using the →Schrödinger equation).

DR A

The conservation of both the angular momentum vector and the LaplaceRunge-Lenz vector results from a special symmetry of the system when only inverse-square central forces act on the particle. Other central forces reduce the system to a simpler rotational symmetry that conserves angular momentum but not the Laplace-Runge-Lenz vector.

Conservation of the Laplace-Runge-Lenz vector By assumption, the force F acting on the particle is a central force F=

dp dt

= f (r) rr

for some function f (r) of the radius r. The angular momentum L ≡ r × p is d conserved under central forces, i.e., dt L = 0. Therefore, h   i dp d r dr m dr 2 dr dt (p × L) = dt × L = f (r) r × r × m dt = f (r) r r r · dt − r dt

where the momentum p ≡ m dr dt as usual and where we have simplified the triple cross product using Lagrange’s formula   dr 2 dr r × r × dr = r r · dt dt − r dt Exploiting the identity d dt

(r · r) = 2r ·

dr dt

=

d dt


r2 = 2r dr dt

Laplace-Runge-Lenz vector

261 we arrive at d dt

(p × L) = −mf (r)r2

h

1 dr r dt

−

r dr r2 dt

i

d = −mf (r)r2 dt −k , r2




this becomes

FT

In the special case of an inverse-square force f (r) =  
mkr d d r d r dt (p × L) = mk dt r = dt

r r

Therefore, A is conserved for inverse-square forces   mkr d d d =0 r dt A ≡ dt (p × L) − dt

Analogous conserved quantities can be defined for other central forces, but none are as simple as the Laplace-Runge-Lenz vector A.

Properties of the Laplace-Runge-Lenz vector

By its definition, A is perpendicular to L, i.e., A · L = 0. (Recall that r · L = 0, because L ≡ r × p).

DR A

The Laplace-Runge-Lenz vector can be used to derive the elliptical orbits of the Kepler problem A · r ≡ Ar cos θ = r · (p × L) − mkr

where θ is the angle between the position and Laplace-Runge-Lenz vectors. Permuting the scalar triple product r·(p × L) = L·(r × p) = L2 and rearranging yields the formula for an ellipse   1 mk A r = L2 1 + mk cos θ

The vector points toward the pericenter, from the geometric center of the orbit to the attracting, central body. The magnitude for a periodic orbit with eccentricity e is given by: |A| = mke

The seven quantities A, L and E are related by the equations A · L = 0 and A2 = m2 k 2 + 2mEL2

giving five independent constants of motion. This is consistent with the six degrees of freedom (the particle’s initial position and velocity vectors, each with three components) that specify the orbit of the particle, after removing the initial time (which is not determined by A, L and E). Laplace-Runge-Lenz vector

262

FT

All central forces conserve angular momentum, but the Laplace-Runge-Lenz vector is conserved only for inverse-square central forces. As shown by Bertrand’s theorem, inverse-square forces can produce closed, elliptical orbits; the constancy of the Laplace-Runge-Lenz vector corresponds to the constancy of the axes of the ellipse. The introduction of even small deviations from the inverse-square force causes the axes of the ellipse to precess, i.e., to vary with time.

History of the Laplace-Runge-Lenz vector

Laplace calculated the components of the vector A explicitly and showed that it was conserved in his Traite de Mecanique Celeste (1799). It was rediscovered independently by Hamilton in 1845. A vector derivation was given in 1900 by Gibbs (the inventor of vectors). This derivation was repeated by Runge in a popular German textbook on vectors, which was referenced by Lenz in a paper on the perturbed hydrogen atom (1924). Thus, it is also known as the Runge-Lenz vector.

DR A

Poisson brackets of the angular momentum and Laplace-Runge-Lenz vector The Poisson brackets of the three components of the angular momentum vector, Li , i = 1, 2, 3 are   Li , Lj = ijk Lk

where ijk is the fully antisymmetric tensor (i.e., the Levi-Civita symbol). The Poisson brackets are represented here with square brackets, both for consistency with the references below and because they will be interpreted as Lie brackets in the next section. Defining a reduced Laplace-Runge-Lenz vector D with the same units as angular momentum D≡ √

A 2m|E|

the Poisson brackets of D with the angular momentum vector L can be written in a similar form   Di , Lj = ijk Dk The Poisson brackets of the reduced Laplace-Runge-Lenz vector D with itself depend on whether the total energy E is negative (closed, elliptical orbits) or

Laplace-Runge-Lenz vector

263

FT

positive (open, hyperbolic orbits). For negative energies, the Poisson brackets are   Di , Dj = ijk Lk whereas, for positive energy, the Poisson brackets have the opposite sign   Di , Dj = −ijk Lk The Casimir invariants for negative energies are defined C1 ≡ D · D + L · L C2 ≡ D · L

which have zero Poisson brackets with all components of D and L [C1 , Li ] = [C1 , Di ] = [C2 , Li ] = [C2 , Di ] = 0

DR A

For the Kepler problem, C2 = 0, since the two vectors are perpendicular.

Quantum mechanics and the Laplace-Runge-Lenz vector A simple prescription for quantizing a classical system is to set the commutation relations of the quantum mechanical operators equal to the Poisson bracket of the corresponding classical variables, multiplied by i~.

By carrying out this quantization and calculating the eigenvalues of the C1 Casimir operator for the Kepler problem, Wolfgang Pauli was able to derive the spectrum of hydrogen-like atoms. Specifically, he was able to show that the energy levels vary as 1/n2 where n is an integer. This elegant derivation was obtained prior to the development of wave mechanics; see the Bohm reference below for details.

Symmetry and the classical Laplace-Runge-Lenz vector Noether’s theorem states that every conserved quantity in a physical system corresponds to a continuous symmetry of the system, and vice versa. For general central forces, one of the conserved quantities is the angular momentum, for which the corresponding Noether symmetry is the rotation group SO(3). For the special case of inverse-square force laws, a greater symmetry is possible, which results in the conservation of both the angular momentum and Laplace-Runge-Lenz vector

264

|e|2 = e21 + e22 + e23 + e24

FT

the Laplace-Runge-Lenz vector. For negative energies, the symmetry is that of four-dimensional rotation group SO(4) which conserves the length of a fourdimensional vector

By contrast, for positive energies, the symmetry is that of the group of Lorentz transformations SO(3,1), which conserves the Minkowski length of a 4-vector ds2 = e21 + e22 + e23 − e24

These symmetries may be identified as follows. The Poisson brackets above can be interpreted as Lie brackets that define a Lie algebra that corresponds to a symmetry group. For negative energies, that group is SO(4) whereas, for positive energies, that group is SO(3,1).

See also

Astrodynamics: Orbit, Eccentricity vector, Orbital elements →Quantum mechanics

DR A

• •

References •

Goldstein H. (1980) Classical Mechanics, 2nd ed., Addison Wesley, pp. 102105, 421-422.

•

Landau LD and Lifshitz EM (1976) Mechanics, 3rd. ed., Pergamon Press. ISBN 0080210228 (hardcover) and ISBN 0080291414 (softcover).

•

Goldstein H. (1975) "Prehistory of the Runge-Lenz vector", Am. J. Phys., 43, 735-738.

•

Goldstein H. (1976) "More on the prehistory of the Runge-Lenz vector", Am. J. Phys., 44, 1123-1124.

•

Lenz W. (1924) "Über den Bewegungsverlauf und Quantenzustaende der gestörten Keplerbewegung", Z. Phys., 24, 197-207.

•

Pauli W., Jr. (1926) "Über das Wasserstoffspektrum vom Standpunkt der neuen Quantenmechanik", Z. Phys., 36, 336–363.

•

Bohm A. (1986) Quantum Mechanics: Foundations and Applications, 2nd ed., Springer Verlag, pp. 208-222. Laplace-Runge-Lenz vector

265 •

John Baez, Mysteries of the gravitational 2-body problem 135

FT

Source: http://en.wikipedia.org/wiki/Laplace-Runge-Lenz_vector Principal Authors: WillowW, 0.39, AugPi, Charles Matthews, Agentsoo

Large Area Neutron Detector

The Large Area Neutron Detector is also know as LAND. It is the name of a detector for neutrons installed at GSI (Institute for Heavy Ion Research) situated in Arheilgen close to the city of Darmstadt, Germany.

DR A

The detector is built of 10 planes with 20 paddles each. The paddles have a size of 10x10x200cm 3 and are composed of a converter (iron) and plastic scintillator material. Within the paddle the 5mm thick converters serve as a dense target for neutrons (uncharged particles) leading to processes that eject charged particles through the process of hadronic showers. The interspersed 5mm plastic scintillator stripes produce light for the passing charged particles. The stripes of one paddle are viewed at both ends by photomultipliers. The detector system was built in 1990.

A research group at GSI is named after this detector. They aim at studying the nuclear structure of radioactive (short-lived) nuclei.

See also •

T. Blaich et al., Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, A314(1992), p. 136-154, Elsevier NORTH-HOLLAND, ISSN: 0168-9002

•

GSI 136

Source: http://en.wikipedia.org/wiki/Large_Area_Neutron_Detector

135 http://math.ucr.edu/home/baez/gravitational.html 136 http://www.gsi.de

Large Area Neutron Detector

266

Lindblad equation

FT

In quantum mechanics, the Lindblad equation or master equation in the Lindblad form is the most general type of master equation allowed by quantum mechanics to describe non-unitary (dissipative) evolution of the density matrix ρ (such as ensuring normalisation and hermiticity of ρ). It reads:  P ρ˙ = − ~i [H, ρ] − ~1 n,m hn,m ρLm Ln + Lm Ln ρ − 2Ln ρLm + h.c.

where ρ is the density matrix, H is the Hamiltonian part, Lm are operators defined by the system to model as are the constants hn,m . It is a quantum analog of the Liouville equation in classical mechanics. A related equation describes the time evolution of the expectation values of observables, it is given by the →Ehrenfest theorem.

DR A

The most common Lindblad equation is that describing the damping of a quantum harmonic oscillator, it has L0 = a, L1 = a† , h0,1 = −(γ/2)(¯ n + 1), h1,0 = −(γ/2)¯ n with all others hn,m = 0. Here n ¯ is the mean number of excitations in the reservoir damping the oscillator and γ is the decay rate. Additional Lindblad operators can be included to model various forms of dephasing and vibrational relaxation. These methods have been incorporated into grid-based density matrix propagation methods.

Source: http://en.wikipedia.org/wiki/Lindblad_equation Principal Authors: Laussy, Charles Matthews, Hansbethe

London moment

The →London moment is a quantum-mechanical phenomenon whereby a spinning superconductive metal sphere generates a magnetic field whose axis lines up exactly with the spin axis. The term may also refer to the magnetic moment of any rotation of any superconductor, caused by the electrons lagging behind the rotation of the object.

Etymology

Named for the physical scientist Fritz London, and moment as in magnetic moment.

London moment

267

• •

London force Proton conductor

References

FT

See also

Source: http://en.wikipedia.org/wiki/London_moment

Many-body problem

This article is about the many-body problem in quantum mechanics. For the n-body problem in classical mechanics, see n-body problem.

DR A

Definition

The many-body problem may be defined as the study of the effects of interaction between bodies on the behaviour of a many-body system, i.e. a closed system which does not contains just a few bodies in action, such as the collisions discussed in classical mechanics. Due to the amount of particles/bodies contained in such a system, we cannot any longer describe the mechanics of the system, by using a small system of equations, as we do in classical mechanics. The sheer number and high-energy chaotic interaction of each body with another forces us to use result in such mathimatical techniques as the canonical transformation technique. Another preferred method for solving the problem, is simply to ignore any interactions present. The many-body problem is usually posed in quantum mechanics, as the question of solving for more complex problems than the hydrogen atom — for example, the chemistry of all realistic molecules, such as a molecule of plastic.

Quotes

"It would indeed be remarkable if Nature fortified herself against further advances in knowledge behind the analytical difficulties of the many-body problem." - Max Born, 1960

Many-body problem

268

See also Hartree-Fock approximation

FT

•

References and further reading •

Jenkins, Stephen. "3. The Many Body Problem and Density Functional Theory". Many Body Problem and Density Functional Theory 137 • D. J. Thouless, "The quantum mechanics of many-body systems", 2d ed., New York, Academic Press, 1972. ISBN 0126915601 • D. J. Thouless, "The quantum mechanics of many-body systems", New York, Academic Press, 1961. LCCN 61012282 /L/r842

External links

Evidence for Efimov quantum states in an ultracold gas of Cs atoms 138

DR A

•

Source: http://en.wikipedia.org/wiki/Many-body_problem

Principal Authors: The Anome, Hephaestos, Omegatron, Grendelkhan, Pt, Conscious, Oneboy, SeventyThree

Mathematical formulation of quantum mechanics

One of the remarkable characteristics of the mathematical formulation of quantum mechanics, which distinguishes it from mathematical formulations of theories developed prior to the early 1900s, is its use of abstract mathematical structures, such as →Hilbert spaces and operators on these spaces. Many of these structures had not even been considered before the twentieth century. In a general sense they are drawn from functional analysis, a subject within pure mathematics that developed in parallel with, and was influenced by the needs of quantum mechanics. In brief, physical observables such as energy and momentum were no longer considered as functions on some phase space, but as eigenvalues of operators which act on such functions.

137 http://newton.ex.ac.uk/research/qsystems/people/jenkins/mbody/mbody3.html 138 http://www2.uibk.ac.at/exphys/ultracold/

Mathematical formulation of quantum mechanics

269

FT

This formulation of quantum mechanics, called canonical quantization, continues to be used today, and still forms the basis of ab-initio calculations in atomic, molecular and solid-state physics. At the heart of the description is an idea of quantum state which, for systems of atomic scale, is radically different from the previous models of physical reality. While the mathematics is a complete description and permits calculation of many quantities that can be measured experimentally, there is a definite limit to access for an observer with macroscopic instruments. This limitation was first elucidated by Heisenberg through a thought experiment, and is represented mathematically by the non-commutativity of quantum observables.

DR A

Prior to the emergence of quantum mechanics as a separate theory, the mathematics used in physics consisted mainly of differential geometry and partial differential equations; probability theory was used in statistical mechanics. Geometric intuition clearly played a strong role in the first two and, accordingly, theories of relativity were formulated entirely in terms of geometric concepts. The phenomenology of quantum physics arose roughly between 1895 and 1915, and for the 10 to 15 years before the emergence of quantum theory (around 1925) physicists continued to think of quantum theory within the confines of what is now called classical physics, and in particular within the same mathematical structures. The most sophisticated example of this is the Sommerfeld-Wilson-Ishiwara quantization rule, which was formulated entirely on the classical phase space.

History of the formalism

The "old quantum theory" and the need for new mathematics Main article: Old quantum theory

In the decade of 1890, Planck was able to derive the blackbody spectrum and solve the classical ultraviolet catastrophe by making the unorthodox assumption that, in the interaction of radiation with matter, energy could only be exchanged in discrete units which he called quanta. Planck postulated a direct proportionality between the frequency of radiation and the quantum of energy at that frequency. The proportionality constant, h is now called Planck’s constant in his honour. In 1905, Einstein explained certain features of the photoelectric effect by assuming that Planck’s light quanta were actual particles, which he called photons.

Mathematical formulation of quantum mechanics

DR A

FT

270

Figure 28

A sketch to justify spectroscopy observations for hydrogen atoms

In 1913, Bohr calculated the spectrum of the hydrogen atom with the help of a new model of the atom in which the electron could orbit the proton only on a discrete set of classical orbits, determined by the condition that angular momentum was an integer multiple of Planck’s constant. Electrons could make quantum leaps from one orbit to another, emitting or absorbing single quanta of light at the right frequency. All of these developments were phenomenological and flew in the face of the theoretical physics of the time. Bohr and Sommerfeld went on to modify classical mechanics in an attempt to deduce the Bohr model from first principles. They proposed that, of all closed classical orbits traced by a mechanical system in its phase space, only the ones that enclosed an area which was a multiple of Planck’s constant were actually allowed. The most sophisticated version of this formalism was the so-called Sommerfeld-Wilson-Ishiwara quantization. Although the Bohr model of the hydrogen atom could be explained in this way, the spectrum of the helium atom (classically an unsolvable 3-body problem) could not be predicted. The mathematical status of quantum theory remained uncertain for some time.

Mathematical formulation of quantum mechanics

271 In 1923 de Broglie proposed that wave-particle duality applied not only to photons but to electrons and every other physical system.

FT

The situation changed rapidly in the years 1925-1930, when working mathematical foundations were found through the groundbreaking work of Erwin Schrödinger and Werner Heisenberg and the foundational work of John von Neumann, Hermann Weyl and Paul Dirac, and it became possible to unify several different approaches in terms of a fresh set of ideas.

The "new quantum theory"

DR A

Erwin Schrödinger’s wave mechanics originally was the first successful attempt at replicating the observed quantization of atomic spectra with the help of a precise mathematical realization of de Broglie’s wave-particle duality. Schrödinger proposed an equation (now bearing his name) for the wave associated to an electron in an atom according to de Broglie, and explained energy quantization by the well-known fact that differential operators of the kind appearing in his equation had a discrete spectrum. However, Schrödinger himself initially did not understand the fundamental probabilistic nature of quantum mechanics, as he thought that the (squared amplitude of the) wavefunction of an electron must be interpreted as the charge density of an object smeared out over an extended, possibly infinite, volume of space. It was Max Born who introduced the probabilistic interpretation of the (squared amplitude of the) wave function as the probability distribution of the position of a pointlike object. With hindsight, Schrödinger’s wave function can be seen to be closely related to the classical Hamilton-Jacobi equation. Werner Heisenberg’s matrix mechanics formulation, introduced contemporaneously to Schrödinger’s wave mechanics and based on algebras of infinite matrices, was certainly very radical in light of the mathematics of classical physics. In fact, at the time linear algebra was not generally known to physicists in its present form.

The reconciliation of the two approaches is generally associated to Paul Dirac, who wrote a lucid account in his 1930 classic Principles of Quantum mechanics. In it, he introduced the bra-ket notation, together with an abstract formulation in terms of the →Hilbert space used in functional analysis, and showed that Schödinger’s and Heisenberg’s approaches were two different representations of the same theory. Dirac’s method is now called canonical quantization. The first complete mathematical formulation of this approach is generally credited to John von Neumann’s 1932 book Mathematical Foundations of Quantum Mechanics, although Hermann Weyl had already referred to Hilbert spaces (which he called unitary spaces) in his 1927 classic book. It was developed in parallel

Mathematical formulation of quantum mechanics

272 with a new approach to the mathematical spectral theory based on linear operators rather than the quadratic forms that were David Hilbert’s approach a generation earlier.

Later developments

FT

Though theories of quantum mechanics continue to evolve to this day, there is a basic framework for the mathematical formulation of quantum mechanics which underlies most approaches and can be traced back to the mathematical work of John von Neumann. In other words, discussions about interpretation of the theory, and extensions to it, are now mostly conducted on the basis of shared assumptions about the mathematical foundations.

DR A

The application of the new quantum theory to electromagnetism resulted in quantum field theory, which was developed starting around 1930. Quantum field theory has driven the development of more sophisticated formulations of quantum mechanics, of which the one presented here is a simple special case. In fact, the difficulties involved in implementing any of the following formulations cannot be said yet to have been solved in a satisfactory fashion except for ordinary quantum mechanics. • • • • •

Feynman path integrals axiomatic, algebraic and constructive quantum field theory geometric quantization quantum field theory in curved spacetime C* algebra formalism

On a different front, von Neumann originally dispatched quantum measurement with his infamous postulate on the collapse of the wavefunction, raising a host of philosophical problems. Over the intervening 70 years, the problem of measurement became an active research area and itself spawned some new formulations of quantum mechanics. • • • •

Relative state/Many-worlds interpretation of quantum mechanics Decoherence Consistent histories formulation of quantum mechanics Quantum logic formulation of quantum mechanics

A related topic is the relationship to classical mechanics. Any new physical theory is supposed to reduce to successful old theories in some approximation. For quantum mechanics, this translates into the need to study the so-called classical limit of quantum mechanics. Also, as Bohr emphasized, human cognitive abilities and language are inextricably linked to the classical realm, and so classical descriptions are intuitively more accessible than quantum ones. In Mathematical formulation of quantum mechanics

273 particular, quantization, namely the construction of a quantum theory whose classical limit is a given and known classical theory, becomes an important area of quantum physics in itself.

• • •

FT

Finally, some of the originators of quantum theory (notably Einstein and Schrödinger) were unhappy with what they thought were the philosophical implications of quantum mechanics. In particular, Einstein took the position that quantum mechanics must be incomplete, which motivated research into so-called hidden-variable theories. The issue of hidden variables has become in part an experimental issue with the help of quantum optics. de Broglie- Bohm- Bell pilot wave formulation of quantum mechanics Bell’s inequalities Kochen-Specker theorem

Mathematical structure of quantum mechanics

DR A

A physical system is generally described by three basic ingredients: states; observables; and dynamics (or law of time evolution) or, more generally, a group of physical symmetries. A classical description can be given in a fairly direct way by a phase space model of mechanics: states are points in a symplectic phase space, observables are real-valued functions on it, time evolution is given by a one-parameter group of symplectic transformations of the phase space, and physical symmetries are realized by symplectic transformations. A quantum description consists of a →Hilbert space of states, observables are self adjoint operators on the space of states, time evolution is given by a oneparameter group of unitary transformations on the Hilbert space of states, and physical symmetries are realized by unitary transformations.

Postulates of quantum mechanics

The following summary of the mathematical framework of quantum mechanics can be partly traced back to von Neumann’s postulates. •

Each physical system is associated with a (topologically) separable complex →Hilbert space H with inner product hφ | ψi. Rays (one-dimensional subspaces) in H are associated with states of the system. In other words, physical states can be identified with equivalence classes of vectors of length 1 in H, where two vectors represent the same state if they differ only by a phase factor. Separability is a mathematically convenient hypothesis, with

Mathematical formulation of quantum mechanics

274

FT

the physical interpretation that countably many observations are enough to uniquely determine the state. • The Hilbert space of a composite system is the Hilbert space tensor product of the state spaces associated with the component systems. For a nonrelativistic system consisting of a finite number of distinguishable particles, the component systems are the individual particles. • Physical symmetries act on the Hilbert space of quantum states unitarily or antiunitarily (supersymmetry is another matter entirely). • Physical observables are represented by densely-defined self-adjoint operators on H.

The expected value (in the sense of probability theory) of the observable A for the system in state represented by the unit vector |ψi ∈ H is hψ | A | ψi

DR A

By spectral theory, we can associate a probability measure to the values of A in any state ψ. We can also show that the possible values of the observable A in any state must belong to the spectrum of A. In the special case A has only discrete spectrum, the possible outcomes of measuring A are its eigenvalues. More generally, a state can be represented by a so-called density operator, which is a trace class, nonnegative self-adjoint operator ρ normalized to be of trace 1. The expected value of A in the state ρ is

tr(Aρ)

If ρψ is the orthogonal projector onto the one-dimensional subspace of H spanned by |ψi, then

tr(Aρψ ) = hψ | A | ψi

Density operators are those that are in the closure of the convex hull of the one-dimensional orthogonal projectors. Conversely, one-dimensional orthogonal projectors are extreme points of the set of density operators. Physicists also call one-dimensional orthogonal projectors pure states and other density operators mixed states.

Mathematical formulation of quantum mechanics

275

FT

One can in this formalism state Heisenberg’s uncertainty principle and prove it as a theorem, although the exact historical sequence of events, concerning who derived what and under which framework, is the subject of historical investigations outside the scope of this article. Superselection sectors. The correspondence between states and rays needs to be refined somewhat to take into account so-called superselection sectors. States in different superselection sectors cannot influence each other, and the relative phases between them are unobservable.

Pictures of dynamics

In the so-called →Schrödinger picture of quantum mechanics, the dynamics is given as follows:

DR A

The time evolution of the state is given by a differentiable function from the real numbers R, representing instants of time, to the Hilbert space of system states. This map is characterized by a differential equation as follows: If |ψ (t)i denotes the state of the system at any one time t, the following →Schrödinger equation holds: d i~ dt |ψ(t)i = H |ψ(t)i

where H is a densely-defined self-adjoint operator, called the system Hamiltonian, i is the imaginary unit and ~ is the reduced Planck constant. As an observable, H corresponds to the total energy of the system. Alternatively, by Stone’s theorem one can state that there is a strongly continuous one-parameter unitary group U (t): H → H such that |ψ(t + s)i = U (t) |ψ(s)i

for all times s, t. The existence of a self-adjoint Hamiltonian H such that U (t) = e−(i/~)tH

is a consequence of Stone’s theorem on one-parameter unitary groups. The →Heisenberg picture of quantum mechanics focuses on observables and instead of considering states as varying in time, it regards the states as fixed and the observables as changing. To go from the Schrödinger to the Heisenberg picture one needs to define time-independent states and time-dependent operators thus: |ψi = |ψ(0)i

Mathematical formulation of quantum mechanics

276 A(t) = U (−t)AU (t)

FT

It is then easily checked that the expected values of all observables are the same in both pictures hψ | A(t) | ψi = hψ(t) | A | ψ(t)i

and that the time-dependent Heisenberg operators satisfy d i~ dt A(t) = [A(t), H]

This assumes A is not time dependent in the Schrödinger picture. Notice the commutator expression is purely formal when one of the operators is unbounded. One would specify a representation for the expression to make sense of it.

DR A

The so-called Dirac picture or interaction picture has time-dependent states and observables, evolving with respect to different Hamiltonians. This picture is most useful when the evolution of the states can be solved exactly, confining any complications to the evolution of the operators. For this reason, the Hamiltonian for states is called "free Hamiltonian" and the Hamiltonian for observables is called "interaction Hamiltonian". In symbols: d i~ dt |ψ(t)i = H0 |ψ(t)i

d i~ dt A(t) = [A(t), Hint ]

The interaction picture does not always exist, though. In interacting quantum field theories, Haag’s theorem states that the interaction picture does not exist. This is because the Hamiltonian cannot be split into a free and an interacting part within a superselection sector. The Heisenberg picture is the closest to classical mechanics, but the Schrödinger picture is considered easiest to understand by most people, to judge from pedagogical accounts of quantum mechanics. The Dirac picture is the one used in perturbation theory, and is specially associated to quantum field theory. Similar equations can be written for any one-parameter unitary group of symmetries of the physical system. Time would be replaced by a suitable coordinate parameterizing the unitary group (for instance, a rotation angle, or a translation distance) and the Hamiltonian would be replaced by the conserved quantity associated to the symmetry (for instance, angular or linear momentum).

Mathematical formulation of quantum mechanics

277

Representations

FT

The original form of the →Schrödinger equation depends on choosing a particular representation of Heisenberg’s canonical commutation relations. The Stone-von Neumann theorem states all irreducible representations of the finitedimensional Heisenberg commutation relations are unitarily equivalent. This is related to quantization and the correspondence between classical and quantum mechanics, and is therefore not strictly part of the general mathematical framework. The quantum harmonic oscillator is an exactly-solvable system where the possibility of choosing among more than one representation can be seen in all its glory. There, apart from the Schrödinger (position or momentum) representation one encounters the Fock (number) representation and the BargmannSegal (phase space or coherent state) representation. All three are unitarily equivalent.

Time as an operator

DR A

The framework presented so far singles out time as the parameter that everything depends on. It is possible to formulate mechanics in such a way that time becomes itself an observable associated to a self-adjoint operator. At the classical level, it is possible to arbitrarily parameterize the trajectories of particles in terms of an unphysical parameter s, and in that case the time t becomes an additional generalized coordinate of the physical system. At the quantum level, translations in s would be generated by a "Hamiltonian" H -E, where E is the energy operator and H is the "ordinary" Hamiltonian. However, since s is an unphysical parameter, physical states must be left invariant by "s-evolution", and so the physical state space is the kernel of H -E (this requires the use of a rigged Hilbert space and a renormalization of the norm). This is related to quantization of constrained systems and quantization of gauge theories. It is also possible to formulate a quantum theory of "events" where time becomes an observable( see D. Edwards ).

The problem of measurement

The picture given in the preceding paragraphs is sufficient for description of a completely isolated system. However, it fails to account for one of the main differences between quantum mechanics and classical mechanics, that is the effects of measurement. The von Neumann description of quantum measurement of an observable A, when the system is prepared in a pure state ψ is the following: •

Let A have spectral resolution

Mathematical formulation of quantum mechanics

278

A=

R

λd EA (λ)

,

FT

where E A is the resolution of the identity (also called projection-valued measure) associated to A. Then the probability of the measurement outcome lying in an interval B of R is |E A (B) ψ| 2. In other words, the probability is obtained by integrating the characteristic function of B against the countably additive measure

hψ | EA ψi .

•

If the measured valued is contained in B, then immediately after the measurement, the system will be in the (generally non-normalized) state E A ψ.

DR A

If the measured value does not lie in B, replace B by its complement for the above state. For example, suppose the state space is the n-dimensinal complex Hilbert space C n and A is a Hermitian matrix with eigenvalues i , with corresponding eigenvectors ψ i . The projection-valued measure associated with A is E A is then

EA (B) = |ψi ihψi | ,

where B is a Borel set containing only the single eigenvalue i . If the system is prepared in state |ψi

Then the probability of a measurement returning the value i can be calculated by integrating the spectral measure

hψ | EA ψi

over B i . This gives trivially

hψ|ψi ihψi | ψi = |hψ | ψi i|2 .

The characteristic property of the von Neumann measurement scheme is that repeating the same measurement will give the same results. This is also called the projection postulate. Mathematical formulation of quantum mechanics

279

FT

A more general formulation replaces the projection-valued measure with a positive-operator valued measure(POVM). To illustrate, take again the finitedimensional case. Here we would replace the rank-1 projections |ψi ihψi | by a finite set of positive operators Fi Fi∗ , whose sum is still the identity operator as before. Just as a set of possible outcomes λ1 · · · λn is associated to a PVM, the same can be said for a POVM. Suppose the measurement outcome is λi . Instead of collapsing to the (unnormalized) state |ψi ihψi |ψi after the measurement, the system now will be in the state Fi |ψi. Since the Fi Fi∗ ’s need not be mutually orthogonal projections, the projection postulate of von Neumann no longer holds. Both of the above approaches apply to general mixed ensembles.

DR A

In von Neumann’s approach, the state transformation due to measurement is distinct from that due to time evolution in several ways. For example, time evolution is determinisic and unitary whereas measurement is non-deterministic and non-unitary. However, since both types of state transformation take one quantum state to another, this difference was viewed by many as unsatisfactory. The POVM formalism views measurement as one among many other quantum operations, which are described by completely positive maps which do not increase the trace.

The relative state interpretation

An alternative interpretation of measurement is Everett’s relative state interpretation, which was later dubbed the "many-worlds interpretation" of quantum mechanics.

List of mathematical tools

Part of the folklore of the subject concerns the mathematical physics textbook Courant-Hilbert, put together by Richard Courant from David Hilbert’s Göttingen University courses. The story is told (by mathematicians) that physicists had dismissed the material as not interesting in the current research areas, until the advent of Schrödinger’s equation. At that point it was realised that the mathematics of the new quantum mechanics was already laid out in it. It is also said that Heisenberg had consulted Hilbert about his matrix mechanics, and Hilbert observed that his own experience with infinite-dimensional matrices had derived from differential equations, advice which Heisenberg ignored, missing the opportunity to unify the theory as Weyl and Dirac did a few years later. Whatever the basis of the anecdotes, the mathematics of theory was conventional at the time, where the physics was radically new. The main tools include:

Mathematical formulation of quantum mechanics

280 • • •

FT

linear algebra: complex numbers, eigenvectors, eigenvalues functional analysis: →Hilbert spaces, linear operators, spectral theory differential equations: Partial differential equations, Separation of variables, Ordinary differential equations, Sturm-Liouville theory, eigenfunctions • harmonic analysis: Fourier transforms See also: list of mathematical topics in quantum theory.

References • • •

DR A

•

S. Auyang, How is Quantum Field Theory Possible?, Oxford University Press, 1995. D. Edwards, The Mathematical Foundations of Quantum Mechanics, Synthese, 42 (1979),pp.1-70. G. Emch, Algebraic Methods in Statistical Mechanics and Quantum Vield Theory, Wiley-Interscience, 1972. R. Jost, The General Theory of Quantized Fields, American Mathematical Society, 1965. A. Gleason, Measures on the Closed Subspaces of a Hilbert Space, Journal of Mathematics and Mechanics, 1957. G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963 (paperback reprint by Dover 2004). J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. Reprinted in paperback form. R. F. Streater and A. S. Wightman, PCT, Spin and Statistics and All That, Benjamin 1964 (Reprinted by Princeton University Press) M. Reed and B. Simon, Methods of Mathematical Physics, vols 1-IV, Academic Press 1972. H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications, 1950.

•

• •

• •

•

Source: http://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics Principal Authors: Miguel, CSTAR, Charles Matthews, Phys, Mct mht

Mathematical formulation of quantum mechanics

281

Matrix mechanics

FT

Matrix mechanics is a formulation of quantum mechanics created by Werner Heisenberg in 1925. Matrix mechanics was the first complete definition of quantum mechanics, its laws, and properties that described fully the behavior of subatomic particles by associating their properties with matrices. It has been shown to be exactly equivalent to the Schroedinger wave formulation of quantum mechanics and is the basis of the bra-ket notation used to summarize quantum mechanical wave functions.

Development of matrix mechanics

When it was introduced by Werner Heisenberg, Max Born and Pascual Jordan in 1925, matrix mechanics was not immediately accepted and was a source of great controversy.

DR A

Schrödinger’s later introduction of wave mechanics was favored because there were no visual aids to fall back on in matrix mechanics and the mathematics were unfamiliar to most physicists.

Matrix mechanics consists of an array of quantities which when appropriately manipulated gave the observed frequencies and intensities of spectral lines. Heisenberg said himself that once and for all he had gotten rid of all electron orbits that did not exist. However, in Heisenberg’s theory, the result of multiplication changes depending on its order. This means that the physical quantities in Heisenberg’s theory are not ordinary variables but mathematical matrices. Heisenberg developed matrix mechanics by interpreting the electron as a particle with quantum behavior. It is based on sophisticated matrix computations which introduce discontinuities and quantum jumps. In atomic physics, through spectroscopy, it was known that observational data related to atomic transitions arise from interactions of the atoms with light quanta. Heisenberg was the first to say that the atomic spectrum which showed spectral lines only in places where photons were being absorbed or emitted as electrons changed orbitals were the only relevant objects to be defined. Heisenberg recognized that the matrix formulation was built on the premise that all physical observables must be represented by matrices. The set of eigenvalues of the matrix representing an observable is the set of all possible values that could arise as outcomes of experiments conducted on a system to measure the observable. Since the outcome of an experiment to measure a real observable must be a real number, Hermitian matrices would represent such observables as their eigenvalues are real. If the result of a measurement is a Matrix mechanics

282 certain eigenvalue, the corresponding eigenvector represents the state of the system immediately after the measurement.

FT

Instead of using three dimensional orbitals, Heisenberg’s matrix mechanics described the space in which the state of a quantum system inhabits as being one-dimensional as in the case of an anharmonic oscillator. To illustrate, consider the simple example of a point particle like an electron that is free to move on a line. An observable in this case could be the position of the particle, represented by the matrix X. Since the particle could be anywhere on the line, the possible outcome of a measurement of its position could be any one of an infinite set of eigenvalues of X, denoted by x. Thus X must be an infinitedimensional matrix, and hence so is the corresponding linear vector space. Thus even one-dimensional motion could have an infinite-dimensional linear vector space associated with it. This made operators, functions, and spaces necessary to describe quantum mechanics.

DR A

The act of measurement in matrix mechanics is taken to ’collapse’ the state of the system to that eigenvector (or eigenstate). Anyone familiar with Schroedinger’s wave equation which came later in 1925 will be familiar with this concept in the form of wavefunction collapse. If one were to make simultaneous measurements of two or more observables, the system will collapse to a common eigenstate of these observables right after the measurement.

Further, from matrix theory we know that eigenvectors corresponding to distinct eigenvalues of a Hermitian matrix are orthogonal to each other which is analogous to the x, y, z axes of the Cartesian coordinate system except with an infinite number of distinct eigenvalues, and hence as many mutually orthogonal eigenvectors directed along different independent directions in the linear vector space. Prior to measurement, the system could have been in a linear superposition of different eigenstates, with impossible to know coefficients. The precise state of the quantum system before the measurement cannot be known. The Copenhagen interpretation is concerned only with outcomes of experiments. Since a single measurement of any observable A yields one of the eigenvalues of A as the outcome, and collapses the state of the system to the corresponding eigenstate, subsequent measurements made immediately thereafter would continue to yield the same eigenvalue. So the correct thing to do would be to prepare a collection of a very large number of identical copies of the system and conduct a single trial on each copy. The arithmetic mean of all the results thus obtained is the average value we see, denoted by (A). The Uncertainty Principle in matrix mechanics stems from the fact that, in general, two matrices A and B do not obey the arithmetical law of commutation. Matrix mechanics

283 The commutator A B - B A = [A, B] does not equal 0. The famous commutation relation that is the basis for Quantum Mechanics and the later Uncertainty Principle is: P

− q(n, k)p(k, n) = h/2πi

FT

k p(n, k)q(k, n)

In 1925, Werner Heisenberg was not yet 24 years old.

Mathematical details

In quantum mechanics in the Heisenberg picture the state vector, |ψ> does not change with time, and an observable A satisfies   d −1 [A, H] + ∂A A = (i~) . dt ∂t classical

In some sense, the Heisenberg picture is more natural and fundamental than the →Schrödinger picture, especially for relativistic theories. Lorentz invariance is manifest in the Heisenberg picture.

DR A

Moreover, the similarity to classical physics is easily seen: by replacing the commutator above by the Poisson bracket, the Heisenberg equation becomes an equation in Hamiltonian mechanics. By the Stone-von Neumann theorem, the Heisenberg picture and the →Schrödinger picture are unitarily equivalent. See also →Schrödinger picture.

Deriving Heisenberg’s equation

Suppose we have an observable A (which is a Hermitian linear operator). The expectation value of A for a given state |ψ(t)> is given by: hAit = hψ(t)|A|ψ(t)i

or if we write following the →Schrödinger equation |ψ(t)i = e−iHt/~ |ψ(0)i

(where H is the Hamiltonian and h\hbar is Planck’s constant divided by 2*pi ) we get hAit = hψ(0)|eiHt/~ Ae−iHt/~ |ψ(0)i

and so we define

A(t) := eiHt/~ Ae−iHt/~

Now,

Matrix mechanics

d dt A(t)

= ~i HeiHt/~ Ae−iHt/~ +



284 

∂A ∂t classical

+ ~i eiHt/~ A · (−H)e−iHt/~

∂A ∂t classical

i ~

(HA(t) − A(t)H) +

FT

(differentiating according to the product rule),   = ~i eiHt/~ (HA − AH) e−iHt/~ + ∂A ∂t classical =  

(the last passage is valid since exp(-iHt/hbar) commutes with H )   = ~i [H, A(t)] + ∂A ∂t classical

(where [X,Y] is the commutator of two operators and defined as [X, Y ] := XY − Y X) So we get d dt A(t)

= ~i [H, A(t)] +





∂A ∂t classical

DR A

Though matrix mechanics does not include concepts such as the wave function of Erwin Schrödinger’s wave equation, the two approaches were proven to be mathematically equivalent by the mathematician John von Neumann.

See also • • • • •

→Interaction picture →Quantum mechanics →Schrödinger equation →Bra-ket notation Basic quantum mechanics

External links • • •

An Overview of Matrix Mechanics 139 Matrix Methods in Quantum Mechanics 140 Heisenberg Quantum Mechanics 141

Source: http://en.wikipedia.org/wiki/Matrix_mechanics Principal Authors: Voyajer, Zowie, Elroch, Snoyes, Neilc

139 http://www.cobalt.chem.ucalgary.ca/ziegler/educmat/chm386/rudiment/tourquan/matmech.htm 140 http://www.cobalt.chem.ucalgary.ca/ziegler/educmat/chm386/rudiment/quanmath/matrix.htm 141 http://www.aip.org/history/heisenberg/p08.htm

Matrix mechanics

285

Matrix model

FT

The term matrix model may refer to one of several concepts: •

In theoretical physics, a matrix model is a system (usually a quantum mechanical system) with matrix-valued physical quantities. See, for example, Lax pair.

•

The "old" matrix models are relevant for string theory in two spacetime dimensions. The "new" matrix model is a synonym for Matrix theory.

•

Matrix population models are used to model wildlife and human population dynamics.

Source: http://en.wikipedia.org/wiki/Matrix_model

DR A

Principal Authors: QFT, Conscious, Charles Matthews, A2Kafir, Shenme

Maxwell-Boltzmann statistics In statistical mechanics, Maxwell-Boltzmann statistics describes the statistical distribution of material particles over various energy states in thermal equilibrium, when the temperature is high enough and density is low enough to render quantum effects negligible. Maxwell-Boltzmann statistics are therefore applicable to almost any terrestrial phenomena for which the temperature is above a few tens of kelvins.

The expected number of particles with energy i for Maxwell-Boltzmann statistics is Ni where: Ni N

=

gi e(i −µ)/kT

=

gi e−i /kT Z

where: • • • • •

Ni is the number of particles in state i i is the energy of the i -th state gi is the degeneracy of state i, the number of microstates with energy i µ is the chemical potential k is Boltzmann’s constant

Maxwell-Boltzmann statistics

286 T is absolute temperature N is the total number of particles N=

•

i Ni

Z is the partition function Z=

•

P

−i /kT i gi e

P

FT

• •

e (...) is the exponential function

Equivalently, the distribution is sometimes expressed as Ni N

=

1 e(i −µ)/kT

=

e−i /kT Z

where the index i now specifies an individual microstate rather than the set of all states with energy i

DR A

Fermi-Dirac and Bose-Einstein statistics apply when quantum effects have to be taken into account and the particles are considered "indistinguishable". The quantum effects appear if the concentration of particles (N/V) ≥ n q (where n q is the quantum concentration). The quantum concentration is when the interparticle distance is equal to the thermal de Broglie wavelength i.e. when the wavefunctions of the particles are touching but not overlapping. As the quantum concentration depends on temperature; high temperatures will put most systems in the classical limit unless they have a very high density e.g. a White dwarf. Fermi-Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), Bose-Einstein statistics apply to bosons. Both Fermi-Dirac and Bose-Einstein become Maxwell-Boltzmann statistics at high temperatures or low concentrations.

Maxwell-Boltzmann statistics are often described as the statistics of "distinguishable" classical particles. In other words the configuration of particle A in state 1 and particle B in state 2 is different from the case where particle B is in state 1 and particle A is in state 2. When this idea is carried out fully, it yields the proper (Boltzmann) distribution of particles in the energy states, but yields non-physical results for the entropy, as embodied in →Gibbs paradox. These problems disappear when it is realized that all particles are in fact indistinguishable. Both of these distributions approach the Maxwell-Boltzmann distribution in the limit of high temperature and low density, without the need

Maxwell-Boltzmann statistics

287

FT

for any ad hoc assumptions. Maxwell-Boltzmann statistics are particularly useful for studying gases F-D statistics are most often used for the study of electrons in solids. As such, they form the basis of semiconductor device theory and electronics.

A derivation of the Maxwell-Boltzmann distribution In this particular derivation, the Boltzmann distribution will be derived using the assumption of distinguishable particles, even though the ad hoc correction for Boltzmann counting is ignored, the results remain valid.

Suppose we have a number of energy levels, labelled by index i , each level having energy i and containing a total of Ni particles. To begin with, lets ignore the degeneracy problem. Assume that there is only one way to put Ni particles into energy level i.

DR A

The number of different ways of performing an ordered selection of one object from N objects is obviously N . The number of different ways of selecting 2 objects from N objects, in a particular order, is thus N (N − 1) and that of selecting n objects in a particular order is seen to be N !/(N − n)!. The number of ways of selecting 2 objects from N objects without regard to order is N (N − 1) divided by the number of ways 2 objects can be ordered, which is 2!. It can be seen that the number of ways of selecting n objects from N objects without regard to order is the binomial coefficient: N !/n!(N − n)!. If we have a set of boxes numbered 1, 2, . . . , k, the number of ways of selecting N1 objects from N objects and placing them in box 1, then selecting N2 objects from the remaining N − N1 objects and placing them in box 2 etc. is       (N −N1 )! Nk ! ! W = N !(NN−N . . . N !0! )! N !(N −N −N )! k 1

= N!

1

2

1

2

Qk

i=1 (1/Ni !)

where the extended product is over all boxes containing one or more objects. If the i -th box has a "degeneracy" of gi , that is, it has gi sub-boxes, such that any way of filling the i -th box where the number in the sub-boxes is changed is a distinct way of filling the box, then the number of ways of filling the i -th box must be increased by the number of ways of distributing the Ni objects in the gi boxes. The number of ways of placing Ni distinguishable objects in gi boxes is giNi . Thus the number of ways (W ) that N atoms can be arranged in energy levels each level i having gi distinct states such that the i -th level has Ni atoms is: Q g Ni W = N ! Ni i !

Maxwell-Boltzmann statistics

288 For example, suppose we have three particles, a, b, and c, and we have three energy levels with degeneracies 1, 2, and 1 respectively. There are 6 ways to arrange the three particles .

.

.

.

.

c. .c b. .b a. .a ab ab ac ac bc bc

FT

.

The six ways are calculated from the formula:  2 1 0 Q g Ni 2 1 W = N ! Ni i ! = 3! 12! 1! 0! = 6

We wish to find the set of Ni for which W is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of W and ln(W ) are achieved by the same values of Ni and, since it is easier to accomplish mathematically, we will maximise the latter function instead. We constrain our solution using Lagrange multipliers forming the function: P P f (Ni ) = ln(W ) + α(N − Ni ) + β(E − Ni i )

DR A

Using Stirling’s approximation for the factorials and taking the derivative with respect to Ni , and setting the result to zero and solving for Ni yields the Maxwell-Boltzmann population numbers: Ni =

gi eα+βi

It can be shown thermodynamically that β = 1/kT where k is Boltzmann’s constant and T is the temperature, and that α = -µ/kT where µ is the chemical potential, so that finally: Ni =

gi e(i −µ)/kT

Note that the above formula is sometimes written: Ni =

gi ei /kT /z

where z = exp(µ/kT ) is the absolute activity. Alternatively, we may use the fact that P i Ni = N to obtain the population numbers as −i /kT

Ni = N gi e Z

where Z is the partition function defined by:

Maxwell-Boltzmann statistics

289 Z=

−i /kT i gi e

P

FT

Another derivation In the above discussion, the Boltzmann distribution function was obtained via directly analysing the multiplicities of a system. Alternatively, one can make use of the canonical ensemble. In a canonical ensemble, a system is in thermal contact with a reservoir. While energy is free to flow between the system and the reservoir, the reservoir is thought to have infinitely large heat capacity as to maintain constant temperature, T, for the combined system.

In the present context, our system is assumed to be have energy levels i with degeneracies gi . As before, we would like to calculate the probability that our system has energy i .

DR A

If our system is in state s1 , then there would be a corresponding number of microstates available to the reservoir. Call this number ΩR (s1 ). By assumption, the combined system (of the system we are interested in and the reservoir) is isolated, so all microstates are equally probable. Therefore, for instance, if ΩR (s1 ) = 2 ΩR (s2 ), we can conclude that our system is twice as likely to be in state s2 than s1 . In general, if P (si ) is the probability that our system is in state si , P (s1 ) P (s2 )

=

ΩR (s1 ) . ΩR (s2 )

Since the entropy of the reservoir SR = k ln ΩR , the above becomes P (s1 ) P (s2 )

=

eSR (s1 ) /k eSR (s1 ) /k

= e(SR (s1 )−SR (s2 ))/k .

Next we recall the thermodynamic identity: dSR =

1 T (dUR

+ P dVR − µdNR ).

In a canonical ensemble, there is no exchange of particles, so the dNR term is zero. Similarly, dVR = 0. This gives (SR (s1 ) − SR (s2 ) =

1 T (UR (s1 ) − UR (s2 ))

= − T1 (E(s1 ) − E(s2 ))

, where UR (si ) and E(si ) denote the energies of the reservoir and the system at si , respectively. For the second equality we have used the conservation of energy. Substituting into the first equation relating P (s1 ), P (s2 ): P (s1 ) P (s2 )

=

e−E(s1 )/kT e−E(s2 )/kT

, which implies, for any state s of the system

Maxwell-Boltzmann statistics

290 P (s) =

1 −E(s)/kT Ze

FT

, where Z is an appropriately chosen "constant" to make total probability 1. (Z is constant provided that the temperature T is invariant.) It is obvious that P Z = s e−E(s)/kT

where the index s run through all microstates of the system. (Z is sometimes called the Boltzmann sum over states.) If we index the summation via the energy eigenvalues instead of all possible states, degeneracy must be taken into account. The probability of our system having energy i is simply the sum of the probabilities of all corresponding microstates: P (i ) =

1 −i /kT Z gi e

where, with obvious modification, Z = before.

This is the same result as

DR A

Comments

−j /kT . j gj e

P

•

Notice that in this formulation, the initial assumption "... suppose the system has total N particles..." is dispensed with. Indeed, the number of particles possessed by the system plays no role in arriving at the distribution. Rather, how many particles would occupy states with energy i follows as an easy consequence.

•

What has been presented above is essentially a derivation of the canonical partition function. As one can tell by comparing the definitions, the Boltzman sum over states is really no different from the canonical partition function.

•

Exactly the same approach can be used to derive Fermi-Dirac and BoseEinstein statistics. However, there one would replace the canonical ensemble with the grand canonical ensemble, since there is exchange of particles between the system and the reservoir. Also, the system one considers in those cases is a single particle state, not a particle. (In the above discussion, we could have assumed our system to be a single atom.)

Maxwell-Boltzmann statistics

291

Limits of applicability The Bose-Einstein and Fermi-Dirac distributions may be written: gi e(i −µ)/kT ±1

FT

Ni =

Assuming the minimum value of i is small, it can be seen that the conditions under which the Maxwell-Boltzmann distribution is valid is when e−µ/kT  1

For an ideal gas, we can calculate the chemical potential using the development in the Sackur-Tetrode article to show that:     ∂E = −kT ln NVΛ3 µ = ∂N S,V

where E is the total internal energy, S is the entropy, V is the volume, and Λ is the thermal de Broglie wavelength. The condition for the applicability of the Maxwell Boltzmann distribution for an ideal gas is again shown to be  1.

DR A

V N Λ3

See also • •

→Bose-Einstein statistics →Fermi-Dirac statistics

Source: http://en.wikipedia.org/wiki/Maxwell-Boltzmann_statistics Principal Authors: Mct mht, PAR, Jheald, LifeStar, Sam Korn

Measurement in quantum mechanics The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications.

Measurement in quantum mechanics

292

Formalism of measurement Measurable quantities ("observables") as operators

FT

An observable quantity is represented mathematically by an Hermitian or self adjoint operator. The set of the operator’s eigenvalues represent the set of possible outcomes of the measurement. For each eigenvalue there is a corresponding eigenstate (or "eigenvector"), which will be the state of the system after the measurement. Some properties of this representation are •

DR A

The eigenvalues of Hermitian matrices are real. The possible outcomes of a measurement are precisely the eigenvalues of the given obervable. • A Hermitian matrix can be unitarily diagonalized (See Spectral theorem), generating an orthonormal basis of eigenvectors which spans the state space of the system. In general, the state of a system can be represented as a linear combination of eigenvectors of any Hermitian operator. Physically, this is to say that any state can be expressed as a superposition of the eigenstates of an observable. Important examples are: •

• •

The Hamiltonian operator, representing the total energy of the system; with ˆ = pˆ2 + the special case of the nonrelativistic Hamiltonian operator: H 2m

V (ˆ x). ∂ The momentum operator: pˆ = ~i ∂x (in the position basis). ∂ The position operator: x ˆ, where x ˆ = −~ i ∂p (in the momentum basis).

Operators can be noncommuting. In the finite dimensional case, two Hermitian operators commute if they have the same set of {normalized} eigenvectors. Noncommuting observables are said to be incompatible and can not be measured simultaneously. This can be seen via the uncertainty principle.

Eigenstates and projection

ˆ be a measurement operator, Assume the system is prepared in state |ψi. Let O an observable, with eigenstates |ni for n = 1, 2, 3, ... and corresponding eigenvalues O1 , O2 , O3 , .... If the measurement outcome is ON , the system will then "collapse" to the state |N i after measurement. The case of a continuous spectrum is more involved, since, physically speaking, the basis has uncountably many eigenstates, but the general concept is the same. In the position representation, for instance, the eigenstates can be represented by the set of delta functions, indexed by all possible positions of Measurement in quantum mechanics

293

Wavefunction collapse

FT

the particle. In the experimental setting, the resolution of any given measurement is finite, and therefore the continuous space may be divided into discrete segments. Another solution is to approximate any lab experiments by a "box" potential (which bounds the volume in which the particle can be found, and thus ensures a discrete spectrum).

Given any quantum state which is a superposition of eigenstates at time t |ψi = c1 e−iE1 t |1i + c2 e−iE2 t |2i + c3 e−iE3 t |3i + · · · ,

if we measure, for example, the energy of the system and receive E 2

DR A

(this result is chosen randomly according to probability given by 2

|c | Pr(En ) = P n

|c |2 k k

),

then the system’s quantum state after the measurement is |ψi = e−iE2 t |2i

so any repeated measurement of energy will yield E 2.

Figure 1. The process of wavefunction collapse illustrated. The process in which a quantum state becomes one of the eigenstates of the operator corresponding to the measured observable is called "collapse", or "wavefunction collapse". The final eigenstate appears randomly with a probability equal to the square of its overlap with the original state. The process of collapse has been studied in many experiments, most famously in the double-slit experiment. The wavefunction collapse raises serious questions of determinism and locality, as demonstrated in the EPR paradox and later in GHZ entanglement. In the last few decades, major advances have been made toward a theoretical understanding of the collapse process. This new theoretical framework, called quantum decoherence, supersedes previous notions of instantaneous collapse

Measurement in quantum mechanics

294

FT

and provides an explanation for the absence of quantum coherence after measurement. While this theory correctly predicts the form and probability distribution of the final eigenstates, it does not explain the randomness inherent in the choice of final state. There are two major approaches toward the "wavefunction collapse": •

•

Accept it as it is. This approach was supported by Niels Bohr and his Copenhagen interpretation which accepts the collapse as one of the elementary properties of nature (at least, for small enough systems). According to this, there is an inherent randomness embedded in nature, and physical observables exist only after they are measured (for example: as long as a particle’s speed isn’t measured it doesn’t have any defined speed). Reject it as a physical process and relate to it only as an illusion. This approach says that there is no collapse at all, and we only think there is. Those who support this approach usually offer another interpretation of quantum mechanics, which avoids the wavefunction collapse.

DR A

von Neumann measurement scheme

The von Neumann measurement scheme, an ancestor of quantum decoherence theory, describes measurements by taking into account the measuring apparatus which is also treated as a quantum object. Let the quantum state be in P the superposition |ψi = n cn |ψn i, where |ψn i are eigenstates of the operator that needs to be measured. In order to make the measurement, the measured system described by |ψi needs to interact with the measuring apparatus described by the quantum state |φi, so that the total wave function before the interaction is |ψi|φi. After the interaction, the total wave function exhibits P the unitary evolution |ψi|φi → n cn |ψn i|φn i, where |φn i are orthonormal states of the measuring apparatus. The unitary evolution above is referred to as premeasurement. One can also introduce the interaction with the environment |ei, so that, after the interaction, the total wave function takes a form P n cn |ψn i|φn i|en i, which is related to the phenomenon of decoherence. The above is completely described by the →Schrödinger equation and there are not any interpretational problems with this. Now the problematic wavefunction collapse does not need to be understood as a process |ψi → |ψn i on the level of the measured system, but can also be understood as a process |φi → |φn i on the level of the measuring apparatus, or as a process |ei → |en i on the level of the environment. Studying these processes provides considerable insight into the measurement problem by avoiding the arbitrary boundary between the quantum and classical worlds, though it does not explain the presence of randomness in the choice of final eigenstate. If the set of states {|ψn i}, {|φn i}, or {|en i} represents a set of states that do not overlap in space, the appearance Measurement in quantum mechanics

295

FT

of collapse can be generated by either the Bohm interpretation or the Everett interpretation which both deny the reality of wavefunction collapse; they both, though, predict the same probabilities for collapses to various states as does the conventional interpretation. The Bohm interpretation is held to be correct only by a small minority of physicists, since there are difficulties with the generalization for use with relativistic quantum field theory. However, there is no proof that the Bohm interpretation is inconsistent with quantum field theory, and work to reconcile the two is ongoing. The Everett interpretation easily accommodates relativistic quantum field theory.

Example

Suppose that we have a particle in a box. If the energy of the particle is 2 2 2 π ~ measured to be EN = N2mL then the corresponding state of the system is 2 q   R N πx 2 |ψN i = |xihx|ψN idx where hx|ψN i = hx|N i = sin , which is L L determined by solving the Time-Independent →Schrödinger equation for the given potential.

DR A

Alternatively, if instead of knowing the energy of the particle the particle’s position is determined to be a distance S from the left wall of the box, the R corresponding system state is |ψS i = |xihx|ψS idx where hx|ψS i = hx|Si = δ(S − x).

These two state functions |ψN i and |ψS i are distinct functions (of the position x after we left multiply by the bra state hx|), but they are in general not orthogonal to each other: q   R RL 2 N πx hψS |ψN i = hS|N i = hS|xihx|N idx = 0 δ(S − x) dx = L sin L q   N πS 2 . L sin L The two systems are therefore distinct; a position measurement is instantaneous whereas a definite value of energy EN is established only in the limit of an infinitely long observation period. Completeness of eigenvectors of Hermitian operators guarantees that either system state, being the eigenvector to one measurement operator, can be expressed as a linear combination of eigenvectors of the other measurement operator:  
RP R P 2 P nπx |Si = n |ni hn|Si = sin nπS dx = n |xihx|ni hn|Si dx = |xi L n sin L L R R |xiδ(S − x)dx, i.e. |Si = |xiδ(S − x)dx and

|N i =

R

|si hs|N i ds =

R

q   |si L2 sin NLπs ds.

Measurement in quantum mechanics

296

FT

The time dependence of the system states is determined by the Time Dependent →Schrödinger equation. In the preceding example, with energy eigenvalues En , it follows that the time dependent solution is P |ψ(t)i = n |nihn|ψS i e−itEn /~ ,

where t represents the time since the particle’s location in space was measured. Consequently q   hn|ψ(t)i = hn|ψS i e−itEn /~ = L2 sin nπS e−itEn /~ 6= 0 L

at least for several distinct energy eigenstates |ni, for all values t, and for all 0 < S < L.

DR A

The particle state |ψS i therefore can not have evolved (in the above technical sense) into state |ψN i (which is orthogonal to all energy eigenstates, except itself), for any duration t. While this conclusion may be characterized accordingly instead as "the wave function of the particle having been projected, or having collapsed into" the energy eigenstate |ψN i, it is perhaps worth emphasizing that any definite value of energy EN can be established only in the limit of a long-lasting trial and never for any finite value of time.

Optimal quantum measurement

What is the optimal quantum measurement to distinguish mixed states from a given ensemble? This is a natural question of which the solution is well understood, and is given by a semidefinite programming.

More specifically, suppose a mixed state ρi is drawn from the ensemble with probability pi , we wish to find a POVM measurement {Πi } so that P i pi tr(Π P i ρi ) is maximized. This is clearly P a semidefinit programming:

max

i

pi tr(Πi ρi ) s.t. Πi ≥ 0,

i

Πi = I.

Interestingly, the dual problem has a nice description:

min tr(X) s.t. X − pi ρi ≥ 0.

ˆ i and X ˆ be the solutions of the primal and the dual, we have Let Π ˆ ˆ Πi · (X − pi ρi ) = 0. ˆ i must also be From this one can conclude that if all ρi are pure states, then Π of rank 1. Furthermore, if ρi ’s are in addition independent, then the optimal measure is a von Neumann measurement.

Philosophical problems of quantum measurements What physical interaction constitutes a measurement? Until the advent of quantum decoherence theory in the late 20th century, a major conceptual problem of quantum mechanics and especially the Copenhagen Measurement in quantum mechanics

297 interpretation was the lack of a distinctive criterion for a given physical interaction to qualify as "a measurement" and cause a wavefunction to collapse. This best illustrated by the →Schrödinger’s cat paradox.

• •

FT

Major philosophical and metaphysical questions surround this issue:

The concept of weak measurements. Macroscopic systems (such as chairs or cats) do not exhibit counterintuitive quantum properties, which can only be observed in microscopic particles such as electrons or photons. This invites the question of when a system is "big enough" to behave classically and not quantum mechanically?

Quantum decoherence theory has successfully addressed other questions that previously haunted quantum measurement theory: •

Does a measurement depend on the existence of a self-aware observer?

DR A

Answer: No. Coupling an isolated quantum system to another quantum system with many degrees of freedom generically transfers the coherence of the first system into mutual coherence of the two systems. The initially isolated quantum system then appears to "collapse." Interpreting the second system as a measurement apparatus, as in the von Neumann scheme, shows that no consciousness or self-awareness is necessary for collapse of the first system. •

What interactions are strong enough to constitute a measurement?

This question is quantitatively answered by decoherence theory, given a model for the measurement apparatus. The scaling of the measurement effects with the system/apparatus interaction strength usually only weakly depends on the choice of a model for the apparatus, so one can give a generic description of the strength of a measurement induced by a given interaction.

Does measurement actually determine the state? The question of whether a measurement actually determines the state, is deeply related to the Wavefunction collapse. Most versions of the Copenhagen interpretation answer this question with an unqualified "yes". See also: • •

Philosophies: Copenhagen interpretation. People (actualist philosophers): Henri Poincaré, Niels Bohr.

Measurement in quantum mechanics

298

The quantum entanglement problem See EPR paradox.

FT

See also •

DR A

Measurement related problems and paradoxes • Afshar experiment • Measurement problem • Wavefunction collapse • EPR paradox • Renninger negative-result experiment • Elitzur-Vaidman bomb-testing problem • →Schrödinger’s cat • Interpretations of quantum mechanics • Transactional interpretation • Copenhagen interpretation • Many-worlds interpretation • Quantum mechanics formalism • →Quantum mechanics • →Mathematical formulation of quantum mechanics • →Schrödinger equation • →Bra-ket notation • Generalized measurement (POVM, Positive operator valued measure)

External links

• Analog: A Farewell to Copenhagen? 142 • " The Double Slit Experiment 143". (physicsweb.org) • Shahriar S. Afshar, " Waving Copenhagen Good-bye: Were the founders of Quantum Mechanics wrong? 144" This link presents a view that is not endorsed by the majority of physicists. • " Variation on the similar two-pin-hole "which-way" experiment 145". (reported in New Scientist; July 24), Reprint at irims.org 146

142 http://www.analogsf.com/0410/altview2.shtml 143 http://physicsweb.org/article/world/15/9/1

144 http://my.harvard.edu/cgi-bin/webevent/webevent.cgi?cmd=showevent&ncmd=calmonth&cal=9719

&y=2004&m=3&d=23&id=10416384&token=G6409379:1&sb=0&cf=cal&lc=calmonth&swe=1&set=0 &sa=0&sort=e,m,t&ws=0&sib=0&de=0&tf=0 145 http://www.sciencefriday.com/images/shows/2004/073004/AfsharExperimentSmall.jpg 146 http://www.irims.org/quant-ph/030503/

Measurement in quantum mechanics

299 " Measurement in Quantum Mechanics 147" Henry Krips in the Stanford Encyclopedia of Philosophy • Decoherence, the measurement problem, and interpretations of quantum mechanics 148 • Measurements and Decoherence 149 • Yonina C. Eldar, Alexandre Megretski, and George C. Verghese. Designing optimal quantum detectors via semidefinite programming. IEEE Transactions on Information Theorey, Vol. 49, No. 4, 1007–1012, 2003.

FT

•

Source: http://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics

Principal Authors: MathKnight, Fwappler, MichaelCPrice, William M. Connolley, Dave Kielpinski

Molecular Hamiltonian

DR A

The non relativistic molecular Hamiltonian for a multi-electron molecule in atomic units is:

H = TN + Hel

where P 1 TN = a − 2M ∇2a a

is the kinetic energy operator corresponding to the molecular dynamics and H el is the electronic molecular Hamiltonian. M a are the masses of the nuclei. ∇2a is the Laplacian with respect to the cartesian nuclear coordinates associated to the molecular geometry. The relativistic molecular hamiltonian differs because it contains terms that depend upon the electron spin and the nuclear spin. The electron spin appears naturally in the solution of the Dirac equation, but the nuclear spins are added in phenomenologically as if the nuclei were "heavy" electrons. The electronic molecular Hamiltonian can be replaced within the BornOppenheimer approximation by the potential energy surfaces. The →Schrödinger equation becomes then an equation describing the "motion" of the nuclei only. The "motion" of the electrons has been taken into account during the diagonalization of the electronic molecular Hamiltonian (see computational chemistry for more details).

147 http://plato.stanford.edu/entries/qt-measurement/ 148 http://arxiv.org/abs/quant-ph/0312059 149 http://arxiv.org/abs/quant-ph/0505070

Molecular Hamiltonian

300

References •

FT

The discrete eigenvalues of the molecular Hamiltonian are called molecular energy levels.

Richard Moss, Advanced Molecular Quantum Mechanics, ISBN 412-104903.

Source: http://en.wikipedia.org/wiki/Molecular_Hamiltonian Principal Authors: Vb, Iain.mcnab, Karol Langner

Multiplicative quantum number

DR A

In quantum field theory, multiplicative quantum numbers are conserved quantum numbers of a special kind. A given quantum number q is said to be additive if in a particle reaction the sum of the q-values of the interacting particles is the same before and after the reaction. Most conserved quantum numbers are additive in this sense; the electric charge is one example. A multiplicative quantum number q is one for which the corresponding product, rather than the sum, is preserved. Any conserved quantum number is a symmetry of the Hamiltonian of the system (see Noether’s theorem). Symmetry groups which are examples of the abstract group called Z 2 give rise to multiplicative quantum numbers. This group consists of an operation, P, whose square is the identity, P 2 = 1. Thus, all symmetries which are mathematically similar to parity (physics) give rise to multiplicative quantum numbers.

In principle, mutliplicative quantum numbers can be defined for any Abelian group. An example would be to trade the electric charge, Q, (related to the Abelian group U(1) of electromagnetism), for the new quantum number exp(2i π Q ). Then this becomes a multiplicative quantum number by virtue of the charge being an additive quantum number. However, this route is usually followed only for discrete subgroups of U(1), of which Z 2 finds the widest possible use.

Multiplicative quantum number

301

See also Parity, C-symmetry, →T-symmetry and G-parity

FT

•

References •

Group theory and its applications to physical problems, by M. Hamermesh (Dover publications, 1990) ISBN 0486661814

Source: http://en.wikipedia.org/wiki/Multiplicative_quantum_number

Principal Authors: Bambaiah, Michael Hardy, Phys, Shimgray, Agentsoo

Neutral particle oscillations

DR A

In particle physics, neutral particle oscillation is the transmutation of a neutral particle with nonzero internal quantum numbers into its antiparticle. These oscillations and the associated mixing of particles gives insight into the realization of discrete parts of the Poincare group, ie, parity (P), charge conjugation (C) and time reversal invariance (T).

The phenomenon

Neutral particles such as the kaon, neutron, bottom quark mesons or neutrinos have internal quantum numbers called flavour. This means that the particle and antiparticle are different. If both particle and antiparticle can decay into the same final state, then it is possible for the decay and its time reversed process to contribute to oscillations— A → F → B → F → A → ...

Neutral particle oscillations

302 where A is the particle, B is the antiparticle, and F is the common set of particles into which both can decay. The example of the neutral kaon is pictured here.

FT

Such a process is actually connected to the mass renormalization of the states A and B in quantum field theory. However, under certain circumstances it can be tackled through a simpler quantum mechanics model which neglects these intermediate multi-particle quantum states and concentrates only on the states A and B.

Quantum mechanical model

Consider a state |ψ(t)> = a(t)|A> + b(t)|B>. Its time evolution is governed by the Hamiltonian, H, through the action of the evolution operator U(t) = exp(iHt) on |ψ(0)>. The 2×2 matrix Hamiltonian can be written as

H=



HAA HBA

HAB HBB



= M − 2i Γ,

DR A

where the Hamiltonian can be decomposed into a mass matrix M and a decay width matrix Γ, both of which are 2×2 Hermitean matrices. We introduce the notation M AB = |M AB| e iα and Γ AB = |Γ AB| e i(α+β ). A and B are both flavour eigenstates. Oscillations mix these states, and the mass eigenstates are the states which propagate without mixing, ie, the eigenvectors of H.

CPT symmetry

The action of the discrete spacetime symmetries are C|A> = -|B>,

P|A> = -|A> and T|A> = +|A>.

If the Hamiltonian is CPT symmetric, then (CPT)H(CPT) -1=H. The transformation properties above imply that CPT|A>=|B>. Then = , so a test of CPT symmetry is that the masses and the decay widths of the particle and the antiparticle are equal. This is a major class of experimental tests of CPT symmetry. Any 2×2 matrix can be written in the form E 0I+Eu.σ, where I is the identity matrix, σ i are the Pauli matrices and u is an unit vector. With CPT symmetry, the diagonal  elements of H are equal, so r sin φ ∗ |MAB |+ 12 |ΓAB |e−i(β+π/2) MAB −iΓ∗ iα iφ AB /2   u = cos φ , , e = MAB −iΓAB /2 = e |MAB |+ 21 |ΓAB |ei(β+π/2) 0

Neutral particle oscillations

303 where φ is a complex angle. H is diagonalized by rotating u into an unit vector in the z-direction. The eigenvectors and eigenvalues are √1 (|Ai ± eiφ |Bi), 2

FT

|1, 2i =

E1,2 = MAA − 12 ΓAA ± eiα |MAB | + 21 |ΓAB |ei(β+π/2) ,

where the plus signs are for the state |1> and minus, for |2>. A change in the phase convention, |B> → e -iθ|B> changes the definition of the eigenstates, but not the eigenvalues. By appropriate choice of this phase, the angle φ can always be set equal to zero, so that the eigenstates are orthogonal.

Oscillations, regeneration and CP violation CPT symmetry breaking

See also

Kaons, B-Bbar oscillations and neutrino oscillations CP violation and CPT symmetry

DR A

• •

Source: http://en.wikipedia.org/wiki/Neutral_particle_oscillations

Principal Authors: Bambaiah, Phys, Pearle, Agentsoo, DavidWBrooks

Normalisable wavefunction

In quantum mechanics, wave functions which describe real particles must be normalisable 1: the probability of the particle to occupy any place must equal 1. Mathematically, in one dimension this is expressed as RB ∗ A ψ (x)ψ(x) dx = 1 in which the integration parameters A and B indicate the interval in which the particle must exist. All wavefunctions which represent real particles must be normalizable, that is, they must have a total probability of one - they must describe the probability of the particle existing as 100%. This trait enables anyone who solves the →Schrödinger equation for certain boundary conditions to discard solutions which do not have a finite integral at a given interval. For example, this Normalisable wavefunction

304 disqualifies periodic functions as wave function solutions for infinite intervals, while those functions can be solutions for finite intervals.

FT

Derivation of normalisation In general, ψ is a complex function. However, ψ ∗ ψ =| ψ |2

is real, greater than zero, and is known as a probability density function. This means that p(−∞ ≤ x ≤ ∞) =

R∞

−∞

| ψ |2 dx. (1)

DR A

where p(x) is the probability of finding the particle at x. Equation (1) is given by the definition of a probability density function. Since the particle exists, its probability of being anywhere in space must be equal to 1. Therefore we integrate over all space: R∞ p(−∞ ≤ x ≤ ∞) = −∞ | ψ |2 dx = 1. (2)

If the integral is finite, we can multiply the wavefunction, ψ, by a constant such that the integral is equal to 1. Alternatively, if the wavefunction already contains an appropriate arbitrary constant, we can solve equation (2) to find the value of this constant which normalises the wavefunction.

Example of normalisation

A particle is restricted to a 1D region between x=0 and x=l; its wavefunction is: (

ψ(x, t) =

Aei(kx−ωt) , 0 ≤ x ≤ l 0, elsewhere.

To normalise the wavefunction we need to find the value of the arbitrary constant A, i.e., solve R∞ 2 −∞ | ψ | dx = 1 to find A.

Substituting ψ into | ψ |2 we get

| ψ |2 = A2 ei(kx−ωt) e−i(kx−ωt) = A2

so

Normalisable wavefunction

305 R0

Rl

−∞ 0dx + 0

A2 dx +

R∞ l

0dx = 1

A2 l = 1 ⇒ A =



1 √

 l

.

FT

therefore

The normalised is:  ( wavefunction 1 √ l

ψ(x, t) =

0,

ei(kx−ωt) ,

0≤x≤l

elsewhere.

Proof that wavefunction normalisation doesn’t change associated properties

DR A

If normalisation of a wavefunction changed the properties associated with the wavefunction, the process becomes pointless as we still cannot yield any information about the properties of the particle associated with the un-normailied wavefunction. It is therefore important to establish that the properties associated with the wavefunction are not altered by normalisation. All properties of the particle such as probability distribution, momentum, energy, expectation value of position etc. are derived from the Schrödinger wave equation. The properties are therefore unchanged if the Schrödinger wave equation is invariant under normalisation. The Schrödinger wave equation is −~ d2 ψ 2m dx2

+ V (x)ψ(x) = Eψ(x).

If ψ is normalised and replaced with Aψ, then   2 d(Aψ) d2 (Aψ) d(Aψ) dψ d = A ddxψ2 . dx = A dx and dx2 = dx dx

The Schrödinger wave equation therefore becomes: −~ d2 ψ 2m A dx2

⇒A

⇒



+ V (x)Aψ(x) = EAψ(x)

−~ d2 ψ 2m dx2

−~ d2 ψ 2m dx2

 + V (x)ψ(x) = A (Eψ(x))

+ V (x)ψ(x) = Eψ(x)

Normalisable wavefunction

306

Note

FT

which is the original Schrödinger wave equation. That is to say, the Schrödinger wave equation is invariant under normalisation, and consequently associated properties are unchanged.

Note 1: The spelling normalisable is a British variant spelling of normalizable.

See also • • •

→Quantum mechanics →Schrödinger equation →Wavefunction

Source: http://en.wikipedia.org/wiki/Normalisable_wavefunction

DR A

Principal Authors: Germen, Charles Matthews, Oleg Alexandrov, Stevertigo, Conscious

Normal mode

Normal modes in an oscillating system are special solutions where all the parts of the system are oscillating with the same frequency (called normal frequencies or allowed frequencies). The concept of normal modes is of vital importance in wave theory, optics and quantum mechanics. Finding normal modes in harmonic oscillation uses the strength of linear algebra and linear sets of differential equations. One can present the problem as a matrix-vector equation and then solve for its eigenvectors. After finding them, the normal modes are the eigenvectors where the normal frequencies are the eigenvalues.

Example - normal modes of coupled oscillators Consider two bodies (not affected by gravity), each of mass M, attached to three springs with stiffness K. They are attached in the following manner:

Normal mode

DR A

FT

307

Figure 29

Various normal modes in a 1D-lattice.

where the edge points are fixed and cannot move. We’ll use x 1(t) to denote the displacement of the leftmost mass, and x 2(t) to denote the displacement of the rightmost. If we denote the second derivative of x(t) with respect to time as x”, the equations of motion are: M x001 = −K(x1 ) − K(x1 − x2 )

M x002 = −K(x2 ) − K(x2 − x1 )

Since we expect oscillatory motion, we try: x1 (t) = A1 eiωt

x2 (t) = A2 eiωt

Substituting these into the equations of motion gives us:

Normal mode

308 −ω 2 M A1 eiωt = −2KA1 eiωt + KA2 eiωt

FT

−ω 2 M A2 eiωt = KA1 eiωt − 2KA2 eiωt Since the exponential factor is common to all terms, we omit it and simplify: (ω 2 M − 2K)A1 + KA2 = 0 KA1 + (ω 2 M − 2K)A2 = 0 And  in matrix representation: 

ω 2 M − 2K K

K ω 2 M − 2K

A1 A2



=0

For this equation to have a non-trivial solution, the determinant of the matrix on the left (the characteristic polynomial of the system) must be equal to 0, so:

DR A

(ω 2 M − 2K)2 − K 2 = 0

Solving for ω, we have: q K ω1 = M ω2 =

q

3K M

If we substitute ω1 into the matrix and solve for (A1 , A2 ), we get (1, 1). If we substitute ω2 , we get (1, -1). (These vectors are eigenvectors, and the frequencies are eigenvalues.) The  firstnormalmode  is:

x1 (t) x2 (t)

= c1

1 cos (ω1 t + φ1 ) 1

and second  normal  the  mode is:

x1 (t) x2 (t)

= c2

1 cos (ω2 t + φ2 ) −1

The general solution is a superposition of the normal modes where c 1, c 2, φ 1, and φ 2, are determined by the initial conditions of the problem.

Normal mode

309

Standing waves

FT

The process demonstrated here can be generalized and formulated using the formalism of Lagrangian mechanics or Hamiltonian mechanics.

DR A

A standing wave is a continuous form of normal mode. In a standing wave, all the space elements (i.e (x,y,z) coordinates) are oscillating in the same frequency and in phase (reaching the equilibrium point together), but each has a different amplitude.

The general form of a standing wave is:

Ψ(t) = f (x, y, z)(A cos(ωt) + B sin(ωt))

where f (x, y, z) represents the dependence of amplitude on location and the cosine\sine are the oscillations in time. Physically, standing waves are formed by the interference (superposition) of waves and their reflections (although one may also say the opposite; that a moving wave is a superposition of standing waves). The geometric shape of the medium determines what would be the interference pattern, thus determines the f (x, y, z) form of the standing wave. This space-dependence is called a normal mode.

Usually, for problems with continuous dependence on (x,y,z) there is no single or finite number of normal mode, but there are infinitely normal modes. If the problem is bounded (i.e it is defined on a finite section of space) there are countably many (a discrete infinity of ) normal modes (usually numbered n = 1,2,3,...). If the problem is not bounded, there is a continuous spectrum of normal modes. The allowed frequencies are dependent on the normal modes, as well on physical constants of the problem (density, tension, pressure, etc.) which sets the Normal mode

310

FT

phase velocity of the wave. The range of all possible normal frequencies is called the frequency spectrum. Usually, each frequency is modulated by the amplitude at which it has arisen, creating a graph of the power spectrum of the oscillations. When relating to music, normal modes of a vibrating instruments (strings, air pipes, drums, etc.) are called "harmonics".

Normal modes in quantum mechanics

In quantum mechanics, a state |ψi of a system is described by a wavefunction ψ(x, t)which solves the →Schrödinger equation. The square of the absolute value of ψ ,i.e. P (x, t) = |ψ(x, t)|2

is the probability (density) to measure the particle in place x at time t.

DR A

Usually, when involving some sort of potential, the wavefunction is decomposed into a superposition of energy eigenstates, each oscillating with frequency of ω = En /~. Thus, we may write P |ψ(t)i = n |ni hn|ψ(t = 0)i e−iEn t/~

The eigenstates have a physical meaning further than an orthonormal basis. When the energy of the system is measured, the wavefunction collapses into one of its eigenstates and so the particle wavefunction is described by the pure eigenstate corresponding to the measured energy.

See also •

Physical applications: • Waves • Optics • harmonic oscillator • vibrational spectroscopy • quantum theory • →Schrödinger equation • →Wavefunction • →Measurement in quantum mechanics • harmonic series (music) • Mathematical tools: • linear algebra • eigenvectors Normal mode

311 differential equations Fourier analysis Sturm-Liouville theory Boundary value problem

External links

FT

• • • •

• Java simulation of coupled oscillators 150. • Java simulation of the normal modes of a string 151, drum 152, and bar 153.

Source: http://en.wikipedia.org/wiki/Normal_mode

Principal Authors: MathKnight, Michael Hardy, Charles Matthews, Pfalstad, Omegatron

DR A

Nuclear physics

Nuclear physics is the branch of physics concerned with the nucleus of the atom. It has three main aspects: probing the fundamental particles (protons and neutrons) and their interactions, classifying and interpreting the properties of nuclei, and providing technological advances. Nuclei do not lend themselves to exact theoretical understanding, because they are composed of many particles (mesons as well as protons and neutrons), but are not large enough to be accurately described as periodic, as done with crystals. So "nuclear models" that, singly or in combination, account for most nuclear behavior are used. Three of the four types of fundamental interaction play important roles in nuclei, the strong, electromagnetic and, on a longer time scale, weak. Nuclei are held together by strong interactions (mostly exchanging pions), but electromagnetic repulsion of the positively charged protons tends to push them apart, according to Coulomb’s law. The stable nuclei all have close to the lowest energy ratio of protons to neutron for their atomic weight. Nuclei near enough to this ratio to be bound but not close enough to be stable, give off electrons or positrons (beta decay) or take in electrons (and also give off neutrinos), to move closer to that ratio. This is the main place where the weak

150 http://www.falstad.com/coupled/ 151 http://www.falstad.com/loadedstring/ 152 http://www.falstad.com/circosc/ 153 http://www.falstad.com/barwaves/

Nuclear physics

312 interactions come in. Nuclei that are too massive to be stable are pulled apart by the coulomb repulsion of their protons and either fission or give off alpha particles.

FT

Though the number of energy levels is not infinite, as it is for the electron wave functions of atoms, most stable or nearly stable nuclei have many bound levels. These usually decay toward the ground state by emitting gamma ray photons. Protons and neutrons are fermions, with different value of the isospin quantum number, so two protons and two neutrons can share the same space wave function. In the rare case of a hypernucleus, a third baryon called a hyperon, with a different value of the strangeness quantum number can also share the wave function. The binding energies of the protons and neutrons are on the order of 1 % of their relativistic rest masses, so non-relativistic quantum mechanics can be used with errors usually smaller than those from other approximations.

DR A

Often, nuclear physicists will use Nuclear Units where h, c, and the mass of the proton m p have been set to unity.

History

Once the chemists of the 18th century had elucidated the chemical elements, the rules governing their combinations in matter, and their systematic classification (Mendeleev’s periodic table of elements) and John Dalton had, in 1803, applied Democritus’s idea of atom to them, it was natural that the next step would be a study of the fundamental properties of individual atoms of the various elements, an activity that we would today classify as atomic physics. These studies led to the discovery in 1896 by Becquerel of the radioactivity of certain species of atoms and to the further identification of radioactive substances by the Curies in 1898. Ernest Rutherford next took up the study of radiation and its properties; once he had achieved an understanding of the nature of the radioactivity, he turned around and used radiated particles to probe the atoms themselves. In the process he proposed in 1911 the existence of the atomic nucleus, the confirmation of which (through the painstaking experiments of Geiger and Marsden) provided a new branch of science, nuclear physics. Investigations into the properties of the nucleus have continued from Rutherford’s time to the present. In the 1940s and 1950s, it was discovered that there was yet another level of structure even more fundamental than the nucleus, which is itself composed of protons and neutrons. Thus nuclear physics can be regarded as the descendant of chemistry and atomic physics and in turn the progenitor of particle physics.

Nuclear physics

313

See also

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Experiments with nuclei continue to contribute to the understanding of basic interactions. Investigation of nuclear properties and the laws governing the structure of nuclei is an active and productive area of research, and practical applications, such as nuclear power, smoke detectors, cardiac pacemakers, and medical imaging devices, have become common.

Important publications in nuclear physics Nuclear fission Nuclear fusion Nuclear reactions Nuclear Structure Radioactivity Radioactive decay Nuclear force

•

Models of the nucleus • Interacting boson model • Liquid drop model • Shell model

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• • • • • • • •

Applications

• • • • • • •

Mossbauer effect Nuclear technology Nuclear engineering Nuclear magnetic resonance Nuclear medicine Nuclear power Nuclear weapons

External links • • •

SCK.CEN Belgian Nuclear Research Centre 154 Mol, Belgium A Free E-Learning Course on Nuclear Physics 155 Nuclear Physics-Nuclear Medicine Information 156

154 http://www.sckcen.be/ 155 http://www.VirtualClassroom.de.vu/

156 http://www.nucmedinfo.com/Pages/physic.html

Nuclear physics

314

References Kenneth S. Krane, "Introductory Nuclear Physics", Wiley & Sons (1988). Tien T. Tong, "Fifty Years of Seeing Atoms", Physics Today, March 2006, p. 31

FT

• •

Source: http://en.wikipedia.org/wiki/Nuclear_physics

Principal Authors: David R. Ingham, Philipum, Heron, Securiger, Icairns, DV8 2XL, Yhr, Zoicon5, Android 93, Stevegiacomelli

Observable

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In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. These operations might involve submitting the system to various electromagnetic fields and eventually reading a value off some gauge. In systems governed by classical mechanics, any experimentally observable value can be shown to be given by a real-valued function on the set of all possible system states. In quantum physics, on the other hand, the relation between system state and the value of an observable is more subtle, requiring some basic linear algebra to explain. In the mathematical formulation of quantum mechanics, states are given by non-zero vectors in a →Hilbert space V (where two vectors are considered to specify the same state if, and only if, they are scalar multiples of each other) and observables are given by self-adjoint operators on V. However, as indicated below, not every self-adjoint operator corresponds to a physically meaningful observable. For the case of a system of particles, the space V consists of functions called wave functions. In quantum mechanics, measurement of observables exhibits some seemingly mysterious phenomena. This often leads to many misconceptions about the nature of quantum mechanics itself. The facts of the matter, however, are far more prosaic. Specifically, if a system is in a state described by a wave function, the measurement process affects the state in a non-deterministic, but statistically predictable way. In particular, after a measurement is applied, the state description by a single wave function may be destroyed, being replaced by a statistical ensemble of wave functions. The irreversible nature of measurement operations in quantum physics is sometimes referred to as the measurement problem and is described mathematically by quantum operations. By the structure of quantum operations, this description is mathematically equivalent to Observable

315 that offered by relative state interpretation where the original system is regarded as a subsystem of a larger system and the state of the original system is given by the partial trace of the state of the larger system.

References •

S. Auyang, How is Quantum Field Theory Possible, Oxford University Press, 1995. G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963. V. Varadarajan, The Geometry of Quantum Mechanics vols 1 and 2, Springer-Verlag 1985.

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•

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Physically meaningful observables must also satisfy transformation laws which relate observations performed by different observers in different frames of reference. These transformation laws are automorphisms of the state space, that is bijective transformations which preserve some mathematical property. In the case of quantum mechanics, the requisite automorphisms are unitary (or antiunitary) linear transformations of the Hilbert space V. Under Galilean relativity or special relativity, the mathematics of frames of reference is particularly simple, and in fact restricts considerably the set of physically meaningful observables.

•

Source: http://en.wikipedia.org/wiki/Observable

Principal Authors: CSTAR, Michael Hardy, Phys, ABCD, Mushin

Oil-drop experiment

The purpose of Robert Millikan’s oil-drop experiment (1909) was to measure the electric charge of the electron. He did this by carefully balancing the gravitational and electric forces on tiny charged droplets of oil suspended between two metal electrodes. Knowing the electric field, the charge on the droplet could be determined. Repeating the experiment for many droplets, it was found that the values measured were always multiples of the same number. This was taken to be the charge on a single electron: 1.602 × 10 -19 coulombs (SI unit for electric charge). In 1923, Millikan won the Nobel Prize for physics in part because of this experiment. This experiment has since been repeated by generations of physics students, although it is rather expensive and difficult to do properly. Oil-drop experiment

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A version of the oil drop experiment has subsequently been used to search for free quarks (which, if they exist, would have a charge of 1/3 e), without success. Current theories of quarks predict that they are tightly bound and will not exist in a free form, however it is interesting to note that in Millikan’s original notebooks he observed and recorded the existence of an oil droplet which had a +1/3 partial charge, which he at the time dismissed as an error.

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Experimental procedure

The apparatus

The diagram shows a simplified version of Millikan’s set up. A uniform electric field is provided by a pair of horizontal parallel plates with a high potential difference between them. A charged drop of oil is allowed to drift in between them. By varying the potential, the drop can be made to rise, descend or stay steady. The plates are held together by a ring of insulating material (not shown in the diagram). There are two holes cut into the ring. A bright light source is shone through one of the holes, and focused on the region where the oil drops drift between the plates. A low-powered microscope is inserted through the other hole. The oil drops reflect the light and look like bright points on a dark field of view through the microscope. The microscope has a graduated scale in the eyepiece which allows for the velocity of the drop to be measured by timing how long it takes to travel from one division to another. The oil used is the type that is usually used in vacuum apparatus. This is because this type of oil has an extremely low vapour pressure. Ordinary oil would evaporate away under the heat of the light source and so the mass of the oil drop would not remain constant over the course of the experiment. Some oil drops will pick up a charge through friction with the nozzle as they are sprayed, but more can be charged by allowing an ionising radiation source (such as an x ray tube) to ionise the air in the chamber.

Oil-drop experiment

317

Method

FT

Initially the oil drops are allowed to fall between the plates with the electric field turned off. They very quickly reach a terminal velocity because of friction with the air in the chamber. The field is then turned on and, if it is large enough, some of the drops (the charged ones) will start to rise. (This is because the upwards electric force F E is greater for them than the downwards gravitational force W ). A likely looking drop is selected and kept in the middle of the field of view by alternately switching off the voltage until all the other drops have fallen. The experiment is then continued with this one drop.

The drop is allowed to fall and its terminal velocity v 1 in the absence of an electric field is calculated. The drag force acting on the drop can then be worked out using Stokes’ law: F = 6πrηv1

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where v 1 is the terminal velocity (i.e. velocity in the absence of an electric field) of the falling drop, η is the viscosity of the air, and r is the radius of the drop.

The weight W is the volume V multiplied by the density ρ and the acceleration due to gravity g. However what is needed is the apparent weight. The apparent weight in air is the true weight minus the upthrust (which equals the weight of air displaced by the oil drop). For a perfectly spherical droplet the apparent weight can be written as: W = 43 πr3 g(ρ − ρair )

Now at terminal velocity the oil drop is not accelerating. So the total force acting on it must be zero. So the two forces F and W must cancel one another out. F = W implies: r2 =

9ηv1 2g(ρ−ρair )

Once r is calculated, W can easily be worked out.

Now the field is turned back on. FE = qE

where q is the charge on the oil drop and E is the electric field between the plates. For parallel plates Oil-drop experiment

318 E=

V d

where V is the potential difference and d is the distance between the plates.

qE − W = 6πrηv2 =

W v2 v1

FT

One conceivable way to work out q would be to adjust V until the oil drop remained steady. Then we could equate F E with W. But in practice this is extremely difficult to do precisely. A more practical approach is to turn V up slightly so that the oil drop rises with a new terminal velocity v 2. Then

Millikan’s experiment and cargo cult science

Richard Feynman said in a commencement lecture he gave at Caltech in 1974

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We have learned a lot from experience about how to handle some of the ways we fool ourselves. One example: Millikan measured the charge on an electron by an experiment with falling oil drops, and got an answer which we now know not to be quite right. It’s a little bit off because he had the incorrect value for the viscosity of air. It’s interesting to look at the history of measurements of the charge of an electron, after Millikan. If you plot them as a function of time, you find that one is a little bit bigger than Millikan’s, and the next one’s a little bit bigger than that, and the next one’s a little bit bigger than that, until finally they settle down to a number which is higher. Why didn’t they discover the new number was higher right away? It’s a thing that scientists are ashamed of - this history - because it’s apparent that people did things like this: When they got a number that was too high above Millikan’s, they thought something must be wrong - and they would look for and find a reason why something might be wrong. When they got a number close to Millikan’s value they didn’t look so hard. And so they eliminated the numbers that were too far off, and did other things like that. We’ve learned those tricks nowadays, and now we don’t have that kind of a disease.

External links and references •

Karlsson, Magnus, " Millikan’s oildrop experiment 157". (Simplified version)

157 http://www.edu.falkenberg.se/gymnasieskolan/fysik/elektron/millikaneng.html

Oil-drop experiment

319

• •

• Graphical simulation of the experiment - examples of the difficulties Thomsen, Marshall, " Good to the Last Drop 158". Millikan Stories as "Canned" Pedagogy. Eastern Michigan University. CSR/TSGC Team, " Quark search experiment 159". The University of Texas at Austin. The oil-drop experiment appears in a list of Science’s 10 Most Beautiful Experiments 160 originally published in the New York Times.

More external links

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•

Delpierre, G.R. and B.T. Sewell, " Millikan’s Oil Drop Experiment 161". 25 April 2005 • Engeness, T.E., " The Millikan Oil Drop Experiment 162". 25 April 2005 • Millikan R. A. (1913). "On the elementary electrical charge and the Avogadro constant" 163. The Physical Review, Series II 2: 109–143., Paper by Millikan discussing modifications to his original experiment to improve its accuracy • Cargo cult science 164, text of the Feynman lecture. • Millikan Oil Drop Experiment in space 165. A variation of this experiment has been suggested for the International Space Station.

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•

Source: http://en.wikipedia.org/wiki/Oil-drop_experiment

Principal Authors: Michael Hardy, Mjmcb1, Reddi, Popefelix, Theresa knott, Raul654, Gene Nygaard, Linas

158 http://www.physics.emich.edu/mthomsen/sege.htm 159 http://www.tsgc.utexas.edu/floatn/1997/teams/UT-austin.html 160 http://physics.nad.ru/Physics/English/top10.htm 161 http://www.physchem.co.za/Static%20Electricity/Millikan.htm

162 http://people.ccmr.cornell.edu/~muchomas/8.04/Lecs/lec_Millikan/Mill.html 163 http://www.aip.org/history/gap/PDF/millikan.pdf 164 http://www.physics.brocku.ca/etc/cargo_cult_science.html 165 http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=APCPCS000504000001000715000001

&idtype=cvips&gifs=yes

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Open quantum system

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In physics, an open quantum system is a quantum system which is found to be in interaction with an external quantum system, the environment. The open quantum system can be viewed as a distinguished part of a larger closed quantum system, the other part being the environment. Open quantum systems are an important concept in quantum optics, measurement theory, quantum statistical mechanics, quantum cosmology and semiclassical approximations.

Source: http://en.wikipedia.org/wiki/Open_quantum_system Principal Authors: Daniel Arteaga, Sn0wflake, Conscious

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Optical theorem

In physics, the optical theorem is a very general law of wave scattering theory, which relates the forward scattering amplitude to the total cross section of the scatterer. It is usually written in the form σtot =

4π k

Im f (0),

where f(0) is the scattering amplitude with an angle of zero, that is, the amplitude of the wave scattered to the center of a distant screen. Because the optical theorem is derived using only conservation of energy, or in quantum mechanics from conservation of probability, the optical theorem is widely applicable and, in quantum mechanics, σtot includes both elastic and inelastic scattering. Note that the above form is for an incident plane wave; a more general form discovered by Werner Heisenberg can be written ˆ0 , k) ˆ = k R f (k ˆ0 , k ˆ00 )f (k ˆ00 , k) ˆ dk ˆ00 . Im f (k 4π

Notice that as a natural consequence of the optical theorem, an object that scatters any light at all ought to have a nonzero forward scattering amplitude, and so a bright central spot.

Optical theorem

321

History

FT

The optical theorem was originally discovered independently by Sellmeier and Lord Rayleigh in 1871. Lord Rayleigh recognized the forward scattering amplitude in terms of the index of refraction as n = 1 + 2πN f (0)/k 2

which he used in a study of the color and polarization of the sky. The equation was later extended to quantum scattering theory by several individuals, and came to be known as the Bohr-Peierls-Placzek relation after a 1939 publication. It was first referred to as the Optical Theorem in print in 1955 by Hans Bethe and de Hoffman, after it had been known as a "well known theorem of optics" for some time.

Derivation

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The theorem can be derived rather directly from a treatment of a scalar wave. If a plane wave is incident on an object, then the wave amplitude a great distance away from the scatterer is given approximately by ikr

ψ(r) ≈ eikz + f (θ) e r .

All higher terms, when squared, vanish more quickly than 1/r2 , and so are negligible a great distance away. Notice that for large values of z and small angles the binomial theorem gives us p 2 +y 2 . r = x2 + y 2 + z 2 ≈ z + x 2z

We would now like to use the fact that the intensity is proportional to the square of the amplitude ψ. Approximating the r in the denominator as z, we have |ψ|2 = |eikz + =1+

f (θ) ikz ik(x2 +y 2 )/2z 2 | r e e

f (θ) ik(x2 +y 2 )/2z z e

+

f ∗ (θ) −ik(x2 +y 2 )/2z z e

+

|f (θ)|2 . z2

If we drop the 1/z 2 term and use the fact that A + A∗ = 2 Re A we have |ψ|2 ≈ 1 + 2 Re

f (θ) ik(x2 +y 2 )/2z . z e

Now suppose we integrate over a screen in the x-y plane, which is small enough for the small angle approximations to be appropriate, but large enough that we can integrate the intensity from −∞ to ∞ with negligible error. In optics, this Optical theorem

322

FT

is equivalent to including many fringes of the diffraction pattern. To further simplify matters, let’s approximate f (θ) = f (0). We quickly obtain R R∞ 2 2 f (0) R ∞ |ψ|2 da ≈ A + 2 Re z −∞ eikx /2z dx −∞ eiky /2z dy where A is the area of the surface integrated over. The exponentials can be treated as Gaussians and so R f (0) |ψ|2 da = A + 2 Re z 2ziπ k =A−

4π k

Im f (0),

which is just the amount of energy that would reach the screen if none was scattered, lessened by an amount (4π/k) Im f (0). Therefore, because of conservation of energy, that must be the total amount of energy scattered, and thus it is the effective scattering cross section of the scatterer.

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References •

R. G. Newton (1976). "Optical Theorem and Beyond". Am. J. Phys 44: 639-642.

•

John David Jackson (1999). Classical Electrodynamics. Hamilton Printing Company. ISBN 047130932X.

Source: http://en.wikipedia.org/wiki/Optical_theorem

Principal Authors: Hyandat, Michael Hardy, Pflatau, That Guy, From That Show!, Phys

Parity (physics)

In physics, a parity transformation (also called parity inversion) is the simultaneous flip in the sign    of all spatial coordinates:

x −x P : y  7→ −y  z −z

A 3×3 matrix representation of P would have determinant equal to –1, and hence cannot reduce to a rotation. In a two-dimensional plane, parity is the same as a rotation by 180 degrees. Parity (physics)

323

Simple symmetry relations

FT

Under rotations, classical geometrical objects can be classified into scalars, vectors, and tensors of higher rank. In classical physics, physical configurations need to transform under representations of every symmetry group.

In a quantum theory states in a →Hilbert space do not need to transform under representations of the group of rotations, but only under projective representations. The word projective refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not an observable, then a projective representation reduces to an ordinary representation. All representations are also projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states.

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The projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example, projective representations of the 3-dimensional rotation group, which is the special orthogonal group SO(3 ), are ordinary representations of the special unitary group SU(2 ). Projective representations of the rotation group that are not representations are called spinors, and so quantum states may transform not only as tensors but also as spinors. If one adds to this a classification by parity, these can be extended, for example, into notions of • •

scalars (P = 1) and pseudoscalars (P = –1) which are rotationally invariant vectors (P = –1) and axial vectors (also called pseudo-vectors) (P = 1) which both transform as vectors under rotation.

One can define reflections such as   

x −x Vx : y  7→  y  , z z

which also have negative determinant. Then, combining them with rotations one can generate the parity transformation defined earlier. In any even number of dimensions, the first definition of parity has positive determinant, and hence can be obtained as some rotation. One then uses reflections to extend the notion of scalars and vectors to pseudo-scalars and pseudo-vectors.

Parity forms the Abelian group Z 2 due to the relation P 2 = 1. All Abelian groups have only one dimensional irreducible representations. For Z 2, there are two irreducible representations: one is even under parity (P φ = φ), the Parity (physics)

324

FT

other is odd (P φ = –φ). These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase.

Classical mechanics

Newton’s equation of motion F = ma (if mass is constant) equates two vectors, and hence is invariant under parity. The law of gravity also involves only vectors and is also, therefore, invariant under parity. However angular momentum is an axial vector. L=r×p, P(L) = (–r) × (–p) = L.

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In classical electrodynamics, charge density ρ is a scalar, the electric field, E, and current j are vectors, but the magnetic field, H is an axial vector. However, Maxwell’s equations are invariant under parity because the curl of an axial vector is a vector.

Quantum mechanics Possible eigenvalues

In quantum mechanics, spacetime transformations act on quantum states. The parity transformation, P, is a unitary operator in quantum mechanics, acting on a state ψ as follows: P ψ(r) = ψ(-r). One must have P 2 ψ(r) = e i φ ψ(r), since an overall phase is unobservable.

The operator P 2, which reverses the parity of a state twice, leaves the spacetime invariant and so is an internal symmetry which rotates its eigenstates by phases e i φ. If P 2 is an element e i Q of a continuous U(1) symmetry group of phase rotations then e -i Q/2 is part of this U(1) and so is also a symmetry. In particular we can define P’=Pe -i Q/2 which is also a symmetry and so we can choose to call P’ our parity operator instead of P. Notice that P’ 2=1 and so P’ has eigenvalues ±1. However when no such symmetry group exists, it may be that all parity transformations have some eigenvalues which are phases other than ±1.

Consequences of parity symmetry

When parity generates the Abelian group Z 2, one can always take linear combinations of quantum states such that they are either even or odd under parity

Parity (physics)

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FT

325

Figure 30 Two dimensional representations of parity are given by a pair of quantum states which go into each other under parity. However, this representation can always be reduced to linear combinations of states, each of which is either even or odd under parity. One says that all irreducible representations of parity are one-dimensional.

(see the figure). Thus the parity of such states is ±1. The parity of a multiparticle state is the product of the parities of each state; in other words parity is a multiplicative quantum number.

In quantum mechanics, Hamiltonians are invariant (symmetric) under a parity transformation if P commutes with the Hamiltonian. In non-relativistic quantum mechanics, this happens for any potential which is scalar, ie, V = V(r), hence the potential is spherically symmetric. The following facts can be easily proven: If |A> and |B> have the same parity, then = 0 where X is the position operator. • For a state |L,m> of orbital angular momentum L with z-axis projection m, P |L,m> = (-1) L|L, m>.

•

Parity (physics)

326 If [H,P] = 0, then no transitions occur between states of opposite parity. If [H,P] = 0, then a non-degenerate eigenstate of H is also an eigenstate of the parity operator; i.e., a non-degenerate eigenfunction of H is either invariant to P or is changed in sign by P.

FT

• •

Some of the non-degenerate eigenfunctions of H are unaffected (invariant) by parity P and the others will be merely be reversed in sign when the Hamiltonian operator and the parity operator commute: P Ψ = c Ψ,

where c is a constant, the eigenvalue of P, P P Ψ = P c Ψ.

Quantum field theory

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The intrinsic parity assignments in this section are true for relativistic quantum mechanics as well as quantum field theory.

If we can show that the vacuum state is invariant under parity (P |0> = |0>), the Hamiltonian is parity invariant ([H,P] = 0) and the quantization conditions remain unchanged under parity, then it follows that every state has good parity, and this parity is conserved in any reaction. To show that quantum electrodynamics is invariant under parity, we have to prove that the action is invariant and the quantization is also invariant. For simplicity we will assume that canonical quantization is used; the vacuum state is then invariant under parity by construction. The invariance of the action follows from the classical invariance of Maxwell’s equations. The invariance of the canonical quantization procedure can be worked out, and turns out to depend on the transformation of the annihilation operator: P a(p,±) P + = -a(-p,±)

where p denotes the momentum of a photon and ± refers to its polarization state. This is equivalent to the statement that the photon has odd intrinsic parity. Similarly all vector bosons can be shown to have odd intrinsic parity, and all axial-vectors to have even intrinsic parity. There is a straightforward extension of these arguments to scalar field theories which shows that scalars have even parity, since P a(p) P + = a(-p).

Parity (physics)

327 This is true even for a complex scalar field. (Details of spinors are dealt with in the article on the Dirac equation, where it is shown that fermions and antifermions have opposite intrinsic parity.)

FT

With fermions, there is a slight complication because there is more than one pin group. (See the article on pin groups for more details.)

Parity in the standard model Fixing the global symmetries

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In the standard model of fundamental interactions there are precisely three global internal U(1) symmetry groups available, with charges equal to the baryon number B, the lepton number L and the electric charge Q. The product of the parity operator with any combination of these rotations is another parity operator. It is conventional to choose one specific combination of these rotations to define a standard parity operator, and other parity operators are related to the standard one by internal rotations. One way to fix a standard parity operator is to assign the parities of three particles with linearly independent charges B, L and Q. In general one assigns the parity of the most common massive particles, the proton, the neutron and the electron, to be +1.

Steven Weinberg has shown that if P 2=(-1) F, where F is the fermion number operator, then, since the fermion number is the sum of the lepton number plus the baryon number, F=B+L, for all particles in the standard model and since lepton number and baryon number are charges Q of continuous symmetries e i Q, it is possible to redefine the parity operator so that P 2=1. However, if there exist Majorana neutrinos, which experimentalists today believe is quite likely, their fermion number is equal to one because they are neutrinos while their baryon and lepton numbers are zero because they are Majorana, and so (-1) F would not be embedded in a continuous symmetry group. Thus Majorana neutrinos would have parity ±i.

Parity of the pion

In the 1954 paper Absorption of Negative Pions in Deuterium: Parity of the Pion 166, William Chinowsky and Jack Steinberger demonstrated that the pion π has negative parity. They studied the decay of an atom made from a deuterium nucleus d and a negatively charged pion π - in a state with zero orbital angular momentum L =0 into two neutrons n d π − −→ n n.

166 http://prola.aps.org/abstract/PR/v95/i6/p1561_1

Parity (physics)

328

Parity violation

FT

Neutrons are fermions and so obey Fermi statistics, which implies that the final state is antisymmetric. Using the fact that the deuteron has spin one and the pion spin zero together with the antisymmetry of the final state they concluded that the two neutrons must have orbital angular momentum L =1. The total parity is the product of the intrinsic parities of the particles and the extrinsic parity (-1) L. Since the orbital momentum changes from zero to one in this process, if the process is to conserve the total parity then the products of the intrinsic parities of the initial and final particles must have opposite sign. A deuteron nucleus is made from a proton and a neutron, and so using the forementioned convention that protons and neutrons have intrinsic parities equal to +1 they argued that the parity of the pion is equal to minus the product of the parities of the two neutrons divided by that of the proton and neutron in deuterium, (-1)(1) 2/(1) 2, which is equal to minus one. Thus they concluded that the pion is a pseudoscalar particle.

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Parity is not a symmetry of our universe. Although it is conserved in electromagnetism, strong interactions and gravity, it turns out to be violated in weak interactions. The Standard Model incorporates parity violation by expressing the weak interaction as a chiral gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in weak interactions in the Standard Model. The history of the discovery of parity violation is interesting. It was suggested several times and in different contexts that parity might not be conserved, but in the absence of compelling evidence these were not taken seriously. A careful review by theoretical physicists Tsung Dao Lee and Chen Ning Yang went further, showing that while parity conservation had been verified in decays by the strong or electromagnetic interactions, it was untested in the weak interaction. They proposed several possible direct experimental tests. They were almost ignored, but Lee was able to convince his Columbia colleague ChienShiung Wu to try it. She needed special cryogenic facilities and expertise, so the experiment was done at the National Bureau of Standards. In 1956-1957 Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson found a clear violation of parity conservation in the beta decay of cobalt-60. As the experiment was winding down, with doublechecking in progress, Wu informed her colleagues at Columbia of their positive results. Three of them, R. L. Garwin, Leon Lederman, and R. Weinrich modified an existing cyclotron experiment and immediately verified parity violation. They delayed publication until after Wu’s group was ready; the two papers appeared back to back.

Parity (physics)

329

FT

After the fact, it was noted that an obscure 1928 experiment had in effect reported parity violation in weak decays, but as the appropriate concepts had not been invented yet, it had no impact. The discovery of parity violation immediately explained the outstanding τ -θ puzzle in the physics of kaons.

Intrinsic parity of hadrons

To every particle one can assign an intrinsic parity as long as nature preserves parity. Although weak interactions do not, one can still assign a parity to any hadron by examining the strong interaction reaction that produces it, or through decays not involving the weak interaction, such as π 0 → γγ.

See also

Charge conjugation, time reversal and CPT symmetry Standard model of particle physics, and the electroweak theory Vector

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• • •

References and external links •

CP violation, by I.I. Bigi and A.I. Sanda 167 [ISBN 0521443490]]

•

Weinberg, S. (1995). The Quantum Theory of Fields. Cambridge University Press. ISBN 0-521-67053-5.

Source: http://en.wikipedia.org/wiki/Parity_%28physics%29

Principal Authors: Xerxes314, JarahE, Bambaiah, Raghunathan, Patrick

167 http://www.amazon.com/exec/obidos/tg/detail/-/0521443490

Parity (physics)

330

Particle in a box

FT

In physics, the particle in a box (also known as the infinite potential well or the infinite square well) is a very simple problem consisting of a single particle bouncing around inside of an immovable box, from which it cannot escape, and which loses no energy when it collides with the walls of the box. In classical mechanics, the solution to the problem is trivial: The particle moves in a straight line, always at the same speed, until it reflects from a wall. When it reflects from a wall, it always reflects at an equal but opposite angle to its angle of approach, and its speed does not change. The problem becomes very interesting when one attempts a quantummechanical solution, since many fundamental quantum mechanical concepts need to be introduced in order to find the solution. Nevertheless, it remains a very simple and solvable problem. This article will only be concerned with the quantum mechanical solution.

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The problem may be expressed in any number of dimensions, but the simplest problem is one dimensional, while the most useful solution is the particle in the three dimensional box. In one dimension this amounts to the particle existing on a line segment, with the "walls" being the endpoints of the segment. In physical terms, the particle in a box is defined as a single point particle, enclosed in a box inside of which it experiences no force whatsoever, i.e. it is at zero potential energy. At the walls of the box, the potential rises to infinity, forming an impenetrable wall. Using this description in terms of potentials allows the →Schrödinger equation to be used to determine the solution.

As mentioned above, if we were studying this system under the rules of classical mechanics we would apply Newton’s laws of motion to the initial conditions and the result would seem reasonable and intuitive. In quantum mechanics, when the →Schrödinger equation is applied to the proposed system, the results are not intuitive. In the first place, the particle can only have certain specific energy levels, and the zero energy level is not one of them. Secondly, the chances of detecting the particle in the box at any specific energy level is not uniform - there are certain locations in the box where the particle might be found, but there are also places where it can never be found. Both of these results differ from the usual way we perceive the world, yet rest on principles that have been extensively experimentally verified.

Formal Introduction

The particle in a box (or the infinite potential well or infinite square well) is a simple idealized system that can be completely solved within quantum Particle in a box

331

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mechanics. It is the situation of a particle confined within a finite region of space (the box) by an infinite potential that exists at the walls of the box. The particle experiences no forces while inside the box, but is constrained by the walls to remain in the box. This is similar to the situation of a gas confined in a container. For simplicity we start with the 1-dimensional case, where all motion is constrained to a single dimension. Later we will extend the discussion to the 2 and 3 dimensional cases. See also the →Particle in a spherically symmetric potential where the case is treated of a particle in a spherical box, or the particle in a ring which shows the case for a particle in a 1D ring. The statistical mechanics of many particles in a box is developed in the gas in a box article.

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As we shall see, the solution of the →Schrödinger equation for the particle in a box problem reveals some decidedly quantum behavior of the particle that agrees with observation but contrasts sharply with the predictions of classical mechanics. This is a particularly useful illustration because this behaviour is not "forced" on the system, it arises naturally from the initial conditions. It neatly demonstrates that quantum behaviour is a natural outcome of any wavelike system, contrary to the common concept of a "quantum leap" where the behavior is almost magical. The quantum behavior in the box includes:

Energy quantization - It is not possible for the particle to have any arbitrary definite energy. Instead only discrete definite energy levels are allowed (if the state is not a steady state, however, any energy past zero-point energy is allowed on average). • Zero-point energy - The lowest possible energy level of the particle, called the zero-point energy, is nonzero. • Nodes - In contrast to classical mechanics the Schrödinger equation predicts that for some energy levels there are nodes, implying positions at which the particle can never be found. •

One can solve analytically the Schrödinger equation for such a simple potential. However trivial, this case is both of great technical value for the insights it allows, and of paramount physical importance. Depending on the boundary conditions, one can use the solutions to describe two important systems. If one considers real valued solutions (of which detailed derivation is given below), one describes actual potentials of heterostructures called quantum wells which trap spatially particles, typically electrons and holes. If one considers complex valued solutions, one describes conveniently a particle propagating freely in a constrained volume (like a solid).

Particle in a box

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Solutions The particle in a 1-dimensional box

2

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For the 1-dimensional case in the x direction, the time-independent Schrödinger equation can be written as: 2

~ d ψ − 2m + V (x)ψ = Eψ dx2

(1)

where ~=

h 2π

h is Planck’s constant

m is the mass of the particle

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ψ is the complex-valued stationery time-independent wavefunction that we want to find V (x) is a function describing the potential at each point x and

E is the energy, a real number.

For the case of the particle in a 1-dimensional box of length L, the potential is zero inside the box, but rises abruptly to infinity at x = 0 and x = L. Thus for the region inside the box V (x) = 0 and Equation 1 reduces to: 2

2

~ d ψ = Eψ − 2m dx2

(2)

This is a well studied differential equation and eigenvalue problem with a general solution of: ψ = A sin(kx) + B cos(kx)

E=

k 2 ~2 2m

(3)

Here, A and B can be any complex numbers, and k can be any real number (k must be real because E is real). Now in order find the specific solution for the problem at hand, we must specify the appropriate boundary conditions and find the values for A and B that Particle in a box

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333

Figure 31

The Potential is 0 inside the box, and infinite elsewhere

satisfy those conditions. One usually resorts to one of the following two choices, describing two kinds of systems. The first case, with which we shall pursue our derivation, demands that ψ equal zero at x = 0 and x = L. A handwaving argument to motivate these boundary conditions is that the particle is unlikely to be found at a location with a high potential (the potential repulses the particle), thus the probability of finding the particle, |ψ| 2, must be small in these regions and decreases with increasing potential. For the case of an infinite potential, |ψ| 2 must infinitesimally small or 0, thus ψ must also be zero in this region. In summary, ψ(0) = ψ(L) = 0

(4)

The second case, to which solutions are given in section free propagation at the end of this article, does not compel the wavefunction to vanish at the boundary. This means that when the particle reaches one border of the well, it instantaneously disappears from this side to reappear on the opposite side, as if the well was some kind of torus. The value of the solutions are discussed in the appropriate section. We now resume derivation with vanishing boundary conditions.

Particle in a box

334

ψ = A sin(kx)

(5)

and at x = L we find: ψ = A sin(kL) = 0

(6)

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Substituting the general solution from Equation 3 into Equation 2 and evaluating at x = 0 (ψ = 0), we find that B = 0 (since sin(0) = 0 and cos(0) = 1). It follows that the wavefunction must be of the form:

One solution for Equation 6 is A = 0, however, this "trivial solution" would imply that ψ = 0 everywhere (I.e. the particle isn’t in the box.) and can be thrown out. If A 6= 0 then sin(kL) = 0, which is only true when: kL = nπ or k =

where nπ L

(7)

n = 1, 2, . . .

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(note that n = 0 is ruled out because then ψ = 0 everywhere, corresponding to the case where the particle is not in the box. Negative values of n are also neglected, since they merely change the sign of sin(nx)). Now in order to find A we must undertake a process called normalising the wavefunction. We recognize that the particle must exist somewhere in space. |ψ|2 is the probability of finding the particle at a particular point in space, so the integral of this value over all x must be equal to 1: R∞ RL 1 = −∞ |ψ|2 dx = |A|2 0 sin2 kx dx = |A|2 L2 or

|A| =

q

2 L

(8)

√ Thus, A may be any complex number with absolute value (2/L); these dif√ ferent values of A yield the same physical state, so we choose A = (2/L) to simplify. Finally, substituting the results from Equations 7 and 8 into Equation 3, the complete set of energy eigenfunctions for the 1-dimensional particle in a box problem is: q
ψn = L2 sin nπx (9) L

Particle in a box

335 En =

n2 ~2 π 2 2mL2

=

n 2 h2 8mL2

(10)

with

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n = 1, 2, 3, . . .

Note, that as mentioned previously, only "quantized" energy levels are possible. Also, since n cannot be zero, the lowest energy from Equation 10 is also non-zero. This zero-point energy, as it is called, can be explained in terms of the uncertainty principle. Because the particle is constrained within a finite region, the variance in its position is upper-bounded. Thus due to the uncertainty principle the variance in the particle’s momentum cannot be zero, so the particle must contain some amount of energy that increases as the length of the box, L, decreases.

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Also, since ψ consists of sine waves, for any value of n greater than one, there are regions within the box for which ψ and thus ψ 2 both equal zero, indicating that for these energy levels, nodes exist in the box where the probability of finding the particle is zero.

The particle in a 2-dimensional or 3-dimensional rectangular box

For the 2-dimensional case the particle is confined to a rectangular surface of length L x in the x-direction and L y in the y-direction. Again the potential is zero inside the "box" and infinite at the walls. For the region inside the box, where the potential is zero, the two dimensional analogue of Equation 2 applies:  2  ∂ ψ ∂2ψ ~2 − 2m = Eψ (11) 2 + 2 ∂x ∂y In this case ψ is a function of both x and y, so ψ=ψ(x,y). In order to solve Equation 11, we use the method of separation of variables. First, we assume that ψ can be expressed as the product of two independent functions, the first depending only on x and the second depending only on y; i.e.: ψ(x, y) = X(x)Y (y)

(12)

Substituting Equation 12 into Equation 11 and evaluating the partial derivatives gives:  2  2 ~2 − 2m Y ∂∂xX2 + X ∂∂yY2 = EXY (13)

which upon dividing by XY and rewriting d 2X /dx 2 as X " and d 2Y /dy 2 as Y " becomes: Particle in a box

336 2

~ − 2m



X 00 X

+

 00

Y Y

=E

(14)

2

00

~ X − 2m X = Ex

−

and

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Now we note that since X "/X is independent of y, varying y can only change the Y "/Y term. However, from Equation 14 we see that changing Y "/Y without varying X "/X, would also change E, but E is a constant, so Y "/Y must also be a constant, independent of y. The same argument can be applied to show that X "/X is independent of x. Since X "/X and Y "/Y are constants, we can write: ~2 Y 00 2m Y

= Ey

(15)

where E x + E y = E. Expanding X " and Y " in terms of the derivatives and rearranging gives: 2

2

2

2

~ ∂ X = Ex X − 2m ∂x2

(16)

~ ∂ Y − 2m = Ey Y ∂y 2

(17)

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each of which are of the same form as the 1-dimensional Schrödinger equation (Equation 2) we solved in the previous section. Thus, adapting the results from the previous section gives: q   Xnx = L2x sin nLx πx (18) x Yny =

q

2 Ly

sin



ny πy Ly



(19)

Finally, since ψ=XY and E = E x + E y, we obtain the solutions:     q ny πy ψnx ,ny = Lx4Ly sin nLx πx sin (20) L x y Enx ,ny =

h2 8m



nx Lx

2

+



ny Ly

2 

(21)

The same separation of variables technique can be applied to the three dimensional case to give the energy eigenfunctions:       q ny πy nz πz ψnx ,ny ,nz = Lx L8y Lz sin nLx πx sin sin (22) L L x y z Enx ,ny ,nz =

~2 π 2 2m



nx Lx

2

+



ny Ly

2

+



nz Lz

2 

with

ni = 1, 2, 3, . . .

Particle in a box

(23)

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Free propagation

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An interesting feature of the above solutions is that when two or more of the lengths are the same (e.g. L x = L y ), there are multiple wavefunctions corresponding to the same total energy. For example the wavefunction with n x = 2, n y = 1 has the same energy as the wavefunction with n x = 1, n y = 2. This situation is called degeneracy and for the case where exactly two degenerate wavefunctions have the same energy that energy level is said to be doubly degenerate. Degeneracy results from symmetry in the system. For the above case two of the lengths are equal so the system is symmetric with respect to a 90◦ rotation.

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If the potential is zero (or constant) everywhere, one describes a free particle. This leads to some difficulties of normalization of the momentum or energy eigenfunctions. One way around is to constrain the particle in a finite volume V of arbitrary (large) extension, in which it is free to propagate. It is expected that in the limit of V → ∞ we recover the free particle while allowing in the intermediate calculations the use of properly normalized states. Also, when describing for instance a particle propagating in a solid, one does not expect spatially localized states but instead completely delocalized states (within the solid), meaning that the particles propagates inside it (since it can be everywhere with the same probability, conversely to the sine solutions we encountered where the particle has favored locations). This understanding follows from the solutions of the Schrödinger equation for zero potential following from the so-called Von-Karman boundary conditions; i.e., the wavefunction assumes same values on opposite sides of the box but it is not required to be zero here. One can then check that the following solutions obey eq. 1:

in 1D : ψk (x) =

√1 eikx ; k L

in 3D : ψk (x) =

2ny π L ; kz

=

2nz π L ;

=

2nπ L ;

√1 eik·r ; kx L3

n∈Z

=

2nx π L ; ky

=

nx , ny , nz ∈ Z

The energy remains ~2 k 2 /2m (cf. eq. 3) but interestingly, now the k are twice as before (cf. eq. 7). This is because in the previous case, n was strictly positive whereas now it can be negative or zero (the ground state). The solutions where the sine does not superpose to itself after a translation of L can not be recovered with exponentials, since in this propagating particle interpretation, the derivative is discontinuous at the border, meaning that the particle acquires infinite velocity here. This shows how the two interpretations bear intrinsically differing behaviours. Particle in a box

338

References Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 0131118927.

See also • •

finite potential well particle in a ring

External links

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•

Scienceworld 168 (Infinite Potential Well)

•

Scienceworld 169 (Finite Potential Well)

•

1-D quantum mechanics java applet 170 simulates particle in a box, as well as other 1-dimensional cases. 2-D particle in a box applet 171

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•

•

Source: http://en.wikipedia.org/wiki/Particle_in_a_box

Principal Authors: PAR, Dgrant, Pfalstad, Heron, AxelBoldt, Paul August, Laussy, JabberWok

Particle in a one-dimensional lattice (periodic potential) In quantum mechanics, the particle in a one-dimensional lattice is problem that occurs in the model of a periodic crystal lattice. The problem can be simplified from the 3D infinite potential barrier (particle in a box) to a onedimensional case. The potential is caused by ions in the periodic structure of the crystal creating an electromagnetic field so electrons are subject to a regular potential inside the lattice. This is an extension of the free electron model that assumes zero potential inside the lattice.

168 http://scienceworld.wolfram.com/physics/InfiniteSquarePotentialWell.html 169 http://scienceworld.wolfram.com/physics/FiniteSquarePotentialWell.html 170 http://www.falstad.com/qm1d/ 171 http://www.falstad.com/qm2dbox/

Particle in a one-dimensional lattice (periodic potential)

339

Problem definition

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When talking about solid materials, the discussion is mainly around crystals - periodic lattices. Here we will discuss a 1-dimensional lattice of positive ions. Assuming the spacing between two ions is a, the potential in the lattice will look something like this:

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The mathematical representation of the potential is a periodic function with a period a. According to Bloch’s theorem, the wavefunction solution of the →Schrödinger equation when the potential is periodic, can be written as: ψ(x) = eikx u(x)

Where u(x) is a periodic function which satisfies: u(x + a) = u(x)

u0 (x + a) = u0 (x)

When nearing the edges of the lattice, there are problems with the boundary condition. Therefore, we can represent the ion lattice as a ring. If L is the length of the lattice so that L » a, then the number of ions in the lattice is so big, that when considering one ion, its surrounding is almost linear, and the wavefuntion of the electron is unchanged. So now, instead of two boundary conditions we get one circular boundary condition: ψ(0) = ψ(L)

If N is the number of Ions in the lattice, then we have the relation: aN = L. Replacing in the boundary condition and applying Bloch’s theorem will result in a quantization for k :

Particle in a one-dimensional lattice (periodic potential)

340 ψ(0) = eik·0 u(0) = eikL u(L) = ψ(L)

⇒ kL = 2πn → k =

2π Ln

Kronig-Penney model



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u(0) = eikL u(N a) → eikL = 1  n = 0, ±1, ±2, ..., ± N2 .

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In order to simplify the problem the potential function is approximated by a

rectangular potential: Using Bloch’s theorem, we only need to find a solution for a single period, make sure it is continuous and smooth, and to make sure the function u(x) is also continuous and smooth. Considering a single period of the potential:

Particle in a one-dimensional lattice (periodic potential)

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341

We

have two regions here. We will solve for each independently: 0<x
−~2 2m ψxx

= Eψ

⇒ ψ = Aeiαx + A0 e−iαx −b < x < 0 :

−~2 2m ψxx



α2 =

2mE ~2



= (E + V0 )ψ

⇒ ψ = Beiβx + B 0 e−iβx



β2 =

2m(E+V0 ) ~2



In order to find u(x) in each region we need to manipulate the probability function:   ψ(0 < x < a − b) = Aeiαx + A0 e−iαx = eikx · Aei(α−k)x + A0 e−i(α+k)x ⇒ u(0 < x < a − b) = Aei(α−k)x + A0 e−i(α+k)x

And in the same manner:

u(−b < x < 0) = Bei(β−k)x + B 0 e−i(β+k)x

To complete the solution we need to make sure the probability function is continuous and smooth, i.e: Particle in a one-dimensional lattice (periodic potential)

342 ψ(0− ) = ψ(0+ ) ψ 0 (0− ) = ψ 0 (0+ ) And that u(x) and u( x) are periodic

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u(−b) = u(a − b) u0 (−b) = u0 (a − b).

These conditions yield the following matrix:  1  α  i(α−k)(a−b)  e i(α−k)(a−b)   (α − k)e 0 0   0 0

1 −α

e−i(α+k)(a−b) (α + k)e−i(α+k)(a−b)

−1 −β

−e−i(β−k)b −(β − k)e−i(β−k)b



 A  A0    = −ei(β+k)b   B  i(β+k)b B0 (β + k)e −1 β

In order for us not to have the trivial solution, the determinant of the matrix must be 0. This leads us to the following expression: α2 +β 2 2αβ

sin(βb) sin[α(a − b)]

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cos(ka) = cos(βb) cos[α(a − b)] −

In order to further simplify the expression, we will perform the following approximations: b → 0 ; V0 → ∞ ; V0 b = constant

⇒ βb → 0 ; β 2 b = constant ; α2 b → 0 ; sin(βb) → βb ; cos(βb) → 1

The expression will now be: cos(ka) = cos(αa) − P

sin(αa) αa



P =

β 2 ab 2



See also • • • • •

Ralph Kronig Kronig-Penney model Free electron model Nearly-free electron model Crystal structure Potential

Particle in a one-dimensional lattice (periodic potential)

343

External links 1-D periodic potential applet 172

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Source: http://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice_%28periodic_potential %29

Principal Authors: Michael Hardy, Pfalstad, Rubber hound, Sverdrup, Salty-horse

Particle in a ring

In quantum mechanics, the case of a particle in a one-dimensional ring is similar to the particle in a box. The →Schrödinger equation for a free particle which is restricted to a ring (technically, whose configuration space is the circle S 1 ) is 2

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~ − 2m ∇2 ψ = Eψ

Using polar coordinates on the 1 dimensional ring, the wave function depends only on the angular coordinate, and so ∇2 =

1 ∂2 r2 ∂θ2

Requiring that the wave function be periodic in θ with a period 2 π (from the demand that the wave functions be single-valued functions on the circle), and that they be normalized leads to the conditions R 2π 2 0 |ψ(θ)| dθ = 1 , and

ψ(θ) = ψ(θ + 2 π)

Under these conditions, the solution to the Schrödinger equation is given by ψ(θ) =

√1 2π

r

e±i ~

√

2mEθ

The energy eigenvalues E are quantized because of the periodic boundary conditions, and they are required to satisfy

172 http://www.falstad.com/qm1dcrystal/

Particle in a ring

344 e±i ~

2mEθ

r

e±i2π ~

√

√ r

= e±i ~

2mE

2mE(θ+2π) ,

or

= 1 = ei2πn

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√ r

This leads to the energy eigenvalues E=

n2 ~2 2mr2

where n = 0, 1, 2, 3, . . .

The full wave functions are, therefore ψ(θ) =

√1 2π

e±inθ

Except for the case n = 0, there are two quantum states for every value of n (corresponding to e±inθ ). Therefore there are 2n+1 states with energies less than an energy indexed by the number n.

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The case of a particle in a one-dimensional ring is an instructive example when studying the quantization of angular momentum for, say, an electron orbiting the nucleus. The azimuthal wave functions in that case are identical to the energy eigenfunctions of the particle on a ring. Interestingly, the statement that any wavefunction for the particle on a ring can be written as a superposition of energy eigenfunctions is exactly identical to Fourier’s theorem about the development of any periodic function in a Fourier series. This simple model can be used to find approximate energy levels of some ring molecules, such as benzene.

See also • • •

Angular momentum Harmonic analysis. One-dimensional periodic case

Source: http://en.wikipedia.org/wiki/Particle_in_a_ring

Principal Authors: AmarChandra, Charles Matthews, Creidieki, Michael Hardy, Idril

Particle in a ring

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Particle in a spherically symmetric potential In quantum mechanics, the particle in a spherically symmetric potential describes the dynamics of a particle in a central force field, i.e. with potential depending only on the distance of the particle to the center of force (radial dependency), having no angular dependency. In its quantum mechanical formulation, it amounts to solving the →Schrödinger equation with potentials V( r) which depend only on r, the modulus of r. Three special cases arise, of special importance: •

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V(r) =0, or solving the vacuum in the basis of spherical harmonics, which serves as the basis for other cases. • V (r) = V0 for r < r0 and 0 (or ∞) elsewhere, or particle in the spherical equivalent of the square well, useful to describe scattering and bound states in a nucleus or quantum dot. • V(r) 1/r to describe bound states of atoms, especially hydrogen. We outline the solutions in these cases, which should be compared to their counterparts in cartesian coordinates, cf. particle in a box. This article relies heavily on Bessel functions.

General considerations

The time independent solution of 3D Schrödinger equation with hamiltonian p2 /2m0 +V (r) where m0 is the particle’s mass, can be separated in the variables r, θ and φ so that the wavefunction ψ reads: ψ(r) = R(r)Ylm (θ, φ)

Ylm are the usual Spherical harmonics, while R needs be solved with the socalled radial equation: h i 2 l(l+1) d d − 2m~ r2 dr (r2 dr ) + ~2 2m r2 + V (r) R(r) = ER(r) 0

0

It has the shape of the 1D Schrödinger equation for the variable u(r) ≡ rR(r), with a centrifugal term ~2 l(l + 1)/2m0 r2 added to V, but r ranges from 0 to ∞ rather than over R. For more information about how one derive Spherical harmonics from spherical symmetry, see Angular momentum, since the spherical harmonics are the eigenstates of the operator L 2. Particle in a spherically symmetric potential

346

Vacuum case

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Let us now consider V(r) =0 (if V0 , replace everywhere E with E − V0 ). Introducing the dimensionless variable q ρ ≡ kr, k ≡ 2m~20 E r the equation becomes a Bessel equation for J defined by J(ρ) ≡ (whence the notational choice of J ): h
i 2 2− l+ 1 2 J =0 ρ2 ddρJ2 + ρ dJ + ρ 2 dρ

√

ρR(r)

which regular solutions for positive energies are given by so-called Bessel functions of the first kind Jl+1/2 (ρ) so that the solutions written for R are the p so-called Spherical Bessel function R(r) = jl (kr) ≡ π/(2kr)Jl+1/2 (kr). The solutions of Schrödinger equation in polar coordinates for a particle of mass m0 in vacuum are labelled by three quantum numbers: discrete indices l and m, and k varying continuously in [0, ∞]:

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ψ(r) = jl (kr)Ylm (θ, φ)

√ where k ≡ 2m0 E/~, jl are the spherical Bessel function and Ylm are the spherical harmonics. These solutions represent states of definite angular momentum, rather than of definite (linear) momentum, which are provided by plane waves exp(ik · r).

Spherical square well

Let us now consider the potential V (r) = V0 for r < r0 , i.e., inside a sphere of radius r0 and zero outside. We first consider bound states, i.e., states which display the particle mostly inside the box (confined states). Those have an energy E less than the potential outside the sphere, i.e., they have negative energy, and we shall see that there are a discrete number of such states, which we shall compare to positive energy with a continuous spectrum, describing scattering on the sphere (of unbound states). Also worth noticing is that unlike Coulomb potential, featuring an infinite number of discrete bound states, the spherical square well has only a finite (if any) number because of its finite range (if it has finite depth). The resolution essentially follows that of the vacuum with normalisation of the total wavefunction added, solving two Schrödinger equations—inside and outside the sphere—of the previous kind, i.e., with constant potential. Also the following constraints hold: Particle in a spherically symmetric potential

347 • •

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•

The wavefunction must be regular at the origin. The wavefunction and its derivative must be continuous at the potential discontinuity. The wavefunction must converge at infinity.

The first constraint comes from the fact that Neumann N and Hankel H functions are nonsingular at the origin. The physical argument that ψ must be defined everywhere selected Bessel function of the first kind J over the other possibilities in the vacuum case. For the same reason, the solution will be of this kind inside the sphere: q  2m0 (E−V0 ) R(r) = Ajl r , r < r0 ~2

with A a constant to be determined later. Note that for bound states, V0 < E < 0.

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Bound states bring the novelty as compared to the vacuum case that E is now negative (in the vacuum it was to be positive). This, along with third constraint, selects Hankel function of the first kind as the only converging solution at infinity (the singularity at the origin of these functions does not matter since we are now outside the sphere): q  (1) 0E R(r) = Bhl i −2m r , r rel="nofollow"> r0 2 ~

Second constraint on continuity of ψ at r = r0 along with normalization allows the determination of constants A and B. Continuity of the derivative (or logarithmic derivative for convenience) requires quantization of energy.

Infinite spherical square well

In case where the potential is infinitely deep, so that we can take V0 = 0 inside the sphere and ∞ outside, the problem becomes that of matching the wavefunction inside the sphere (the spherical Bessel functions) with identically zero wavefunction outside the sphere. Allowed energies are those for which the radial wavefunction vanishes at the boundary. Thus, we use the zeros of the spherical Bessel functions to find the energy spectrum and wavefunctions. Calling ul,k the k th zero of jl , we have: El =

u2l,k ~2 2m0 r02

So that one is reduced to the computations of these zeros ul,k and to their ordering them (as illustrated graphically below) (note that zeros of j are the same as those of J ). Particle in a spherically symmetric potential

348

Zeros of the first spherical Bessel equations

FT

Calling s, p, d, f, g, h, etc., states with l =0, 1, 2, 3, 4, 5, etc., respectively, we obtain the following spectrum: Spectrum of the infinitely deep spherical square well

Source: http://en.wikipedia.org/wiki/Particle_in_a_spherically_symmetric_potential Principal Authors: Laussy, Oleg Alexandrov, Charles Matthews, Fibonacci, Starwed

Path integral formulation

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This article is about a formulation of quantum mechanics. For integrals along a path, also known as line or contour integrals, see Line integral.

The path integral formulation of quantum mechanics was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier, in the course of his doctoral thesis work with John Archibald Wheeler. It is a description of quantum theory which generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique history for a system with a sum, or functional integral, over an infinity of possible histories to compute a quantum amplitude. This formulation has proved crucial to the subsequent development of theoretical physics, since it provided the basis for the grand synthesis of the 1970’s called the renormalization group which unified quantum field theory with statistical mechanics. It is no surprise, therefore, that path integrals have also been used in the study of Brownian motion and diffusion.

Formulating quantum mechanics

The path integral method is an alternative formulation of quantum mechanics. The canonical approach, pioneered by Schrödinger, Heisenberg and Paul Dirac paid great attention to wave-particle duality and the resulting uncertainty principle by replacing Poisson brackets of classical mechanics by commutators between operators in quantum mechanics. The →Hilbert space of quantum states and the superposition law of quantum amplitudes follows. The path integral starts from the superposition law, and exploits wave-particle duality to build a generating function for quantum amplitudes. Path integral formulation

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349

Figure 32 These are just three of the paths that contribute to the quantum amplitude for a particle moving from point A at some time t 0 to point B at some other time t 1.

Quantum amplitudes

Feynman proposed the following postulates:

1. The probability for any fundamental event is given by the absolute square of a complex amplitude. 2. The amplitude for some event is given by adding together all the histories which include that event.

3. R The amplitude a certain history contributes is proportional to R i ˙ e ~ [L(q,q,t)]dt , where [L(q, q, ˙ t)]dt is the action of that history, or time integral of the Lagrangian.

In order to find the overall probability amplitude for a given process, then, one adds up, or integrates, the amplitude of postulate 3 over the space of all possible histories of the system in between the initial and final states, including histories that are absurd by classical standards. In calculating the amplitude Path integral formulation

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for a single particle to go from one place to another in a given time, it would be correct to include histories in which the particle describes elaborate curlicues, histories in which the particle shoots off into outer space and flies back again, and so forth. The path integral includes them all. Not only that, it assigns all of them, no matter how bizarre, amplitudes of equal magnitude; only the phase, or argument of the complex number, varies. The contributions wildly different from the classical history are suppressed only by the interference of similar histories (see below). Feynman showed that his formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics. An amplitude computed according to Feynman’s principles will also obey the →Schrödinger equation for the Hamiltonian corresponding to the given action.

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Feynman’s postulates are somewhat ambiguous in that they do not define what an "event" is or the exact proportionality constant in postulate 3. The proportionality problem can be solved by simply normalizing the path integral by dividing the amplitude by the square root of the total probability for something to happen (resulting in that the total probability given by all the normalized amplitudes will be 1, as we would expect). Generally speaking one can simply define the "events" in an operational sense for any given experiment. The equal magnitude of all amplitudes in the path integral tends to make it difficult to define it such that it converges and is mathematically tractable. For purposes of actual evaluation of quantities using path-integral methods, it is common to give the action an imaginary part in order to damp the wilder contributions to the integral, then take the limit of a real action at the end of the calculation. In quantum field theory this takes the form of Wick rotation. There is some difficulty in defining a measure over the space of paths. In particular, the measure is concentrated on "fractal-like" distributional paths.

Recovering the action principle

Feynman was initially attempting to make sense of a brief remark by Paul Dirac about the quantum equivalent of the action principle in classical mechanics. In the limit of action that is large compared to Planck’s constant ~, the path integral is dominated by solutions which are stationary points of the action, since there the amplitudes of similar histories will tend to constructively interfere with one another. Conversely, for paths that are far from being stationary points of the action, the complex phase of the amplitude calculated according to postulate 3 will vary rapidly for similar paths, and amplitudes will tend to cancel. Therefore the important parts of the integral—the significant possibilities—in the limit of large action simply consist of solutions of the Euler-Lagrange equation, and classical mechanics is correctly recovered. Path integral formulation

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Time Slicing Definition

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Action principles can seem puzzling to the student of physics because of their seemingly teleological quality: instead of predicting the future from initial conditions, one starts with a combination of initial conditions and final conditions and then finds the path in between, as if the system somehow knows where it’s going to go. The path integral is one way of understanding why this works. The system doesn’t have to know in advance where it’s going; the path integral simply calculates the probability amplitude for a given process, and the stationary points of the action mark neighborhoods of the space of histories for which quantum-mechanical interference will yield large probabilities.

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For a particle in a smooth potential, the path integral is approximated by Feynman as the small-step limit over zig-zag paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position x0 at time 0 to xn at time t, the time interval can be divided up into little segments of fixed duration ∆t. This process is called time slicing. The path integral can be computed as proportional to R R +∞ R +∞ R +∞ R +∞ i (H(x1 ,...,xj ,t)dt) ~ lim dx dx dx . . . dx e 1 2 3 n−1 −∞ −∞ −∞ −∞ ∆t→0,n→∞,n∆t=t

where H is the entire history in which the particle zigzags from its initial to its final position linearly between all the values of xj = x(j∆t).

In the limit of ∆t going to zero, this becomes a functional integral. This limit does not, however, exist for the most important quantum-mechanical systems, the atoms, due to the singularity of the Coulomb potential e2 /r at the origin. The problem was solved in 1979 by Duru and Kleinert (see here 173 and here 174) by choosing ∆t proportional to r and going to new coordinates whose square length is equal to r (→Duru-Kleinert transformation).

Particle in Curved Space

For a particle in curved space the kinetic term depends on the position and the above time slicing cannot be applied, this being a manifestation of the notorious operator ordering problem in Schrödinger quantum mechanics. One may, however, solve this problem by transforming the time-sliced flat-space

173 http://www.physik.fu-berlin.de/~kleinert/kleiner_re65/65.pdf 174 http://www.physik.fu-berlin.de/~kleinert/kleiner_reb5/psfiles/pthic13.pdf

Path integral formulation

352 path integral to curved space using a multivalued coordinate transformation (nonholonomic mapping explained here 175).

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The path integral and the partition function The path integral is just the generalization of the integral above to all quantum mechanical problems— R RT Z = DxeiS[x]/~ where S[x] = 0 dtL[x(t)]

is the action of the classical problem in which one investigates the path starting at time t=0 and ending at time t=T, and Dx denotes integration over all paths. In the classical limit, ~ → 0, the path of minimum action dominates the integral, because the phase of any path away from this fluctuates rapidly and different contributions cancel.

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The connection with statistical mechanics follows. Perform the Wick rotation t→it, i.e., make time imaginary. Then the path integral resembles the partition function of statistical mechanics defined in a canonical ensemble with temperature 1/T ~.

Clearly, such a deep analogy between quantum mechanics and statistical mechanics cannot be dependent on the formulation. In the canonical formulation, one sees that the unitary evolution operator of a state is given by |α; ti = eiHt/~ |α; 0i

where the state α is evolved from time t=0. If one makes a Wick rotation here, and finds the amplitude to go from any state, back to the same state in (imaginary) time iT is given by Z = Tr[e−HT /~ ]

which is precisely the partition function of statistical mechanics for the same system at temperature quoted earlier. One aspect of this equivalence was also known to Schrödinger who remarked that the equation named after him looked like the diffusion equation after Wick rotation.

Quantum field theory

Today, the most common use of the path-integral formulation is in quantum field theory.

175 http://www.physik.fu-berlin.de/~kleinert/b5/psfiles/pthic10.pdf

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The propagator

Functionals of fields

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A common use of the path integral is to calculate hq1 , t1 |q0 , t0 i, a quantity (here written in bra-ket notation) known as the propagator. As such it is very useful in quantum field theory, where the propagator is an important component of Feynman diagrams. One way to do this, which Feynman used to explain photon and electron/positron propagators in quantum electrodynamics, is to apply the path integral to the motion of a single particle—one, however, that can roam back and forth through time as well as space in the course of its wanderings. (Such behavior can be reinterpreted as the contribution of the creation and annihilation of virtual particle-antiparticle pairs, so in this sense the singleparticle restriction has already been loosened.)

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However, the path-integral formulation is also extremely important in direct application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a field over all space. The action is referred to technically as a functional of the field: S[φ] where the field φ(xµ ) is itself a function of space and time, and the square brackets are a reminder that the action depends on all the field’s values everywhere, not just some particular value. In principle, one integrates Feynman’s amplitude over the class of all possible combinations of values that the field could have anywhere in space-time. Much of the formal study of QFT is devoted to the properties of the resulting functional integral, and much effort (not yet entirely successful) has been made toward making these functional integrals mathematically precise.

Such a functional integral is extremely similar to the partition function in statistical mechanics. Indeed, it is sometimes called a partition function, and the two are essentially mathematically identical except for the factor of i in the exponent in Feynman’s postulate 3. Analytically continuing the integral to an imaginary time variable (called a Wick rotation) makes the functional integral even more like a statistical partition function, and also tames some of the mathematical difficulties of working with these integrals.

Expectation values

In quantum field theory, if the action is given by the functional S of field configurations (which only depends locally on the fields), then the time ordered vacuum expectation value of polynomially bounded functional F, , is given by

hF i =

R

DφF [φ]eiS[φ] R DφeiS[φ]

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R

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The symbol Dφ here is a concise way to represent the infinite-dimensional integral over all possible field configurations on all of space-time. As stated above, we put the unadorned path integral in the denominator to normalize everything properly.

Schwinger-Dyson equations

Since this formulation of quantum mechanics is analogous to classical action principles, one might expect that identities concerning the action in classical mechanics would have quantum counterparts derivable from a functional integral. This is often the case.

In the language of functional analysis, we can write the Euler-Lagrange equaδ tions as δφ S[φ] = 0 (the left-hand side is a functional derivative; the equation means that the action is stationary under small changes in the field configuration). The quantum analogues of these equations are called the SchwingerDyson equations.

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If the functional measure Dφ turns out to be translationally invariant (we’ll assume this for the rest of this article, although this does not hold for, let’s say nonlinear sigma models) and if we assume that after a Wick rotation eiS[φ] ,

which now becomes e−H[φ]

for some H, goes to zero faster than any reciprocal of any polynomial for large values of φ, we can integrate by parts (after a Wick rotation, followed by a Wick rotation back) to get the following Schwinger-Dyson equations: E D E D δ δ F [φ] = −i F [φ] S[φ] δφ δφ for any polynomially bounded functional F.



 F,i = −i F S,i in the deWitt notation.

These equations are the analog of the on shell EL equations. If J (called the source field) is an element of the dual space of the field configurations (which has at least an affine structure because of the assumption of

Path integral formulation

355 the translational invariance for the functional measure), then, the generating functional R Z of the source fields is defined to be:

FT

Dφei(S[φ]+hJ,φi)

Z[J] =

Note that δn Z [J] δJ(x1 )···δJ(xn )

= in Z[J] hφ(x1 ) · · · φ(xn )iJ

or

 Z ,i1 ...in [J] = in Z[J] φi1 · · · φin J where

hF iJ =

R

DφF [φ]ei(S[φ]+hJ,φi) R Dφei(S[φ]+hJ,φi)

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Basically, if DφeiS[φ] is viewed as a functional distribution (this shouldn’t be taken too literally as an interpretation of QFT, unlike its Wick rotated statistical mechanics analogue, because we have time ordering complications here!), then hφ(x1 ) · · · φ(xn )i are its moments and Z is its Fourier transform. If F is a functional of φ, then for an operator K, F[K] is defined to be the operator which substitutes K for φ. For example, if F [φ] =

∂ k1 k ∂x11

φ(x1 ) · · ·

∂ kn ∂xknn

φ(xn )

and G is a functional of J, then h i k1 δ δ F −i δJ ··· G[J] = (−i)n ∂ k1 δJ(x ) ∂x1

1

∂ kn δ G[J]. ∂xknn δJ(xn )

Then, from the properties of the functional integrals, we get the "master" Schwinger-Dyson equation: h i δS δ −i δJ Z[J] + J(x)Z[J] = 0 δφ(x) or

S,i [−i∂]Z + Ji Z = 0

If the functional measure is not translationally invariant, it might be possible to express it as the product M [φ] Dφ where M is a functional and Dφ is a translationally invariant measure. This is true, for example, for nonlinear sigma models where the target space is diffeomorphic to R n. However, if the

Path integral formulation

356 target manifold is some topologically nontrivial space, the concept of a translation does not even make any sense.

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In that case, we would have to replace the S in this equation by another functional Sˆ = S − i ln(M )

If we expand this equation as a Taylor series about J=0, we get the entire set of Schwinger-Dyson equations.

Functional identity

If we perform a Wick rotation inside the functional integral, professors J. Garcia and Gerard T´Hooft showed using a functional differential equation that: R P n+1 δ n e−J/~ D[x]e−S[x]/~ = −A[x] ∞ n=0 (~)

where :S is the Wick-rotated classical action of the particle,J is the classical action with an extra term "x" and delta here is the functional derivative operator R :A[x] = exp(1/~ dtX(t)

Ward-Takahashi identities

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See main article Ward-Takahashi identity

Now how about the on shell Noether’s theorem for the classical case? Does it have a quantum analog as well? Yes, but with a caveat. The functional measure would have to be invariant under the one parameter group of symmetry transformation as well.

Let’s just assume for simplicity here that the symmetry in question is local (not local in the sense of a gauge symmetry, but in the sense that the transformed value of the field at any given point under an infinitesimal transformation would only depend on the field configuration over an arbitrarily small neighborhood of the point in question). Let’s also assume that the action is local in the sense that it is the integral over spacetime of a Lagrangian, and that Q[L(x)] = ∂µ f µ (x) for some function f where f only depends locally on φ (and possibly the spacetime position). If we don’t assume any special boundary conditions, this would not be a "true" symmetry in the true sense of the term in general unless f=0 or something. Here, Q is a derivation which generates the one parameter group in question. We could have antiderivations as well, such as BRST and supersymmetry.

R

Let’s also assume DφQ[F ][φ] = 0 for any polynomially bounded functional F. This property is called the invariance of the measure. And this does not hold in general. See anomaly (physics) for more details. Then, R

DφQ[F eiS ][φ] = 0

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357 , which implies

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R  hQ[F ]i + i F ∂V f µ dsµ = 0 where the integral is over the boundary. Noether’s theorem.

This is the quantum analog of

R d Now, let’s assume even further that Q is a local integral Q = d xq(x) where q(x)[φ(y)]=δ (d)(x-y)Q[φ(y)] so that q(x)[S] = ∂µ j µ (x) where

j µ (x) = f µ (x) −

∂ ∂(∂µ φ) L(x)Q[φ]

(this is assuming the Lagrangian only depends on φ and its first partial derivatives! More general Lagrangians would require a modification to this definition!). Note that we’re NOT insisting that q(x) is the generator of a symmetry (i.e. we’re NOT insisting upon the gauge principle), but just that Q is. And let’s also assume the even stronger assumption R that the functional measure is locally invariant:

Dφq(x)[F ][φ] = 0

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. Then, we’d have

hq(x)[F ]i + i hF q(x)[S]i = hq(x)[F ]i + i hF ∂µ j µ (x)i = 0

Alternatively,

δ δ q(x)[S][−i δJ ]Z[J] + J(x)Q[φ(x)][−i δJ ]Z[J] δ J(x)Q[φ(x)][−i δJ ]Z[J] = 0

=

δ ∂µ j µ (x)[−i δJ ]Z[J] +

The above two equations are the Ward-Takahashi identities.

Now for the case where f=0, we can forget about all the boundary conditions and locality assumptions. We’d simply have hQ[F ]i = 0.

Alternatively, R d δ d xJ(x)Q[φ(x)][−i δJ ]Z[J] = 0

The path integral in quantum-mechanical interpretation In one philosophical interpretation of quantum mechanics, the "sum over histories" interpretation, the path integral is taken to be fundamental and reality Path integral formulation

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is viewed as a single indistinguishable "class" of paths which all share the same events. For this interpretation, it is crucial to understand what exactly an event is. The sum over histories method gives identical results to canonical quantum mechanics, and Sinha and Sorkin (what is the reference?) claim the interpretation explains the Einstein-Podolsky-Rosen paradox without resorting to nonlocality. Some advocates of interpretations of quantum mechanics emphasizing decoherence have attempted to make more rigorous the notion of extracting a classical-like "coarse-grained" history from the space of all possible histories.

Suggested reading •

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Feynman, R. P., and Hibbs, A. R., Quantum Physics and Path Integrals, New York: McGraw-Hill, 1965 [ISBN 0-070-20650-3]. The historical reference, written by the Master himself and one of his students. Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore, 2004); Paperback ISBN 981-238-107-4 (also available online: PDF-files 176) Zinn Justin, Jean ; Path Integrals in Quantum Mechanics, Oxford University Press (2004), [ISBN 0-19-856674-3]. A highly readable introduction to the subject. Schulman, Larry S. ; Techniques & Applications of Path Integration, Jonh Wiley & Sons (New York-1981) [ISBN ]. The modern reference on the subject. Grosche, Christian & Steiner, Frank ; Handbook of Feynman Path Integrals, Springer Tracts in Modern Physics 145, Springer-Verlag (1998) [ISBN 3540-57135-3] Ryder, Lewis H. ; Quantum Field Theory (Cambridge University Press, 1985), [ISBN 0-521-33859-X] Highly readable textbook, certainly the best introduction to relativistic Q.F.T. for particle physics. Rivers, R.J. ; Path Integrals Methods in Quantum Field Theory, Cambridge University Press (1987) [ISBN 0-521-22979-7] Albeverio, S. & Hoegh-Krohn. R. ; Mathematical Theory of Feynman Path Integral, Lecture Notes in Mathematics 523, Springer-Verlag (1976) [ISBN ]. Glimm, James, and Jaffe, Arthur, Quantum Physics: A Functional Integral Point of View, New York: Springer-Verlag, 1981. [ISBN 0-387-90562-6].

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176 http://www.physik.fu-berlin.de/~kleinert/b5

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Gerald W. Johnson and Michel L. Lapidus ; The Feynman Integral and Feynman’s Operational Calculus, Oxford Mathematical Monographs, Oxford University Press (2002) [ISBN 0-19-851572-3]. Etingof, Pavel ; Geometry and Quantum Field Theory 177, M.I.T. OpenCourseWare (2002). This course, designed for mathematicians, is a rigorous introduction to perturbative quantum field theory, using the language of functional integrals.

Papers on-line •

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Grosche, Christian ; An Introduction into the Feynman Path Integral, lecture given at the graduate college Quantenfeldtheorie und deren Anwendung in der Elementarteilchen- und Festkörperphysik, Universität Leipzig, 16-26 November 1992. Full text available at : hep-th/9302097 178. • MacKenzie, Richard ; Path Integral Methods and Applications, lectures given at Rencontres du Vietnam: VIth Vietnam School of Physics, Vung Tau, Vietnam, 27 December 1999 - 8 January 2000. Full text available at : quant-ph/0004090 179. • DeWitt-Morette, Cécile ; Feynman’s path integral - Definition without limiting procedure, Communication in Mathematical Physics 28(1) (1972) pp. 47–67. Full text available at : Euclide Project 180. • Cartier, Pierre & DeWitt-Morette, Cécile ; A new perspective on Functional Integration, Journal of Mathematical Physics 36 (1995) pp. 2137-2340. Full text available at : funct-an/9602005 181.

Source: http://en.wikipedia.org/wiki/Path_integral_formulation

Principal Authors: Matt McIrvin, Karl-H, AcidFlask, Bambaiah, Charles Matthews

177 http://ocw.mit.edu/OcwWeb/Mathematics/18-238Fall2002/CourseHome/index.htm 178 http://arxiv.org/abs/hep-th/9302097 179 http://arxiv.org/abs/quant-ph/0004090 180 http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.cmp/1103858329 181 http://fr.arxiv.org/abs/funct-an/9602005

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Penrose Interpretation

Figure 33 Diagram illustrating a less complex version of the experiment to test penrose Interpretation

Penrose Interpretation is an interpretation of quantum mechanics formulated by Sir Roger Penrose. This theory is a possible step toward Quantum gravity, as it describes quantum mechanics in terms of General relativity. It states that a system requires energy to allow it to exist in more than one location. A macroscopic system, such as a human, connot exist in more than one position because its significant gravitational field requires it to have a large amount of energy to sustain, and will settle into one position within a trilionth of a second. However a microscopic system or particle (such as an electron) has an insignificant gravitational field, and therefore requires so little energy that it could exist in more than one location almost indefinitly; This is called superposition. In Einstein’s theory, any object that has mass causes a warp in the structure of space and time round it. This warping produces the effect we experience as gravity. Penrose points out that tiny objects-dust specks, atoms, electrons-produce space-time warps as well. Ignoring these warps is where Penrose Interpretation

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most physicists go awry, he believes. If a dust speck is in two locations at the same time, each one should create its own distortions in space-time, yielding two superposed gravitational fields. According to Penrose’s theory, it takes energy to sustain these dual fields. The stability of a system depends on the amount of energy involved: The higher the energy required to sustain a system, the less stable it is. Over time, an unstable system tends to settle back to its simplest, lowest-energy state-in this case, one object in one location producing one gravitational field. If Penrose is right, gravity yanks objects back into a single location, without any need to invoke observers or parallel universes.

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Penrose believes that the transition between macroscopic and quantum begins on the scale of dust particles, that could exist in more than one location for as long as one second (a very long time compared to larger objects). An experiment has been developed to test this theory, in which a X-ray laser in space is directed toward a tiny miror , and fissioned by a beam spliter from thousands of miles away, in which the photons are directed toward to other mirrors and reflected back. According to modern physics one photon will stike the tiny mirror moving en route to another mirror and move the tiny mirror back as it returns, so the tiny mirror exists in two location at one time. If gravity effects the mirror, it will be unable to exist in two locations at once because gravity holds it in place.

See also •

Interpretations of Quantum Mechanics

References •

Folger, Tim. "If an Electron Can Be in 2 Places at Once, Why Can’t I?" Discover. Vol. 25 No. 6 (June 2005). 32.

External links • • •

Molecules - Quantum Interpretations 182 QM- the Penrose Interpretation 183 Roger Penrose discusses his experiment on the BBC (25 minutes in) 184

182 http://universe-review.ca/F12-molecule.htm#interpretations 183 http://sci4um.com/about2884.html

184 http://www.bbc.co.uk/radio4/history/inourtime/inourtime_20020502.shtml

Penrose Interpretation

362 Source: http://en.wikipedia.org/wiki/Penrose_Interpretation

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Peres-Horodecki criterion

The Peres-Horodecki criterion is a necessary condition, for the joint density matrix ρ of two systems A and B, to be separable. It is also called the PPT criterion, for Positive Partial Transpose. In the 2x2 and 2x3 dimensional cases the condition is also sufficient. The criterion reads:

If ρ is separable, then the partial transpose σm µ n ν := ρn µ m ν

of ρ, taken in some basis |miA ⊗ |µiB , has non negative eigenvalues.

In matrix notation, if we write a N X M mixed state ρ as block matrix:



··· .. . ···

 ρ1n ..  . 

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ρ11  .. ρ= .

ρn1

ρnn

,where each ρi:j is M X M and n runs from 1 to N. The partial transpose of ρ is then given by

ρP T

ρT11  .. = . 

ρTn1

··· .. . ···

 ρT1n ..  . 

ρTnn

So ρ is PPT if (I ⊗ T )(ρ) is positive, where T is the transposition map on matrices. That necessity of PPT for separability follows immediately from the fact that if ρ is separable, then (I ⊗ Φ)(ρ) must be positive for all positive map Φ. The transposition map is clearly a positive map. Showing that being PPT is also sufficient for in the 2 X 2 and 2 X 3 (therefore 3 X 2) cases is more involved. It was shown by the Horodecki’s that for every entangled state there exists an entanglement witness. This is a result of geometric nature and invokes the Hahn-Banach theorem (see reference below). From the existence of entanglement witnesses, one can show that (I ⊗ Φ)(ρ) being positive for all positive map Φ is not only necessary but also sufficient for Peres-Horodecki criterion

363

Λ = Λ1 + Λ 2 ◦ T

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separability of ρ. Furthermore, every positive map from the C*-algebra of 2 × 2 matrices to 2 × 2 or 3 × 3 matrices can be decomposed into a sum of completely positive and completely copositive maps. In other words, every such map can be written as

,where Λ1 and Λ2 are completely positive and T is the transposition map. Combining the above two facts, we can conclude PPT is also sufficient for separability in the 2 X 2 and 2 X 3 cases. Due to the existence of non-decomposable positive maps in higher dimensions, PPT is no longer sufficient in higher dimensions. In higher dimension, there are entangled states which are PPT. Such states have some interesting propeties including the fact that thay are bound entangled, i.e. they can not be distilled for quantum communication purposes.

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References • •

Asher Peres, Separability Criterion for Density Matrices, Phys. Rev. Lett. 77, 1413–1415 (1996) M. Horodecki, P. Horodecki, R. Horodecki, Separability of Mixed States: Necessary and Sufficient Conditions, Physics Letters A 210, 1996.

Source: http://en.wikipedia.org/wiki/Peres-Horodecki_criterion

Principal Authors: Mct mht, Tinissimo, Stevey7788, Matthew Mattic, Charles Matthews

Perturbation theory (quantum mechanics) In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. The idea is to start with a simple system and gradually turn on an additional "perturbing" Hamiltonian representing a weak disturbance to the system. If the disturbance is not too large, the various physical quantities associated with the perturbed system (e.g. its energy levels and eigenstates) will be continuously generated from those of the simple system. We can therefore study the former based on our knowledge of the latter.

Perturbation theory (quantum mechanics)

364

Applications of perturbation theory

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Perturbation theory is an extremely important tool for describing real quantum systems, as it turns out to be very difficult to find exact solutions to the →Schrödinger equation for Hamiltonians of even moderate complexity. The Hamiltonians to which we know exact solutions, such as the hydrogen atom, the quantum harmonic oscillator and the particle in a box, are too idealized to adequately describe most systems. Using perturbation theory, we can use the known solutions of these simple Hamiltonians to generate solutions for a range of more complicated systems. For example, by adding a perturbative electric potential to the quantum mechanical model of the hydrogen atom, we can calculate the tiny shifts in the spectral lines of hydrogen caused by the presence of an electric field (the →Stark effect). This is only approximate because the sum of a Coulomb potential with a linear potential is unstable although the tunneling time (decay rate) is very long. This shows up as a broadening of the energy spectrum lines, something which perturbation theory fails to notice entirely.

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The expressions produced by perturbation theory are not exact, but they can lead to accurate results as long as the expansion parameter, say α, is very small. Typically, the results are expressed in terms of finite power series in α that seem to converge to the exact values when summed to higher order. After a certain order n ∼ 1/α, however, the results become increasingly worse since the series are usually divergent, being asymptotic series). There exist ways to convert them into convergent series, which can be evalauted for largeexpansion parameters, most efficiently by variational perturbation theory. In the theory of quantum electrodynamics (QED), in which the electron-photon interaction is treated perturbatively, the calculation of the electron’s magnetic moment has been found to agree with experiment to eleven decimal places. In QED and other quantum field theories, special calculation techniques known as Feynman diagrams are used to systematically sum the power series terms. Under some circumstances, perturbation theory is an invalid approach to take. This happens when the system we wish to describe cannot be described by a small perturbation imposed on some simple system. In quantum chromodynamics, for instance, the interaction of quarks with the gluon field cannot be treated perturbatively at low energies because the coupling constant (the expansion parameter) becomes too large. Perturbation theory also fails to describe states that are not generated adiabatically from the "free model", including bound states and various collective phenomena such as solitons. Imagine, for example, that we have a system of free (i.e. non-interacting) particles, to which an attractive interaction is introduced. Depending on the form of the interaction, this may create an entirely new set of eigenstates corresponding to Perturbation theory (quantum mechanics)

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groups of particles bound to one another. An example of this phenomenon may be found in conventional superconductivity, in which the phonon-mediated attraction between conduction electrons leads to the formation of correlated electron pairs known as Cooper pairs. When faced with such systems, one usually turns to other approximation schemes, such as the variational method and the WKB approximation. This is because there is no analogue of a bound particle in the unperturbed model and the energy of a soliton typically goes as the inverse of the expansion parameter. However, if we "integrate" over the solitonic phenomena, the nonperturbative corrections in this case will be tiny; of the 2 order of e−1/g or e−1/g in the perturbation parameter g. Perturbation theory can only detect solutions "close" to the unperturbed solution, even if there are other solutions (which typically blow up as the expansion parameter goes to zero).

DR A

The problem of non-perturbative systems has been somewhat alleviated by the advent of modern computers. It has become practical to obtain numerical non-perturbative solutions for certain problems, using methods such as density functional theory. These advances have been of particular benefit to the field of quantum chemistry. Computers have also been used to carry out perturbation theory calculations to extraordinarily high levels of precision, which has proven important in particle physics for generating theoretical results that can be compared with experiment.

Time-independent perturbation theory There are two categories of perturbation theory: time-independent and timedependent. In this section, we discuss time-independent perturbation theory, in which the perturbation Hamiltonian is static (i.e., possesses no time dependence.) Time-independent perturbation theory was invented by Erwin Schrödinger in 1926, shortly after he produced his theories in wave mechanics.

We begin with an unperturbed Hamiltonian H 0, which is also assumed to have no time dependence. It has known energy levels and eigenstates, arising from the time-independent Schrödinger equation: (0)

H0 |n(0) i = En |n(0) i ,

n = 1, 2, 3, · · ·

For simplicity, we have assumed that the energies are discrete. The (0) superscripts denote that these quantities are associated with the unperturbed system.

We now introduce a perturbation to the Hamiltonian. Let V be a Hamiltonian representing a weak physical disturbance, such as a potential energy produced by an external field. (Thus, V is formally a Hermitian operator.) Let λ be a Perturbation theory (quantum mechanics)

366 dimensionless parameter that can take on values ranging continuously from 0 (no perturbation) to 1 (the full perturbation). The perturbed Hamiltonian is

FT

H = H0 + λV The energy levels and eigenstates of the perturbed Hamiltonian are again given by the Schrödinger equation: (H0 + λV ) |ni = En |ni

Our goal is to express E n and |n> in terms of the energy levels and eigenstates of the old Hamiltonian. If the perturbation is sufficiently weak, we can write them as power series in : (0)

(1)

(2)

En = En + λEn + λ2 En + · · ·

|ni = |n(0) i + λ|n(1) i + λ2 |n(2) i + · · ·

DR A

When = 0, these reduce to the unperturbed values, which are the first term in each series. Since the perturbation is weak, the energy levels and eigenstates should not deviate too much from their unperturbed values, and the terms should rapidly become smaller as we go to higher order. Plugging the power series into the Schrödinger equation, we obtain


(0) (1) (H0 +  λV ) |n i + λ|n i + · · · 
(0) (1) (2) = En + λEn + λ2 En + · · · |n(0) i + λ|n(1) i + · · ·

Expanding this equation and comparing coefficients of each power of results in an infinite series of simultaneous equations. The zeroth-order equation is simply the Schrödinger equation for the unperturbed system. The first-order equation is (0)

(1)

H0 |n(1) i + V |n(0) i = En |n(1) i + En |n(0) i

Multiply through by
En = hn(0) |V |n(0) i

This is simply the expected value of the perturbation Hamiltonian while the system is in the unperturbed state. This result can be interpreted in the following way: suppose the perturbation is applied, but we keep the system in Perturbation theory (quantum mechanics)

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FT

the quantum state |n (0)>, which is a valid quantum state though no longer an energy eigenstate. The perturbation causes the average energy of this state to increase by . However, the true energy shift is slightly different, because the perturbed eigenstate is not exactly the same as |n (0)>. These further shifts are given by the second and higher order deviations. To obtain the first-order deviation in the energy eigenstate, we insert our expression for the first-order energy shift back into the above equation between the first-order coefficients of . We then make use of the resolution of the identity,  P (0) (0) (0) V |n(0) i = k |k ihk | V |n i The result is     P (0) En − H0 |n(1) i = k6=n hk (0) |V |n(0) i |k (0) i

DR A

For the moment, suppose that this energy level is not degenerate, i.e. there is no other eigenstate with the same energy. The operator on the left hand side therefore has a well-defined inverse, and we get P hk (0) |V |n(0) i |n(1) i = k6=n (0) (0) |k (0) i En −Ek

The first-order change in the n-th energy eigenket has a contribution from each of the energy eigenstates k 6= n. Each term is proportional to the matrix element , which is a measure of how much the perturbation mixes eigenstate n with eigenstate k ; it is also inversely proportional to the energy difference between eigenstates k and n, which means that the perturbation deforms the eigenstate to a greater extent if there are more eigenstates at nearby energies. We see also that the expression is singular if any of these states have the same energy as state n, which is why we assumed that there is no degeneracy. We can find the higher-order deviations by a similar procedure, though the calculations become quite tedious with our current formulation. For example, the second-order energy shift is P (2) |hk (0) |V |n(0) i|2 En = k6=n (0) (0) En −Ek

Effects of degeneracy

Suppose that two or more energy eigenstates are degenerate. Our above calculation for the first-order energy shift is unaffected, but the calculation of the change in the eigenstate is problematic because the operator Perturbation theory (quantum mechanics)

368 (0)

En − H 0 does not have a well-defined inverse.

FT

This is actually a conceptual, rather than mathematical, problem. Imagine that we have two or more perturbed eigenstates with different energies, which are continuously generated from an equal number of unperturbed eigenstates that are degenerate. Let D denote the subspace spanned by these degenerate eigenstates. The problem lies in the fact that there is no unique way to choose a basis of energy eigenstates for the unperturbed system. In particular, we could construct a different basis for D by choosing different linear combinations of the spanning eigenstates. In such a basis, the unperturbed eigenstates would not continuously generate the perturbed eigenstates. We thus see that, in the presence of degeneracy, perturbation theory does not work with an arbitrary choice of energy basis. We must instead choose a basis so that the perturbation Hamiltonian is diagonal in the degenerate subspace D. In other words, ∀ |k (0) i ∈ D

DR A

V |k (0) i = k |k (0) i + (terms not in D)

In that case, our equation for the first-order deviation in the energy eigenstate reduces to     P (0) En − H0 |n(1) i = k6∈D hk (0) |V |n(0) i |k (0) i The operator on the left hand side is not singular when applied to eigenstates outside D, so we can write P hk (0) |V |n(0) i |n(1) i = k6∈D (0) (0) |k (0) i En −Ek

Time-dependent perturbation theory

Time-dependent perturbation theory, developed by Paul Dirac, studies the effect of a time-dependent perturbation V(t) applied to a time-independent Hamiltonian H 0. Since the perturbed Hamiltonian is time-dependent, so are its energy levels and eigenstates. Therefore, the goals of time-dependent perturbation theory are slightly different from time-independent perturbation theory. We are interested in the following quantities: •

The time-dependent expected value of some observable A, for a given initial state. • The time-dependent amplitudes of those quantum states that are energy eigenkets (eigenvectors) in the unperturbed system. Perturbation theory (quantum mechanics)

369

FT

The first quantity is important because it gives rise to the classical result of an A measurement performed on a macroscopic number of copies of the perturbed system. For example, we could take A to be the displacement in the x-direction of the electron in a hydrogen atom, in which case the expected value, when multiplied by an appropriate coefficient, gives the time-dependent electrical polarization of a hydrogen gas. With an appropriate choice of perturbation (i.e. an oscillating electric potential), this allows us to calculate the AC permittivity of the gas. The second quantity looks at the time-dependent probability of occupation for each eigenstate. This is particularly useful in laser physics, where one is interested in the populations of different atomic states in a gas when a timedependent electric field is applied. These probabilities are also useful for calculating the "quantum broadening" of spectral lines (see line broadening).

DR A

We will briefly examine the ideas behind Dirac’s formulation of time-dependent perturbation theory. Choose an energy basis {|n>} for the unperturbed system. (We will drop the (0) superscripts for the eigenstates, because it is not meaningful to speak of energy levels and eigenstates for the perturbed system.)

If the unperturbed system is in eigenstate |j > at time t = 0, its state at subsequent times varies only by a phase (we are following the →Schrödinger picture, where state vectors evolve in time and operators are constant): |j(t)i = e−iEj t/~ |ji

We now introduce a time-dependent perturbing Hamiltonian V(t). The Hamiltonian of the perturbed system is H = H0 + V (t)

Let |ψ(t) > denote the quantum state of the perturbed system at time t. It obeys the time-dependent Schrödinger equation, ∂ H|ψ(t)i = i~ ∂t |ψ(t)i

The quantum state at each instant can be expressed as a linear combination of the basis {|n>}. We can write the linear combination as P |ψ(t)i = n cn (t)e−iEn t/~ |ni

where the c n(t) s are undetermined complex functions of t which we will refer to as amplitudes (strictly speaking, they are the amplitudes in the Dirac picture.) We have explicitly extracted the exponential phase factors exp(iE nt/<strike>h) on the right hand side. This is only a matter of Perturbation theory (quantum mechanics)

370

FT

convention, and may be done without loss of generality. The reason we go to this trouble is that when the system starts in the state |j > and no perturbation is present, the amplitudes have the convenient property that, for all t, c j(t) = 1 and c n(t) = 0 if n6=j. The absolute square of the amplitude c n(t) is the probability that the system is in state n at time t, since |cn (t)|2 = |hn|ψ(t)i|2

Plugging into the Schrödinger equation and using the fact that ∂/∂t acts by a chain rule, we obtain  P  ∂cn −iEn t/~ |ni = 0 n i~ ∂t − cn (t)V (t) e

DR A

By resolving the identity in front of V, this can be reduced to a set of partial differential equations for the amplitudes: ∂cn −i P −i(Ek −En )t/~ k hn|V (t)|ki ck (t) e ∂t = ~

The matrix elements of V play a similar role as in time-independent perturbation theory, being proportional to the rate at which amplitudes are shifted between states. Note, however, that the direction of the shift is modified by the exponential phase factor. Over times much longer than the energy difference E k-E n, the phase winds many times. If the time-dependence of V is sufficiently slow, this may cause the state amplitudes to oscillate. Such oscillations are useful for managing radiative transitions in a laser.

Up to this point, we have made no approximations, so this set of differential equations is exact. By supplying appropriate initial values c n(0), we could in principle find an exact (i.e. non-perturbative) solution. This is easily done when there are only two energy levels (n = 1, 2), and the solution is useful for modelling systems like the ammonia molecule. However, exact solutions are difficult to find when there are many energy levels, and one instead looks for perturbative solutions, which may be obtained by putting the equations in an integral form: P Rt 0 0 0 −i(Ek −En )t0 /~ cn (t) = cn (0) + −i k 0 dt hn|V (t )|ki ck (t ) e ~

By repeatedly substituting this expression for c n back into right hand side, we get an iterative solution (0)

(1)

(2)

cn (t) = cn + cn + cn + · · ·

where, for example, the first-order term is

Perturbation theory (quantum mechanics)

371 (1)

cn (t) =

−i ~

P Rt k 0

0

dt0 hn|V (t0 )|ki ck (0) e−i(Ek −En )t /~

FT

Many further results may be obtained, such as →Fermi’s golden rule, which relates the rate of transitions between quantum states to the density of states at particular energies, and the Dyson series, obtained by applying the iterative method to the time evolution operator, which is one of the starting points for the method of Feynman diagrams.

Source: http://en.wikipedia.org/wiki/Perturbation_theory_%28quantum_mechanics%29 Principal Authors: CYD, Phys, A. Wilson, Karol Langner, Sigfpe

Photoelectric effect

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The photoelectric effect is the emission of electrons from matter upon the absorption of electromagnetic radiation, such as ultraviolet radiation or x-rays. An older term for the photoelectric effect was the Hertz effect, though this phrase has fallen out of current use. 185

Introduction

Upon exposing a metallic surface to electromagnetic radiation that is above the threshold frequency (which is specific to the type of surface and material), the photons are absorbed and current is produced. No electrons are emitted for radiation with a frequency below that of the threshold, as the electrons are unable to gain sufficient energy to overcome the electrostatic barrier presented by the termination of the crystalline surface (the material’s work function). In 1905 it was known that the energy of the photoelectrons increased with increasing frequency of incident light, but the manner of the increase was not experimentally determined to be linear until 1915 when Robert Andrews Millikan showed that Einstein was correct [3]. By conservation of energy, the energy of the photon is absorbed by the electron and, if sufficient, the electron can escape from the material with a finite kinetic energy. A single photon can only eject a single electron, as the energy of one photon may only be absorbed by one electron. The electrons that are emitted are often termed photoelectrons. The photoelectric effect helped further wave-particle duality, whereby physical systems (such as photons, in this case) display both wave-like and particle-like

185

http://scienceworld.wolfram.com/physics/HertzEffect.html

Photoelectric effect

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372

DR A

Figure 34 The photoelectric effect. Incoming EM radiation on the left ejects electrons, depicted as flying off to the right, from a substance.

properties and behaviours, a concept that was used by the creators of quantum mechanics. The photoelectric effect was explained mathematically by Albert Einstein, who extended the work on quanta developed by Max Planck.

Explanation

The photons of the light beam have a characteristic energy given by the wavelength of the light. In the photoemission process, if an electron absorbs the energy of one photon and has more energy than the work function, it is ejected from the material. If the photon energy is too low, however, the electron is unable to escape the surface of the material. Increasing the intensity of the light beam does not change the energy of the constituent photons, only their number, and thus the energy of the emitted electrons does not depend on the intensity of the incoming light. Electrons can absorb energy from photons when irradiated, but they follow an "all or nothing" principle. All of the energy from one photon must be absorbed and used to liberate one electron from atomic binding, or the energy is reemitted. If the photon is absorbed, some of the energy is used to liberate it from the atom, and the rest contributes to the electron’s kinetic (moving) energy as a free particle.

Photoelectric effect

373

Equations In analysing the photoelectric effect quantitatively using Einstein’s method, the following equivalent equations are used:

Algebraically: hf = φ + Ekmax where • • •

FT

Energy of photon = Energy needed to remove an electron + Kinetic energy of the emitted electron

DR A

h is Planck’s constant, f is the frequency of the incident photon, φ = hf0 is the work function, or minimum energy required to remove an electron from atomic binding, 2 is the maximum kinetic energy of ejected electrons, • Ekmax = 12 mvm • f 0 is the threshold frequency for the photoelectric effect to occur, • m is the rest mass of the ejected electron, and • vm is the velocity of the ejected electron.

Note: If the photon’s energy (hf ) is not greater than the work function (φ), no electron will be emitted. The work function is sometimes denoted W .

History

Early observations

In 1839, Alexandre Edmond Becquerel observed the photoelectric effect via an electrode in a conductive solution exposed to light. In 1873, Willoughby Smith found that selenium is photoconductive.

Hertz’s spark gaps

Heinrich Hertz, in 1887, made observations of the photoelectric effect and of the production and reception of electromagnetic (EM) waves, published in the journal Annalen der Physik. His receiver consisted of a coil with a spark gap, whereupon a spark would be seen upon detection of EM waves. He placed the apparatus in a darkened box in order to see the spark better; he observed, however, that the maximum spark length was reduced when in the box. A glass panel placed between the source of EM waves and the receiver absorbed ultraviolet radiation that assisted the electrons in jumping across the gap. When removed, the spark length would increase. He observed no decrease in spark

Photoelectric effect

374 length when he substituted quartz for glass, as quartz does not absorb UV radiation.

FT

Hertz concluded his months of investigation and reported the results obtained. He did not further pursue investigation of this effect, nor did he make any attempt at explaining how the observed phenomenon was brought about.

JJ Thomson: electrons

DR A

In 1899, Joseph John Thomson investigated ultraviolet light in Crookes tubes. Influenced by the work of James Clerk Maxwell, Thomson deduced that cathode rays consisted of negatively charged particles, later called electrons, which he called "corpuscles". In the research, Thomson enclosed a metal plate (a cathode) in a vacuum tube, and exposed it to high frequency radiation. It was thought that the oscillating electromagnetic fields caused the atoms’ field to resonate and, after reaching a certain amplitude, caused a subatomic "corpuscle" to be emitted, and current to be detected. The amount of this current varied with the intensity and color of the radiation. Larger radiation intensity or frequency would produce more current.

Tesla’s radiant energy

On November 5 1901, Nikola Tesla received the U.S. Patent 685957 186 (Apparatus for the Utilization of Radiant Energy) that describes radiation charging and discharging conductors by "radiant energy". Tesla used this effect to charge a capacitor with energy by means of a conductive plate. The patent specified that the radiation included many different forms.

Von Lenard’s observations

In 1902, Philipp von Lenard observed 187 the variation in electron energy with light frequency. He used a powerful electric arc lamp which enabled him to investigate large changes in intensity, and had sufficient power to enable him to investigate the variation of potential with light frequency. His experiment directly measured potentials, not electron kinetic energy: he found the electron energy by relating it to the maximum stopping potential (voltage) in a phototube. He found that the calculated maximum electron kinetic energy is determined by the frequency of the light. For example, an increase in frequency results in an increase in the maximum kinetic energy calculated for an electron upon liberation - ultraviolet radiation would require a higher applied stopping potential to stop current in a phototube than blue light. However Lenard’s results were qualitative rather than quantitative because of the difficulty in

186 http://patft.uspto.gov/netacgi/nph-Parser?patentnumber=685957 187 http://www.phys.virginia.edu/classes/252/photoelectric_effect.html

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Einstein: light quanta

FT

performing the experiments: the experiments needed to be done on freshly cut metal so that the pure metal was observed, but it oxidised in a matter of minutes even in the partial vacuums he used. The current emitted by the surface was determined by the light’s intensity, or brightness: doubling the intensity of the light doubled the number of electrons emitted from the surface. Lenard did not know of photons.

DR A

Albert Einstein’s mathematical description in 1905 of how it was caused by absorption of what were later called photons, or quanta of light, in the interaction of light with the electrons in the substance, was contained in the paper named "On a Heuristic Viewpoint Concerning the Production and Transformation of Light". This paper proposed the simple description of "light quanta" (later called "photons") and showed how they could be used to explain such phenomena as the photoelectric effect. The simple explanation by Einstein in terms of absorption of single quanta of light explained the features of the phenomenon and helped explain the characteristic frequency. Einstein’s explanation of the photoelectric effect won him the Nobel Prize of 1921.

The idea of light quanta was motivated by Max Planck’s published law of blackbody radiation ("On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik 4 (1901)) by assuming that Hertzian oscillators could only exist at energies E proportional to the frequency f of the oscillator by E = hf, where h is Planck’s constant. Einstein, by assuming that light actually consisted of discrete energy packets, wrote an equation for the photoelectric effect that fit experiments. This was an enormous theoretical leap and the reality of the light quanta was strongly resisted. The idea of light quanta contradicted the wave theory of light that followed naturally from James Clerk Maxwell’s equations for electromagnetic behavior and, more generally, the assumption of infinite divisibility of energy in physical systems. Even after experiments showed that Einstein’s equations for the photoelectric effect were accurate there was resistance to the idea of photons, since it appeared to contradict Maxwell’s equations, which were believed to be well understood and well verified. Einstein’s work predicted that the energy of the ejected electrons would increase linearly with the frequency of the light. Perhaps surprisingly, that had not yet been tested. In 1905 it was known that the energy of the photoelectrons increased with increasing frequency of incident light, but the manner of the increase was not experimentally determined to be linear until 1915 when Robert Andrews Millikan showed that Einstein was correct 188.

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376

Effect on wave-particle question

Uses and effects

FT

The photoelectric effect helped propel the then-emerging concept of the dual nature of light, that light exhibits characteristics of waves and particles at different times. The effect was impossible to understand in terms of the classical wave description of light, as the energy of the emitted electrons did not depend on the intensity of the incident radiation. Classical theory predicted that the electrons could ’gather up’ energy over a period of time, and then be emitted. For such a classical theory to work a pre-loaded state would need to persist in matter. The idea of the pre-loaded state was discussed in Millikan’s book Electrons (+ & -) and in Compton and Allison’s book X-Rays in Theory and Experiment. These ideas were abandoned.

Solar cells (used in solar power) and light-sensitive diodes use the photoelectric effect. They absorb photons from light and put the energy into electrons, creating electric current.

DR A

Electroscopes

Electroscopes are fork-shaped, hinged metallic leaves placed in a vacuum jar, partially exposed to the outside environment. When an electroscope is charged positively or negatively, the two leaves separate, as charge distributes evenly along the leaves causing repulsion between two like poles. When ultraviolet radiation (or any radiation above threshold frequency) shines onto the metallic outside of the electroscope, a negatively charged scope will discharge and the leaves will collapse, while nothing will happen to a positively charged scope (besides charge decay). The reason is that electrons will be liberated from the negatively charged one, gradually making it neutral, while liberating electrons from the positively charged one will make it even more positive, keeping the leaves apart.

Photoelectron spectroscopy

Since the energy of the photoelectrons emitted is exactly the energy of the incident photon minus the material’s work function or binding energy, the work function of a sample can be determined by bombarding it with a monochromatic X-ray source or UV source (typically a helium discharge lamp), and measuring the kinetic energy distribution of the electrons emitted. This must be done in a high vacuum environment, since the electrons would be scattered by air.

188 http://spiff.rit.edu/classes/phys314/lectures/photoe/photoe.html

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Spacecraft

FT

A typical electron energy analyzer is a concentric hemispherical analyser (CHA), which uses an electric field to divert electrons different amounts depending on their kinetic energies. For every element and core atomic orbital there will be a different binding energy. The many electrons created from each will then show up as spikes in the analyzer, and can be used to determine the elemental composition of the sample. 189

The photoelectric effect will cause spacecraft exposed to sunlight to develop a positive charge. This can get up to the tens of volts. This can be a major problem, as other parts of the spacecraft in shadow develop a negative charge (up to several kilovolts) from nearby plasma, and the imbalance can discharge through delicate electrical components. The static charge created by the photoelectric effect is self-limiting, though, because a more highly-charged object gives up its electrons less easily. 190

Moon dust

DR A

Light from the sun hitting lunar dust causes it to become charged through the photoelectric effect. The charged dust then repels itself and lifts off the surface of the Moon by electrostatic levitation. This manifests itself almost like an "atmosphere of dust", visible as a thin haze and blurring of distant features, and visible as a dim glow after the sun has set. This was first photographed by the Surveyor program probes in the 1960s. It is thought that the smallest particles are repelled up to kilometers high, and that the particles move in "fountains" as they charge and discharge. 191 192

189 Photoelectron spectroscopy(http://www.chem.qmw.ac.uk/surfaces/scc/scat5_3.htm) 190 Spacecraft charging(http://www.eas.asu.edu/~holbert/eee460/spc-chrg.html) 191 - Moon fountains(http://www.firstscience.com/site/articles/moonfountains.asp) 192 - Dust gets a charge in a vacuum(http://www.spacer.com/news/dust-00a.html)

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See also Electronics:

People:

• • • • •

• • • • • • •

Physics: Atom Corona discharge →Double-slit experiment Electron Gamma ray Nobel Prize in Physics Optical phenomenon →Planck’s law of black body radiation Photon →Quantum mechanics Radiant energy →Wave-particle duality

Lists: • • • • • •

List of electronics topics List of optical topics List of physics topics Timeline of solar cells Scientific method list Timeline of mechanics and physics

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• • • • • • • • • • • •

Aleksandr Grigorievich Stoletov Albert Einstein Heinrich Hertz Ernest Lawrence Robert Millikan Max Planck Joseph John Thomson

FT

Photocurrent Photomultiplier Solar cell Solar power Transducer

External links and references General

Nave, R., " Wave-Particle Duality 193". HyperPhysics. Jpaul’s " Photovoltaics: Theory and Practice 194". Photoelectric effect 195. " Photoelectric effect 196". Physics 2000. University of Colorado, Boulder, Colorado. • ACEPT W3 Group, " The Photoelectric Effect 197". Department of Physics and Astronomy, Arizona State University, Tempe, AZ. • Haberkern, Thomas, and N Deepak " Grains of Mystique: Quantum Physics for the Layman 198". Einstein Demystifies Photoelectric Effect 199, Chapter 3. • Department of Physics, " The Photoelectric effect 200". Physics 320 Laboratory, Davidson College, Davidson.

• • •

193 http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html 194 http://www.students.uiuc.edu/~jpaul/theory.htm

195 http://www.students.uiuc.edu/~jpaul/photoelectric.htm 196 http://www.colorado.edu/physics/2000/quantumzone/photoelectric.html 197 http://acept.la.asu.edu/PiN/rdg/photoelectric/photoelectric.shtml 198 http://www.faqs.org/docs/qp/ 199 http://www.faqs.org/docs/qp/chap03.html 200 http://www.phy.davidson.edu/ModernPhysicsLabs/hovere.html

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• •

Fowler, Michael, " The Photoelectric Effect 201". Physics 252, University of Virginia. Brandl, Michael, " MISN-0-213 The Photoelectric Effect 202" (PDF file), Project PHYSNET 203. Quantum Chemistry I Lecute 204

Applets • • •

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•

Curull, Xavi Espinal, " Photoelectric effect Applet 205". (Java) Fendt, Walter, and Taha Mzoughi, " The Photoelectric Effect 206". (Java) " Applet: Photo Effect 207". Open Source Distributed Learning Content Management and Assessment System. (Java)

Notes

Source: http://en.wikipedia.org/wiki/Photoelectric_effect

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Principal Authors: Reddi, Enochlau, William M. Connolley, Heron, Omegatron

Planck particle

A Planck particle is a hypothetical subatomic particle, defined as a tiny black hole whose →Compton wavelength is the same as its Schwarzschild radius. Its mass is thus (by definition) equal to the Planck mass, and its Compton wavelength and Schwarzschild radius are equal (also by definition) to the Planck length.

See also •

Micro black hole

Source: http://en.wikipedia.org/wiki/Planck_particle Principal Authors: Jaraalbe, Hidaspal

201 http://www.phys.virginia.edu/classes/252/photoelectric_effect.html 202 http://35.9.69.219/home/modules/pdf_modules/m213.pdf 203 http://www.physnet.org 204 http://cinarz.zdo.com/moodle/mod/resource/view.php?id=15 205 http://www.ifae.es/xec/phot2.html 206 http://www.walter-fendt.de/ph14e/photoeffect.htm

207 http://lectureonline.cl.msu.edu/~mmp/kap28/PhotoEffect/photo.htm

Planck particle

380

Planck postulate

FT

The Planck Postulate (or Planck’s Postulate) was used by Max Planck in his derivation of his law of black body radiation. It is the postulate that the energy of oscillators in a black body is quantised by: E = nhν ,

where n = 1, 2, 3, ..., h is Planck’s constant, and ν is the frequency.

External links and sources •

Planck Postulate 208 — from Eric Weisstein’s World of Physics

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Source: http://en.wikipedia.org/wiki/Planck_postulate

Planck’s law of black body radiation In physics, the spectral intensity of electromagnetic radiation from a black body at temperature T is given by Planck’s law of black body radiation: I(ν, T ) =

1 2hν 3 hν c2 e kT −1

where the following table provides the definition and SI units of measure for each symbol: Symbol Meaning

SI units of measure

I

spectral radiance, energy per unit time per unit surface area per unit solid angle per unit frequency

J·s -1·m -2·sr -1·Hz -1

ν

frequency

hertz

T

temperature of the black body

kelvin

h

Planck’s constant

joule per hertz

c

speed of light

meter per second

e

base of the natural logarithm, 2.718282...

dimensionless

k

Boltzmann’s constant

joule per kelvin

208 http://scienceworld.wolfram.com/physics/PlanckPostulate.html

Planck’s law of black body radiation

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FT

381

Figure 35

Black body spectrum as a function of wavelength

The wavelength is related to the frequency by λ = νc .

The law is sometimes written in terms of the spectral energy density u(ν, T ) =

4π c I(ν, T )

=

1 8πhν 3 hν c3 e kT −1

which has units of energy per unit volume per unit frequency (joule per cubic meter per hertz). The spectral energy density can also be expressed as a function of wavelength: u(λ, T ) =

1 8πhc λ5 ehc/λkT −1

as shown in the derivation below.

Max Planck originally produced this law in 1900 (published in 1901) in an attempt to improve upon an expression proposed by Wilhelm Wien which fit the experimental data at short wavelengths but deviated from it at long wavelengths. He found that the above function, Planck’s function, fit the data for Planck’s law of black body radiation

382

E = hν .

FT

all wavelengths remarkably well. In constructing a derivation of this law, he considered the possible ways of distributing electromagnetic energy over the different modes of charged oscillators in matter. Planck’s law emerged when he assumed that the energy of these oscillators was limited to a set of discrete, integer multiples of a fundamental unit of energy, E, proportional to the oscillation frequency ν:

DR A

Planck made this quantization assumption five years before Albert Einstein hypothesized the existence of photons as a means of explaining the photoelectric effect. At the time, Planck believed that the quantization applied only to the tiny oscillators that were thought to exist in the walls of the cavity (what we now know to be atoms), and made no assumption that light itself propagates in discrete bundles or packets of energy. Moreover, Planck did not attribute any physical significance to this assumption, but rather believed that it was merely a mathematical device that enabled him to derive a single expression for the black body spectrum that matched the empirical data at all wavelengths. Ultimately, Planck’s assumption of energy quantization and Einstein’s photon hypothesis became the fundamental basis for the later development of Quantum Mechanics. Both scientists would eventually receive (separate) Nobel prizes in recognition of these major contributions to the advancement of physics.

Derivation (Statistical Mechanics)

(See also the gas in a box article for a general derivation.)

Consider a cube of side L with conducting walls filled with electromagnetic radiation. At the walls of the cube, the parallel component of the electric field and the orthogonal component of the magnetic field must vanish. Analogous to the wave function of a particle in a box, one finds that the fields are superpositions of periodic functions. The wavelength λi in the three directions i = 1 . . . 3 orthogonal to the walls can be: λi =

2L ni

where the ni are integers. For each set of integers ni there are two linear independent solutions (modes). According to quantum theory, the energy levels of a mode are given by:
hc q 2 En1 ,n2 ,n3 (r) = r + 12 2L n1 + n22 + n23 (1)

Planck’s law of black body radiation

383

FT

The quantum number r can be interpreted as the number of photons in the mode. The two modes for each set of ni correspond to the two polarization states of the photon which has a spin of 1. Note that for r = 0 the energy of the mode is not zero. This vacuum energy of the electromagnetic field is responsible for the Casimir effect. In the following we will calculate the internal energy of the box at temperature T relative to the vacuum energy. According to statistical mechanics, the probability distribution over the energy levels of a particular mode is given by: Pr =

exp(−βE(r)) Z(β)

Here β ≡ 1/ (kT ).

DR A

The denominator Z (β), is the partition function of a single mode and makes Pr properly normalized: P 1 Z (β) = ∞ r=0 exp [−βE (r)] = 1−exp[−βε] Here we have defined q hc n21 + n22 + n23 ε ≡ 2L

which is the energy of a single photon. As explained here, the average energy in a mode can be expressed in terms of the partition function: hEi = −

d log(Z) dβ

=

ε exp(βε)−1

This formula is a special case of the general formula for particles obeying →Bose-Einstein statistics. Since there is no restriction on the total number of photons, the chemical potential is zero. The total energy in the box now follows by summing hEi over all allowed single photon states. This can be done exactly in the thermodynamic limit L → ∞. In this limit, ε becomes continuous and we can then integrate hEi over this parameter. To calculate the energy in the box in this way, we need to evaluate how many photon states there are in a given energy range. If we write the total number of single photon states with energies between ε and ε + dε as g (ε) d, where g (ε) is the density of states which we’ll evaluate in a moment, then we can write: R∞ ε U = 0 exp(βε)−1 g (ε) dε (2)

Planck’s law of black body radiation

384 To calculate the density of states we rewrite equation (1) as follows: ε≡

hc 2L n

FT

where n is the norm of the vector ~n = (n1 , n2 , n3 ): q n = n21 + n22 + n23

For every vector n with integer components larger or equal than zero there are two photon states. This means that the number of photon states in a certain region of n-space is twice the volume of that region. An energy range of dε corresponds to shell of thickness dn = 2L hc dε in n-space. Because the components of ~n have to be positive, this shell spans an octant of a sphere. The number of photon states g (ε) d in an energy range dε is thus given by: g (ε) d = 2 81 4πn2 dn =

8πL3 2 ε dε h3 c3

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Inserting this in Eq. (2) gives: R∞ ε3 dε (3) U = L3 h8π 3 3 0 exp(βε)−1 c

From this equation one easily derives the spectral energy density as a function of frequency u(ν, T ) and as a function of wavelength u(λ, T ): R∞ U = 0 u(ν, T )dν L3 where:

u(ν, T ) =

1 8πhν 3 c3 ehν/kT −1

u(ν, T ) is known as the black body spectrum. It is a spectral energy density function with units of energy per unit frequency per unit volume. And:

U L3

=

R∞ 0

u(λ, T )dλ

where

u(λ, T ) =

8πhc 1 λ5 ehc/λkT −1

This is also a spectral energy density function with units of energy per unit wavelength per unit volume. Integrals of this type for Bose and Fermi gases can be expressed in terms of polylogarithms. In this case, however, it is possible to

Planck’s law of black body radiation

385 calculate the integral in closed form. Let’s first make the integration variable in Eq. (3) dimensionless by substituting ε = kT x: 8π(kT )4 J (hc)3

Here J is given by: R∞ x3 J = 0 exp(x)−1 dx =

π4 15

FT

u(T ) =

We prove this result in the Appendix below. The total electromagnetic energy inside the box is thus given by: U V

=

8π 5 (kT )4 15(hc)3

DR A

where V = L3 is the volume of the box. (Note - This is not the StefanBoltzmann law, which is the total energy radiated by a black body. See that article for an explanation.) Since the radiation is the same in all directions, and propagates at the speed of light (c), the spectral intensity (energy/time/area/solid angle/frequency) is I(ν, T ) =

u(ν,T ) c 4π

which yields

I(ν, T ) =

1 2hν 3 c2 ehν/kT −1

Derivation (Thermodynamics)

The fact that the energy density of the box containing radiation is proportional to T 4 was derived by Ludwig Boltzmann in 1884 using thermodynamics. It follows from classical electrodynamics that the radiation pressure P is related to the internal energy density: P =

u 3

The total internal energy of the box containing radiation can thus be written as: U = 3P V

Inserting this in the fundamental law of thermodynamics dU = T dS − P dV

yields the equation:

dS = 4 PT dV + 3 VT dP

We can now use this equation to derive a Maxwell relation. We read off that: Planck’s law of black body radiation

386 ∂S P ∂V = 4 T

V

FT

P

And V ∂S ∂P = 3 T

The symmetry of second derivatives of S w.r.t. P and V then implies: ∂( P ) ∂( V ) 4 ∂PT = 3 ∂VT V

P

Because the pressure is proportional to the internal energy density it depends only on the temperature and not on the volume. In the derivative on the r.h.s. the temperature is thus a constant. Evaluating the derivatives gives the differential equation: 1 dP P dT

=

4 T

This implies that u = 3P ∝ T 4

History

DR A

Many popular science accounts of quantum theory, as well as some physics textbooks, contain some serious errors in their discussions of the history of Planck’s Law. Although these errors were pointed out over forty years ago by historians of physics, they have proved to be difficult to eradicate. The article by Helge Kragh cited below gives a lucid account of what actually happened.

Contrary to popular opinion, Planck did not quantize light. This is evident from his original 1901 paper and the references therein to his earlier work. It is also plainly explained in his book "Theory of Heat Radiation," where he explains that his constant refers to Hertzian oscillators. The idea of quantization was developed by others into what we now know as quantum mechanics. The next step along this road was made by Albert Einstein, who, by studying the photoelectric effect, proposed a model and equation whereby light was not only emitted but also absorbed in packets or photons. Then, in 1924, Satyendra Nath Bose developed the theory of the statistical mechanics of photons, which allowed a theoretical derivation of Planck’s law. Contrary to another myth, Planck did not derive his law in an attempt to resolve the "ultraviolet catastrophe", the name given to the paradoxical result that the total energy in the cavity tends to infinity when the equipartition theorem of classical statistical mechanics is applied to black body radiation. Planck did not consider the equipartion theorem to be universally valid, so he never noticed any sort of "catastrophe" — it was only discovered some five years later by Einstein, Lord Rayleigh, and Sir James Jeans.

Planck’s law of black body radiation

387

Appendix

FT

A simple way to calculate the integral R∞ x3 J = 0 exp(x)−1 dx is as follows. After multiplying the numerator and denominator of the integrand we can expand the integrand in powers of exp(−x). R ∞ x3 exp(−x) R∞ 3 P P∞ 1 π4 J = 0 1−exp(−x) dx = ∞ n=1 0 x exp (−nx) dx = 6 n=1 n4 = 15 Here we have used that

P∞

1 n=1 n4

the argument 4, which is given by the contour integral H π cot(πz) CR

z4

is the Riemann zeta function evaluated for π4 90 .

This fact can be proven by considering

DR A

Where CR is a contour of radius R around the origin. In the limit R → ∞ the integral approaches zero. Using the residue theorem the integral can also be written as a sum of residues at the poles of the integrand. The poles are at zero, the positive and negative integers. The sum of the residues yields precisely twice the desired summation plus the residue at zero. This means that P∞ 1 π4 3 n=1 n4 equals minus 2 times the coefficient of x of the series expansion of the series expansion of the cotangent function.

External link and references

Planck’s original 1901 paper 209 Planck, Max, "On the Law of Distribution of Energy in the Normal Spectrum". Annalen der Physik, vol. 4, p. 553 ff (1901). • Radiation of a Blackbody 210 - interactive simulation to play with Planck’s law • Scienceworld entry on the Planck Law 211 • Kragh, Helge Max Planck: The reluctant revolutionary 212 Physics World, December 2000 • •

Source: http://en.wikipedia.org/wiki/Planck%27s_law_of_black_body_radiation

209 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html 210 http://www.vias.org/simulations/simusoft_blackbody.html 211 http://scienceworld.wolfram.com/physics/PlanckLaw.html 212 http://www.physicsweb.org/articles/world/13/12/8/1

Planck’s law of black body radiation

388 Principal Authors: Metacomet, PAR, Unc.hbar, Diegueins, Rparson

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FT

Plum pudding model

Figure 36

A schematic representation of the plum pudding model of the atom.

In physics, the Plum pudding model of the atom was proposed by J. J. Thomson, the discoverer of the electron in 1897. The plum pudding model was proposed in March, 1904 before the discovery of the atomic nucleus. In this model, the atom is composed of electrons surrounded by a soup of positive charge to balance the electron’s charge, like plums surrounded by pudding. The electrons were thought to be positioned throughout the atom, but with many electron structures possible, particularly rotating rings (see below). Instead of a soup, the atom was also sometimes said to have had a cloud of positive charge. The model was disproved by the 1909 gold foil experiment, which was interpreted by Ernest Rutherford in 1911 to imply a very small nucleus of the atom containing its full positive charge, thus leading implicitly to the →Rutherford Plum pudding model

389 model of the atom, and eventually, by 1913, to the solar-system-like (but quantum-limited) →Bohr model of the atom.

FT

Thomson’s model was compared (though not by Thomson) to a British treat called plum pudding, hence the name. It has also been called the chocolate chip cookie model, but only by those who have not read Thomson’s original paper (On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure), published in the Philosophical Magazine (the leading British science journal of the day). For an excerpt see 213.

DR A

A little-known (or now forgotten) fact about the original Thomson "plum pudding" model is that it was dynamic, not static. The electrons were free to rotate within the blob or cloud of positive substance. These orbits were stabilized in the model by the fact that when an electron moved farther from the center of the positive cloud, it felt a larger net positive inward force, because there was more material of opposite charge, inside its orbit (A particle like a small black hole would feel the same restorative force if it penetrated the body of the Earth; such a particle would feel only the gravity of the Earth inside its radius). In Thomson’s model, electrons were free to rotate in rings which were further stabilized by interactions between the electrons, and spectra were to be accounted for by energy differences of different ring orbits. Thomson attempted to make his model account for some of the major spectral lines known for some elements, but was not notably successful at this. Still, Thomson’s model (along with a similar Saturnian ring model for atomic electrons, put forward by Nagaoka after the Maxwell model of Saturn’s rings), were earlier harbingers of the later and more successful solar-system like →Bohr model of the atom.

External links •

"On the Structure of the Atom: an Investigation of the Stability and Periods of Oscillation of a number of Corpuscles arranged at equal intervals around the Circumference of a Circle; with Application of the Results to the Theory of Atomic Structure" 214 — J.J. Thomson’s 1904 paper proposing the plum pudding model.

Source: http://en.wikipedia.org/wiki/Plum_pudding_model

Principal Authors: Linas, Salsb, Sbharris, Fastfission, Ragesoss

213 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Thomson-Structure-Atom.html 214 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Thomson-Structure-Atom.html

Plum pudding model

390

Definition

FT

Position operator In quantum mechanics, the position operator corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L 2(R), the Hilbert space of complex-valued and square-integrable (with respect to the Lebesgue measure) functions on the real line. The position operator, Q, is then defined by Q(ψ)(x) = x · ψ(x),

with domain D(Q) = {ψ ∈ L2 |Qψ ∈ L2 }.

DR A

Since all continuous functions with compact support lie in D(Q), Q is densely defined. Q, being simply multiplication by x, is a self adjoint operator, thus satisfying the requirement of a quantum mechanical observable. Immediately from the definition we can deduce that the spectrum consists of the entire real line and that Q has purely continuous spectrum, therefore no eigenvalues. The three dimensional case is defined analogously. We shall keep the onedimensional assumption in the following discussion.

Measurement

As with any observable, In order to discuss measurement, we need to calculate the spectral resolution of Q : R Q = λdΩQ (λ).

Since Q is just multiplication by x, its spectral resolution is simple. For a Borel subset B of the real line, let C B denote the indicator function of B. We see that the projection-valued measure ΩQ are given by ΩQ (B)ψ = C B · ψ

,i.e. ΩQ (B) is multiplication by the indicator fuction of B. Therefore, if the system is prepared in state Ψ, then the probability of the measured position of the particle being in a Borel set B is R |ΩQ (B)ψ|2 = |C B · ψ|2 = B |ψ|2 dµ

,µ being the Lebesgue measure. After the measurement, the wave function Ω (B)ψ collapses to ||ΩQ (B)ψ|| , where || · || is the Hilbert space norm. Q

Position operator

391

Unitary equivalence with momentum operator For a particle on a line, the momentum operator P is defined by

FT

∂ P ψ = −i~ ∂x ψ

,with appropriate domain. P and Q are unitarily equivalent, with the unitary operator being given explicitly by the Fourier transform. Thus they have the same spectrum. In physical language, P acting on momentum space wave functions is the same as Q acting on position space wave functions (under the image of Fourier transform).

Source: http://en.wikipedia.org/wiki/Position_operator

Potential energy surface

DR A

A potential energy surface is generally used within the adiabatic or BornOppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems. There is a natural correspondence between potential energy surfaces as they exist (as polynomial surfaces) and their application in potential theory, which associates and studies harmonic functions in relation to these surfaces. For example, the Morse potential and the simple harmonic potential well are common one-dimensional potential energy surfaces (potential energy curves) in applications of quantum chemistry and physics.

Source: http://en.wikipedia.org/wiki/Potential_energy_surface Principal Authors: V8rik, Cypa, Charles Matthews

Potential well

A potential well is the region surrounding a local minimum of potential energy. Energy captured in a potential well is unable to convert to another type of energy (kinetic energy in the case of a gravitational potential well) because it is captured in the local minimum of a potential well. Therefore, a body may not proceed to the global minimum of potential energy, as it would naturally tend to due to entropy.

Potential well

392

Overview

FT

Energy may be released from a potential well if sufficient energy is added to the system such that the local minimum is surmounted. In quantum physics, potential energy may escape a potential well without added energy due to the probabilistic characteristics of quantum particles; in these cases a particle may be imagined to tunnel through the walls of a potential well.

The graph of a 2D potential energy function is a potential energy surface that can be imagined as the Earth’s surface in a landscape of hills and valleys. Then a potential well would be a valley surrounded on all sides with higher terrain, which thus could be filled with water (i.e., be a lake) without any water flowing away toward another, lower minimum (i.e. sea level). In the case of gravity, the region around a mass is a gravitional potential well, unless the density of the mass is so low that tidal forces from other masses are greater than the gravity of the body itself.

DR A

A potential hill is the opposite of a potential well, the region surrounding a local maximum.

Quantum confinement

Quantum confinement is when electrons and holes in a semiconductor are confined by a potential well in 1D (quantum well), 2D (quantum wire), or 3D (quantum dot). That is, quantum confinement occurs when one or more of the dimensions of a nanocrystal is made very small so that it approaches the size of an exciton in bulk crystal, called the Bohr exciton radius. A quantum well is a structure where the height is approximately the Bohr exciton radius while the length and breadth can be large. A quantum wire is a structure where the height and breadth is made small while the length can be long. A quantum dot is a structure where all dimensions are near the Bohr exciton radius, typically a small sphere.

See also •

a Graphical representation of a potential well

References •

W. E. Buhro and V. L. Colvin, Semiconductor nanocrystals: matters 215, Nat. Mater., 2003, 2, 138 139.

Shape

215 http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=12612665

Potential well

393

Source: http://en.wikipedia.org/wiki/Potential_well

POVM

FT

Principal Authors: Patrick, Laurascudder, Linas, Fasten, Bantman

In functional analysis and quantum measurement theory, a POVM (Positive Operator Value Measure) is a measure whose values are non-negative self-adjoint operators on a →Hilbert space. It is the most general formulation of a measurement in the theory of quantum physics. The need for the POVM formalism arises from the fact that projective measurements on a larger system will act on a subsystem in ways that cannot be described by projective measurement on the subsystem alone. They are used in the field of Quantum information.

DR A

In rough analogy, a POVM is to a projective measurement what a density matrix is to a pure state. Density matrices can describe part of a larger system that is in a pure state (see purification of quantum state); analogously, POVMs on a physical system can describe the effect of a projective measurement performed on a larger system.

Definition

In the simplest case, a POVM is a set of Hermitian positive semidefinite operators P {Fi } on a Hilbert space H that sum to unity, n i=1

Fi = IH .

This formula is similar to the decomposition of a →Hilbert space into a set of orthogonal projectors,

PN

i=1

Ei = IH ,

and if i 6= j,

Ei Ej = 0.

An important difference is that the elements of a POVM are not necessarily orthogonal, with the consequence that the number of elements in the POVM, n, can be larger than the dimension, N, of the →Hilbert space they act in.

In general, POVMs can be defined in situations where outcomes can occur in a non-discrete space. The relevant fact is that measurements determine probability measures on the outcome space: POVM

394

E 7→ hF (E)ξ | ξi

FT

Definition. Let (X, M ) be measurable space; that is M is a σ-algebra of subsets of X. A POVM is a function F defined on M whose values are bounded nonnegative self-adjoint operators on a Hilbert space H such that F(X ) = I H and for every ξ ∈ H,

is a non-negative countably additive measure on the σ-algebra M.

POVMs and measurement

As in the theory of projective measurement, the probability the outcome associated with measurement of operator Fi occurs is, P (i) = T r(Fi ρ),

where ρ is the density matrix describing the state of the measured system. An element of a POVM can always be written as,

DR A

Fi = Mi† Mi ,

for some operator Mi , known as a Kraus Operator. The state of the system after the measurement ρ0 is transformed according to, ρ0 =

Mi ρMi† . tr(Mi ρMi† )

Neumark’s dilation theorem

An alternate spelling of this is Naimark’s Theorem

Neumark’s dilation theorem states that measuring a POVM consisting of a set of n>N operators acting on a N-dimensional →Hilbert space can always be achieved by performing a projective measurement on a Hilbert space of dimension n then consider the reduced state. In practice, however, obtaining a suitable projection-valued measure from a given POVM is usually done by coupling to the original system an ancilla. Consider a Hilbert space HA that is extended by HB . The state of total system is ρAB and ρA = T rA (ρAB ). The probability the projective measurement π ˆi succeeds is, P (i) = T rA (T rB (ˆ πi ρAB )).

POVM

395 An implication of Neumark’s theorem is that the associated POVM in subspace A, Fi , must have the same probability of success.

FT

P (i) = T rA (Fi ρA )).

An example: Unambiguous quantum state discrimination

DR A

The task of unambiguous quantum state discrimination (UQSD) is to discern conclusively which state, of given set of pure states, a quantum system (which we call the input) is in. The impossibility of perfectly discriminating between a set of non-orthogonal states is the basis for quantum information protocols such as quantum cryptography, quantum coin-flipping, and quantum money. This example will show that a POVM has a higher success probability for performing UQSD than any possible projective measurement.

Figure 37 The projective measurement strategy for unambiguously discriminating between nonorthogonal states.

First let us consider a trivial case. Take a set that consists of two orthogonal states |ψi and |ψ T i. A projective measurement of the form,

POVM

396 Aˆ = a|ψ T ihψ T | + b|ψihψ|,

|hφ|ψi| = cos(θ),

FT

will result in eigenvalue a only when the system is in |ψ T i and eigenvalue b only when the system is in |ψi. In addition, the measurement always discriminates between the two states (i.e. with 100% probability). This latter ability is unnecessary for UQSD and, in fact, is impossible for anything but orthogonal states. Now consider a set that consists of two states |ψi and |φi in two-dimensional Hilbert space that are not orthogonal. i.e.,

for θi0. These states could a system, such as the spin of spin-1/2 particle (e.g. an electron), or the polarization of a photon. Assuming that the system has an equal likelihood of being in each of these two states, the best strategy for UQSD using only projective measurement is to perform each of the following measurements,

DR A

π ˆψT = |ψ T ihψ T |, π ˆφT = |φT ihφT |,

50% of the time. If π ˆφT is measured and results in an eigenvalue of 1, than it is certain that the state must have been in |ψi. However, an eigenvalue of zero is now an inconclusive result since this can come about from the system could being in either of the two states in the set. Similarly, a result of 1 for π ˆψT indicates conclusively that the system is in |φi and 0 is inconclusive. The probability that this strategy returns a conclusive result is, Pproj =

1−|hφ|ψi|2 . 2

In contrast, a strategy based on POVMs has a greater probability of success given by, PP OV M = 1 − |hφ|ψi|.

This is the minimum allowed by the rules of quantum indeterminacy and the uncertainty principle. This strategy is based on a POVM consisting of, 1−|φihφ| Fˆψ = 1+|hφ|ψi|

Fˆφ =

1−|ψihψ| 1+|hφ|ψi|

POVM

397 Fˆinconcl. = 1 − Fˆψ − Fˆφ ,

DR A

FT

where the result associated with Fˆi indicates the system is in state i with certainty.

Figure 38 The POVM strategy for unambiguously discriminating between nonorthogonal states.

These POVMs can be created by extending the two-dimensional Hilbert space. This can be visualized as follows: The two states fall in the x-y plane with an angle of θ between them and the space is extended in the z-direction. (The total space is the direct sum of spaces defined by the z-direction and the x-y plane.) The measurement first unitarily rotates the states towards the z-axis so that |ψi has no component along the y-direction and |φi has no component along the x-direction. At this point, the three elements of the POVM correspond to projective measurements along x-direction, y-direction and z-direction, respectively. For a specific example, take a stream of photons, each of which are polarized along either along the horizontal direction or at 45 degrees. On average there are equal numbers of horizontal and 45 degree photons. The projective strategy corresponds to passing the photons through a polarizer in either the vertical

POVM

398

See also • • • • • •

Quantum measurement →Mathematical formulation of quantum mechanics Quantum logic Density matrix Quantum operation Projection-valued measure

References •

FT

direction or -45 degree direction. If the photon passes through the vertical polarizer it must have been at 45 degrees and vice versa. The success probability is (1 − 1/2)/2 = 25%. The POVM strategy for this example is more complicated and requires another optical mode (known as an ancilla). It has a success √ probability of 1 − 1/ 2 = 29.3%.

DR A

POVMs • J.Preskill, Lecture Note for Physics: Quantum Information and Computation, http://theory.caltech.edu/people/preskill • K.Kraus, States, Effects, and Operations, Lecture Notes in Physics 190, Springer (1983) • E.B.Davies, Quantum Theory of Open Systems, Academic Press (1976). • Neumark’s theorem • A. Peres. Neumark’s theorem and quantum inseparability. Foundations of Physics, 12:1441–1453, 1990. • A. Peres. Quantum Theory: Concepts and Methods. Kluwer Academic Publishers, 1993. • I. M. Gelfand and M. A. Neumark, On the imbedding of normed rings into the ring of operators in Hilbert space, Rec. Math. [Mat. Sbornik] N.S. 12(54) (1943), 197–213. • Unambiguous quantum state-discrimination • I. D. Ivanovic, Phys. Lett. A 123 257 (1987). • D. Dieks, Phys. Lett. A 126 303 (1988). • A. Peres, Phys. Lett. A 128 19 (1988).

Source: http://en.wikipedia.org/wiki/POVM

POVM

399

Probability amplitude

FT

In quantum mechanics, a probability amplitude is a complex-valued function that describes an uncertain or unknown quantity. For example, each particle has a probability amplitude describing its position. This amplitude is then called wave function. This is a complex-valued function of the position coordinates.

For a probability amplitude ψ, the associated probability density function is ψ*ψ, which is equal to |ψ| 2. This is sometimes called just probability density 1, and may be found used without normalization.

DR A

If |ψ| 2 has a finite integral over the whole of three-dimensional space, then it is possible to choose a normalising constant, c, so that by replacing ψ by cψ the integral becomes 1. Then the probability that a particle is within a particular region V is the integral over V of |ψ| 2. Which means, according to the Copenhagen interpretation of quantum mechanics, that, if some observer tries to measure the quantity associated with this probability amplitude, the result of the measurement will lie within  with a probability P() given by R P () =  |ψ(x)|2 dx Probability amplitudes which are not square integrable are usually interpreted as the limit of a series of functions which are square integrable. For instance the probability amplitude corresponding to a plane wave corresponds to the ’non physical’ limit of a monochromatic source of particles. Another example: The Siegert wave functions describing a resonance are the limit for t → ∞ of a time-dependent wave packet scattered at an energy close to a resonance. In these cases, the definition of P() given above is still valid. The change over time of this probability (in our example, this corresponds to a description of how the particle moves) is expressed in terms of ψ itself, not just the probability function |ψ| 2. See →Schrödinger equation. In order to describe the change over time of the probability density it is acceptable to define the probability flux (also called probability current). The probability flux j is defined as: j=

~ m

·

1 2i

(ψ ∗ ∇ψ − ψ∇ψ ∗ ) =

~ ∗ m Im (ψ ∇ψ)

and measured in units of (probability)/(area*time) = r -2t -1. The probability flux satisfies a quantum continuity equation, i.e.:

Probability amplitude

400 ∇·j+

∂ ∂t P (x, t)

=0

FT

where P(x,t) is the probability density and measured in units of (probability)/(volume) = r -3. This equation is the mathematical equivalent of probability conservation law. It is easy to show that for a plane wave function, |ψi = A exp (ikx − iωt) the probability flux is given by j(x, t) = |A|2 k~ m

The bi-linear form of the axiom has interesting consequences as well.

Notes

Note 1: Max Born was awarded part of the 1954 Nobel Prize in Physics for this work.

DR A

•

Source: http://en.wikipedia.org/wiki/Probability_amplitude

Principal Authors: Charles Matthews, P3d0, That Guy, From That Show!, Onebyone, Michael Hardy, RJFJR, NymphadoraTonks, Conscious, Paul A

Probability current

In quantum mechanics, the probability current (sometimes called probability flux) is a useful concept which describes the flow of probability density. In particular, if one pictures the probability density as an inhomogeneous fluid, then the probability current is the rate of flow of this fluid (the density times the velocity).

Definition

The probability current, ~j, is defined as   ~ − Ψ∇Ψ ~ ∗ ~j = ~ Ψ∗ ∇Ψ 2mi

in the position basis and satisfies the quantum mechanical continuity equation

Probability current

401 ∂ρ ∂t

~ · ~j = 0 +∇

ρ = |Ψ|2 .

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with the probability density ρ defined as

The divergence theorem implies the continuity equation is equivalent to the integral equation R R ∂ ~ =0 |Ψ|2 dV + ~j · dA ∂t V

S

where the V is any volume and S is the boundary of V . This is the conservation law for probability density in quantum mechanics.

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In particular, if Ψ is a wavefunction describing a single particle, the integral in the first term of the preceding equation (without the time derivative) is the probability of obtaining a value within V when the position of the particle is measured. The second term is then the rate at which probability is flowing out of the volume V . Altogether the equation states that the time derivative of the change of the probability of the particle being measured in V is equal to the rate at which probability flows into V .

Examples Plane wave

The probability current associated with the (three dimensional) plane wave ~

Ψ = eik·~r

is

~j =

~ 2mi



 ~ ~ i~k·~r ~ ~ −i~k·~r = e−ik·~r ∇e − eik·~r ∇e

~~k m.

This is just the particle’s momentum p~ = ~~k

divided by its mass, i.e. its "velocity" (insofar as a quantum mechanical particle has one). Note that the probability current is nonzero despite the fact that plane waves are stationary states and hence d|Ψ|2 dt

=0

Probability current

402 everywhere. This demonstrates that a particle may be in "motion" even if its spacial probability density has no explicit time dependence.

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Particle in a box The energy eigenstates of a particle in a box of one spatial dimension and of length L are q
Ψn = L2 sin nπ L x The associated probability currents are   ∂Ψ∗n ~ n jn = 2mi Ψ∗n ∂Ψ =0 ∂x − Ψn ∂x since Ψn = Ψ∗n .

Derivation of continuity equation

In this section the continuity equation is derived from the definition of probability current and the basic principles of quantum mechanics.

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Suppose Ψ is the wavefunction for a single particle in the position basis (i.e. Ψ is a function of x, y, and z). Then R P = V |Ψ|2 dV

is the probability that a measurement of the particle’s position will yield a value within V. The time derivative of this is  R R  ∂Ψ ∗ ∂ dP ∂Ψ∗ 2 dV dt = ∂t V |Ψ| dV = V ∂t Ψ + Ψ ∂t

where the last equality follows from the product rule and the fact that the shape of V is presumed to be independent of time (i.e. the time derivative can be moved through the integral). In order to simplify this further consider the time dependent →Schrödinger equation i~ ∂Ψ ∂t =

−~2 2 2m ∇ Ψ + V Ψ

and use it to solve for the time derivative of Ψ : ∂Ψ ∂t

=

i~ 2 i 2m ∇ Ψ − ~ V Ψ

When substituted back into the preceding equation for
R ~ dP ∗ 2 2 ∗ dt = − V 2mi Ψ ∇ Ψ − Ψ∇ Ψ dV .

Now from the product rule for the divergence operator Probability current

dP dt

this gives

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~ − Ψ∇Ψ ~ ∗ = ∇Ψ ~ ∗ · ∇Ψ ~ + Ψ∗ ∇2 Ψ − ∇Ψ ~ · ∇Ψ ~ ∗ − Ψ∇ ~ 2 Ψ∗ ∇ · Ψ∗ ∇Ψ

dt

V

2mi

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and since the first and third terms cancel:   R dP ~ · ~ Ψ∗ ∇Ψ ~ − Ψ∇Ψ ~ ∗ dV =− ∇

If we now recall the expression for P and note that the argument of the divergence operator is just ~j this becomes  R  ∂|Ψ|2 ~ · ~j dV = 0 + ∇ V ∂t which is the integral form of the continuity equation. The differential form follows from the fact that the preceding equation holds for all V , and hence the integrand must vanish everywhere: ∂|Ψ|2 ∂t

~ · ~j = 0. +∇

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Source: http://en.wikipedia.org/wiki/Probability_current

Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space P (H ) of a complex →Hilbert space is the set of equivalence classes of vectors v in H, with v 6= 0, for the relation given by v w when v = w

with a scalar, that is, a complex number (which must therefore be non-zero). Here the equivalence classes for are also called projective rays.

This is the usual construction of projective space, applied to a Hilbert space. The physical significance of the projective Hilbert space is that in quantum theory, the wave functions ψ and ψ represent the same physical state, for any 6= 0. There is not a unique normalized wavefunction in a given ray, since we can multiply by with absolute value 1. This freedom means that projective representations enter quantum theory. The same construction can be applied also to real Hilbert spaces.

In the case H is finite-dimensional, that is, H = Hn , the set of projective rays may be treated just as any other projective space; it is a homogeneous space for

Projective Hilbert space

404 a unitary group or orthogonal group, in the complex and real cases respectively. For the finite-dimensional complex Hilbert space, one writes

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P (Hn ) = CP n−1 so that, for example, the two-dimensional projective Hilbert space (the space describing one qubit) is the complex projective line CP 1 . This is known as the Bloch sphere, which treats the subject in greater detail. Complex projective Hilbert space may be given a natural metric, the FubiniStudy metric. The product of two projective Hilbert space is given by the Segre mapping.

Source: http://en.wikipedia.org/wiki/Projective_Hilbert_space Principal Authors: Charles Matthews, Asbestos, Linas

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Pure gauge

In physics, pure gauge is the set of field configurations obtained by a gauge transform on the null field configuration. So it is a particular "gauge orbit" in the field configuration’s space. In the abelian case, where Aµ (x) → A0µ (x) = Aµ (x)+∂µ f (x), the pure gauge is the set of field configurations A0µ (x) = ∂µ f (x) for all f (x).

Source: http://en.wikipedia.org/wiki/Pure_gauge

Principal Authors: LeeHunter, Michael Hardy, Sn0wflake, Oleg Alexandrov, Jag123

Quantum

In physics, a quantum refers to an indivisible and perhaps elementary entity. For instance, a "light quantum", being a unit of light (that is, a photon). In combinations like "quantum mechanics", "quantum optics", etc., it distinguishes a more specialized field of study. The word comes from the Latin "quantus", for "how much". Behind this, one finds the fundamental notion that a physical property may be "quantized", referred to as "quantization". This means that the magnitude can take on only certain numerical values, rather than any value, at least within a Quantum

405 range. For example, the energy of an electron bound to an atom (at rest) is quantized. This accounts for the stability of atoms, and matter in general.

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An entirely new conceptual framework was developed around this idea, during the first half of the 1900s. Usually referred to as quantum "mechanics", it is regarded by virtually every professional physicist as the most fundamental framework we have for understanding and describing nature, for the very practical reason that it works. It is "in the nature of things", not a more or less arbitrary human preference.

Discovery of quantum theory

Quantum theory, the branch of physics based on quantization, began in 1900 when Max Planck published his theory explaining the emission spectrum of black bodies. In that paper Planck used the Natural system of units invented by him the previous year.

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The consequences of the differences between classical and quantum mechanics quickly became obvious. But it was not until 1926, by the work of Werner Heisenberg, Erwin Schrödinger, and others, that quantum mechanics became correctly formulated and understood mathematically. Despite tremendous experimental success, the philosophical interpretations of quantum theory are still widely debated.

Planck was reluctant to accept the new idea of quantization, as were many others. But, with no acceptable alternative, he continued to work with the idea, and found his efforts were well received. Eighteen years later, he called it, "a few weeks of the most strenuous work" of his life, when he accepted the Nobel Prize in Physics for his contributions. During those few weeks, he even had to discard much of his own theoretical work from the preceding years. Quantization turned out to be the only way to describe the new, and detailed experiments which were just then being performed. He did this practically overnight, openly reporting his change of mind to his scientific colleagues, in the October, November, and December meetings of the German Physical Society, in Berlin, where the black body work was being intensely discussed. In this way, careful experimentalists (including F. Paschen, O.R. Lummer, Ernst Pringsheim, Heinrich Rubens, and F. Kurlbaum), and a reluctant theorist, ushered in the greatest revolution science has ever seen.

The quantum black-body radiation formula When a body is heated, it emits radiant heat, a form of electromagnetic radiation in the infrared region of the EM spectrum. All of this was well understood at the time, and of considerable practical importance. When the body becomes

Quantum

406

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red-hot, the red wavelength parts start to become visible. This had been studied over the previous years, as the instruments were being developed. However, most of the heat radiation remains infrared, until the body becomes as hot as the surface of the Sun (about 6000 ◦ C, where most of the light is green in color). This was not achievable in the laboratory at that time. What is more, measuring specific infrared wavelengths was only then becoming feasible, due to newly developed experimental techniques. Until then, most of the electromagnetic spectrum was not measurable, and therefore blackbody emission had not been mapped out in detail. The quantum black-body radiation formula, being the very first piece of quantum mechanics, appeared Sunday evening October 7, 1900, in a so-called backof-the-envelope calculation by Planck. It was based on a report by Rubens (visiting with his wife) of the very latest experimental findings in the infrared. Later that evening, Planck sent the formula on a postcard, which Rubens had the following morning. A couple of days later, he could tell Planck that it worked perfectly. As it does to this day.

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At first, it was just a fit to the data. Only weeks later did it turn out to enforce quantization.

That the latter became possible involved a certain amount of luck (or skill, even though Planck himself called it "a fortuitous guess at an interpolation formula"). It only had that drastic "side effect" because the formula happened to become fundamentally correct, in regard to the as yet non-existent quantum theory. And normally, that much is not at all expected. The skill lay in simplifying the mathematics, so that this could happen. And here Planck used hard won experience from the previous years. Briefly stated, he had two mathematical expressions:

• •

(i) from the previous work on the red parts of the spectrum, he had x; (ii) now, from the new infrared data, he got x 2.

Combining these as x(a+x), he still has x, approximately, when x is much smaller than a ( the red end of the spectrum). But now also x 2, again approximately, when x is much larger than a (in the infrared). The luck part is that, this procedure turned out to actually give something completely right, far beyond what could reasonably be expected. The formula for the energy E, in a single mode of radiation at frequency f, and temperature T, can be written E=

hf

hf e kT

−1

This is (essentially) what is being compared with the experimental measurements. There are two parameters to determine from the data, written in the Quantum

407

FT

present form by the symbols used today: h is the new Planck’s constant, and k is Boltzmann’s constant. Both have now become fundamental in physics, but that was by no means the case at the time. The "elementary quantum of energy" is hf. But such a unit does not normally exist, and is not required for quantization.

The birthday of quantum mechanics

From the experiments, Planck deduced the numerical values of h and k. Thus he could report, in the German Physical Society meeting on December 14, 1900, where quantization (of energy) was revealed for the first time, values of the Avogadro-Loschmidt number, the number of real molecules in a mole, and the unit of electrical charge, which were more accurate than those known until then. This event has been referred to as "the birthday of quantum mechanics".

Quantization in antiquity

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In a sense, it can be said that the quantization idea is very old. A string under tension, and fixed at both ends, will vibrate at certain quantized frequencies, corresponding to various standing waves. This, of course, is the basis of music. The basic idea was regarded as essential by the Pythagoreans, who are reported to have held numbers in high esteem.

It is a curious fact that the famous formula, named after Pythagoras, for the side lengths of a right triangle, today serves as a cornerstone of quantum mechanics as well. The very existence of atoms and molecules can be ascribed to various forms of quantization contrary to notions of matter as some form of continuous medium. This was also understood already in antiquity, particularly by Leucippus and Democritus, although not generally appreciated, even by physicists, until the late 19th- and early 20th- centuries, shortly before the invention of quantum mechanics. It should be mentioned, though, that later works within the Epicurean school of thought played a significant role in forming the physics and chemistry of the Renaissance period in Europe. In particular the famous tutorial poem "De rerum natura" by the Roman author Titus Lucretius Carus. Ancient India had a very highly developed atomic doctrine in the school of Vaisheshika associated with the sage Kanada.

Quantum

408

References •

See also →Quantum mechanics →Quantum state Quantum number Quantum cryptography Quantum electronics Quantum computer Quantum immortality Magnetic flux quantum Quantization →Subatomic particle →Elementary particle

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• • • • • • • • • • •

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J. Mehra and H. Rechenberg, The Historical Development of Quantum Theory, Vol.1, Part 1, Springer-Verlag New York Inc., New York 1982. • Lucretius, "On the Nature of the Universe", transl. from the Latin by R.E. Latham, Penguin Books Ltd., Harmondsworth 1951. There are, of course, many translations, and the translation’s title varies. Some put emphasis on how things work, others on what things are found in nature. • M. Planck, A Survey of Physical Theory, transl. by R. Jones and D.H. Williams, Methuen & Co., Ltd., London 1925 (Dover editions 1960 and 1993) including the Nobel lecture.

Source: http://en.wikipedia.org/wiki/Quantum Principal Authors:

Dennis Estenson II, Finn-Zoltan, Deathphoenix, Bensaccount, Stevertigo,

Freakofnurture, Scorpionman, Truthflux, Salvatore Ingala, Ashmoo

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409

Quantum 1/f noise

Other noise data sets

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Quantum 1/f noise is an intrinsic part of quantum mechanics. It comes from scattering of different particles of one another in solid state physics. Quantum 1/f noise is a source of Chaos in such systems.

1/f noise has also recently been discovered in higher ordered self constructing functions, as well as complex systems, both biological, chemical, and physical.

The theory

The basic derivation of Quantum 1/f was made by Peter Handel from the University of Missouri - St. Louis, and published in Physical Review Letters A, in August 1980.

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For more on Quantum 1/f, see: P.H. Handel: "1/f Macroscopic Quantum Fluctuations of Electric Currents Due to Bremsstrahlung with Infrared Radiative Corrections", Zeitschrift fuer Naturforschung 30a, p.1201 (1975)

See also • • • • • • • • • • • • •

shot noise 1/f noise white noise Johnson-Nyquist noise signal-to-noise ratio noise level noise power noise-equivalent power phase noise thermal noise list of noise topics audio system measurements Colors of noise

Source: http://en.wikipedia.org/wiki/Quantum_1/f_noise

Principal Authors: Gordon Stangler, Charles Matthews, Wendell, Ardric47, Conscious

Quantum 1/f noise

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Quantum acoustics

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In physics, quantum acoustics is the study of sound under conditions such that quantum mechanical effects are germane. For most applications, classical mechanics are sufficient to accurately describe the physics of sound. However very high frequency sounds, or sounds made at very low temperatures may be subject to quantum effects. A symposium on quantum acoustics is held in Poland each year

See also •

Superfluid

References •

216

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Quantum acoustics by Humphrey J. Maris in the McGraw-Hill Encyclopedia of Science & Technology Online 217 • Handbook of Acoustics by Malcolm Crocker has a chapter on quantum acoustics.

Source: http://en.wikipedia.org/wiki/Quantum_acoustics

Quantum biology

Quantum biology is the science of studying biological processes in terms of quantum mechanics. In exploring quantum mechanical explanations for biological phenomena, the nascent science of quantum biology represents one of the first efforts to apply quantum theory to systems much more macroscopic than the atomic or subatomic realms generally described by quantum theory. The following biological phenomena have been described in terms of quantum processes (although, to the extent that quantum theory is correct, all macroscopic phenenoma would be the result of quantum processes):

216 http://gnom.matfiz.polsl.gliwice.pl/afik/2005/index.html 217 http://www.accessscience.com/Encyclopedia/5/56/Est_562350_frameset.html?doi

Quantum biology

411 •

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the absorbance of frequency-specific radiation (i.e., photosynthesis and vision); • the conversion of chemical energy into motion; • magnetoreception in animals. Quantum biological research is extremely limited by computer processing power; the analytical power required to model quantum effects increases exponentially with the number of particles involved.

References • •

nanoword.net 218 Theoretical and Computational Biophysics Group, University of Illinois at Urbana-Champaign 219

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Source: http://en.wikipedia.org/wiki/Quantum_biology

Quantum chaos

Quantum chaos is an interdisciplinary branch of physics, arising from so-called semi-classical models. Classical mechanics has historically been one of the fundamental theories of physics, and is complete in the sense that all its axioms are mutually consistent and not in need of further incremental refinement. However, many of the most difficult unsolved problems in contemporary physics and applied mathematics in fact originate in classical mechanics, particularly in the field of deterministic chaos. Laws of classical mechanics govern the macroscopic world of everyday experience. An important question of quantum mechanics is how to obtain the laws of classical mechanics as limiting cases of the more fundamental laws governing the microscopic constituents of matter. The correspondence principle is an expression of this goal, which strongly influenced the early development of quantum mechanical theories and their applications. However, the classical limit of a quantum description may lead to a mechanical system with chaotic dynamics.

218 http://www.nanoword.net/library/weekly/aa062500a.htm 219 http://www.ks.uiuc.edu/Research/quantum_biology/

Quantum chaos

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During the first half of the twentieth century, chaotic behavior in mechanics was recognized (in celestial mechanics), but not well-understood. The foundations of modern quantum mechanics were laid in that period, essentially leaving aside the issue of the quantum-classical correspondence in systems whose classical limit exhibits chaos. This question defines the field of quantum chaos, which has emerged in the second half of the twentieth century, aided to a large extent by renewed interest in classical nonlinear dynamics (chaos theory), and by quantum experiments bordering on the macroscopic size regime where laws of classical mechanics are expected to emerge. This transition regime between classical and quantum systems is also called semiclassical physics.

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Similar questions arise in many different branches of physics, ranging from nuclear to atomic, molecular and solid-state physics, and even to acoustics, microwaves and optics. This is what makes quantum chaos an interdisciplinary field, unified by wave phenomena that can be interpreted as fingerprints of classical chaos. Such phenomena can be identified in spectroscopy by analyzing the statistical distribution of spectral lines. Other phenomena show up in the time evolution of a quantum system, or in its response to various types of external forces. In some contexts, such as acoustics or microwaves, wave patterns are directly observable and exhibit irregular amplitude distributions.

Important observations often associated with classically chaotic quantum systems are level repulsion in the spectrum, dynamical localization in the time evolution (e.g. ionization rates of atoms), and enhanced stationary wave intensities in regions of space where classical dynamics exhibits only unstable trajectories (wave function scarring). An alternative name for quantum chaos, proposed by Sir Michael Berry, is quantum chaology.

History

Important methods applied in the theoretical study of quantum chaos include random-matrix theory (significant contributions by Oriol Bohigas, see also American Scientist 220) and periodic-orbit theory (pioneered by Martin Gutzwiller).

220 http://www.americanscientist.org/template/AssetDetail/assetid/21879/page/1;jsessionid=aaa-ZYP5NrRxh8

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References •

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A. Einstein (1917). "Zum Quantensatz von Sommerfeld und Epstein". Verhandlungen der Deutschen Physikalischen Gesellschaft 19: 82-92. Reprinted in The Collected Papers of Albert Einstein, A. Engel translator, (1997) Princeton University Press, Princeton. 6 p.434. (Provides an elegant reformulation of the Bohr-Sommerfeld quantization conditions, as well as an important insight into the quantization of non-integrable (chaotic) dynamical systems.) • Joeseph B. Keller (1958). "". Annals of Physics (NY) 4: 180. (An independent rediscovery of the A. Einstein quantization conditions.) • Joeseph B. Keller (1960). "". Annals of Physics (NY) 9: 24. • Martin C. Gutzwiller (1971). "". Journal of Mathematical Physics 12: 343. • Martin C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (1990) Springer-Verlag, New York ISBN=0-387-97173-4.

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Source: http://en.wikipedia.org/wiki/Quantum_chaos

Principal Authors: Linas, ChicXulub, Rmrfstar, Daniel tzvi, Neilc

Quantum Critical Point

The Quantum critical point is the lowest temperature point (1-3 degrees Kelvin) at which a change of state-of-existence occurs. Experimentally, this has been demonstrated with the Han purple pigment. 221 When exposed to both super-low temperatures and very high magnetic fields (above 23 Tesla), the Han Purple pigment actually loses a dimension, transforming from 3D to 2D.

References

Source: http://en.wikipedia.org/wiki/Quantum_Critical_Point

221

National high magnetic field laboratory, press release May 31, 2006(http://www.magnet.fsu.edu /news/pressreleases/2006may31.html) (accessed June 15, 2005)

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Quantum entanglement

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Quantum entanglement is a quantum mechanical phenomenon in which the quantum states of two or more objects have to be described with reference to each other, even though the individual objects may be spatially separated. This leads to correlations between observable physical properties of the systems. For example, it is possible to prepare two particles in a single quantum state such that when one is observed to be spin-up, the other one will always be observed to be spin-down and vice versa, this despite the fact that it is impossible to predict, according to quantum mechanics, which set of measurements will be observed. As a result, measurements performed on one system seem to be instantaneously influencing other systems entangled with it. Quantum entanglement does not enable the transmission of classical information faster than the speed of light (see discussion in next section below).

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Quantum entanglement is closely concerned with the emerging technologies of quantum computing and quantum cryptography, and has been used to experimentally realize quantum teleportation. At the same time, it prompts some of the more philosophically oriented discussions concerning quantum theory. The correlations predicted by quantum mechanics, and observed in experiment, reject the principle of local realism, which is that information about the state of a system should only be mediated by interactions in its immediate surroundings. Different views of what is actually occurring in the process of quantum entanglement can be related to different interpretations of quantum mechanics.

Background

Entanglement is one of the properties of quantum mechanics which caused Einstein and others to dislike the theory. In 1935, Einstein, Podolsky, and Rosen formulated the EPR paradox, a quantum-mechanical thought experiment with a highly counterintuitive and apparently nonlocal outcome. Einstein famously derided entanglement as "spooky action at a distance." On the other hand, quantum mechanics has been highly successful in producing correct experimental predictions, and the strong correlations associated with the phenomenon of quantum entanglement have in fact been observed. One apparent way to explain quantum entanglement is an approach known as "hidden variable theory", in which unknown deterministic microscopic parameters would cause the correlations. However, in 1964 Bell derived an upper limit, known as Bell’s inequality, on the strength of correlations for any

Quantum entanglement

415

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theory obeying "local realism" (see principle of locality). Quantum entanglement can lead to stronger correlations that violate this limit, so that quantum entanglement is experimentally distinguishable from a broad class of local hidden-variable theories. Results of subsequent experiments have overwhelmingly supported quantum mechanics. It is known that there are a number of loopholes in these experiments. High efficiency and high visibility experiments are now in progress which should accept or reject those loopholes. For more information, see the article on Bell test experiments. Observations on entangled states naively appear to conflict with the property of Einsteinian relativity that information cannot be transferred faster than the speed of light. Although two entangled systems appear to interact across large spatial separations, no useful information can be transmitted in this way, so causality cannot be violated through entanglement. This occurs for two subtle reasons: (i) quantum mechanical measurements yield probabilistic results, and (ii) the no cloning theorem forbids the statistical inspection of entangled quantum states.

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Although no information can be transmitted through entanglement alone, it is possible to transmit information using a set of entangled states used in conjunction with a classical information channel. This process is known as quantum teleportation. Despite its name, quantum teleportation cannot be used to transmit information faster than light, because a classical information channel is involved.

Pure States

The following discussion builds on the theoretical framework developed in the articles bra-ket notation and mathematical formulation of quantum mechanics. Consider two noninteracting systems A and B, with respective →Hilbert spaces HA and HB . The Hilbert space of the composite system is the tensor product HA ⊗ HB

If the first system is in state |ψiA and the second in state |φiB , the state of the composite system is |ψiA ⊗ |φiB ,

which is often also written as |ψiA |φiB .

States of the composite system which can be represented in this form are called separable states, or product states. Quantum entanglement

416

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Pick observables (and corresponding Hermitian operators) ΩA acting on HA , and ΩB acting on HB . According to the spectral theorem, we can find a basis {|iiA } for HA composed of eigenvectors of ΩA , and a basis {|jiB } for HB composed of eigenvectors of ΩB . We can then write the above pure state as P  P ( i ai |iiA ) j bj |jiB ,

for some choice of complex coefficients ai and bj . This is not the most general state of HA ⊗ HB , which has the form P i,j cij |iiA ⊗ |jiB . If such a state is not separable, it is known as an entangled state.

For example, given two basis vectors {|0iA , |1iA } of HA and two basis vectors {|0iB , |1iB } of HB , the following is an entangled state:   √1 |0iA ⊗ |1iB − |1iA ⊗ |0iB . 2

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If the composite system is in this state, it is impossible to attribute to either system A or system B a definite pure state. Instead, their states are superposed with one another. In this sense, the systems are "entangled".

Now suppose Alice is an observer for system A, and Bob is an observer for system B. If Alice performs the measurement ΩA , there are two possible outcomes, occurring with equal probability: • •

Alice measures 0, and the state of the system collapses to |0iA |1iB Alice measures 1, and the state of the system collapses to |1iA |0iB .

If the former occurs, any subsequent measurement of ΩB performed by Bob always returns 1. If the latter occurs, Bob’s measurement always returns 0. Thus, system B has been altered by Alice performing her measurement on system A., even if the systems A and B are spatially separated. This is the foundation of the EPR paradox. The outcome of Alice’s measurement is random. Alice cannot decide which state to collapse the composite system into, and therefore cannot transmit information to Bob by acting on her system. (There is a possible loophole: if Bob could make multiple duplicate copies of the state he receives, he could obtain information by collecting statistics. This loophole is closed by the no cloning theorem, which forbids the creation of duplicate states.) Causality is thus preserved, as claimed above. In more formal mathematical settings, it is noted that the correct setting for pure states in quantum mechanics is projective Hilbert space endowed with Quantum entanglement

417

Ensembles

FT

the Fubini-Study metric. The product of two pure states is then given by the Segre embedding.

As mentioned above, a state of a quantum system is given by a unit vector in a Hilbert space. More generally, if one has a large number of copies of the same system, then the state of this ensemble is described by a density matrix, which is a positive matrix (or trace class, when the state space is infinite dimensional) and has trace 1. Again, by the spectral theorem, such a matrix takes the general form: P ρ = i wi |αi ihαi |,

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where the wi ’s sum up to 1 (in the infinite dimensional case, we would take the closure of such states in the trace norm). We can interpret ρ as representing an ensemble where wi is the proportion of the ensemble whose states are |αi i. When a mixed state has rank 1, it therefore describes a pure ensemble. When there is less than total information about the state of a quantum system we need density matrices to represent the state (see experiment discussed below). Following the definition in previous section, for a bipartite composite system, mixed states are just density matrices on HA ⊗ HB . Extending the definition of separability from the pure case, we say that a mixed state is separable if it can be written as P B ρ = i p i ρA i ⊗ ρi B ,where ρA i ’s and ρi ’s are they themselves states on the subsystems A and B respectively. In other words, a state is separable if it is probability distribution over uncorrelated states, or product states. We can assume without loss of genB erality that ρA i and ρi are pure ensembles. A state is then said to be entangled if it is not separable. In general, finding out whether or not a mixed state is entangled is considered difficult. Formally, it has been shown to be NP-hard. For the 2 × 2 and 2 × 3 cases, a necessary and sufficient criterion for separability is given by the famous PPT (Positive Partial Transpose) condition.

Experimentally, a mixed ensemble might be realized as follows. Consider a "black-box" apparatus that spits electrons towards an observer. The electrons’ Hilbert spaces are identical. The apparatus might produce electrons that are all in the same state; in this case, the electrons received by the observer are then a pure ensemble. However, the apparatus could produce electrons in different states. For example, it could produce two populations of electrons: one with state |z+i (spins aligned in the positive z direction), and the other with state Quantum entanglement

418 |y−i (spins aligned in the negative y direction.) Generally, there can be any number of populations, each corresponding to a different state. Therefore we now have a mixed ensemble.

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Reduced Density Matrices

Consider as above systems A and B each with a Hilbert space HA , HB . Let the state of the composite system be |Ψi ∈ HA ⊗ HB .

As indicated above, in general there is no way to associate a pure state to the component system A. However, it still is possible to associate a density matrix. Let ρT = |Ψi hΨ|.

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which is the projection operator onto this state. The state of A is the partial trace of ρT over the basis of system B: P ρA ≡ j hj|B (|ΨihΨ|) |jiB = TrB ρT .

ρA is sometimes called the reduced density matrix of ρ on subsystem A. Colloquially, we "trace out" system B to obtain the reduced density matrix on A. For example, the density matrix of A for the entangled state discussed above is   ρA = (1/2) |0iA h0|A + |1iA h1|A

This demonstrates that, as expected, the reduced density matrix for an entangled pure ensemble is a mixed ensemble. Also not surprisingly, the density matrix of A for the pure product state |ψiA ⊗ |φiB discussed above is ρA = |ψiA hψ|A .

Entropy

In this section we briefly discuss entropy of a mixed state and how it can be viewed as a measure of entanglement.

In classical information theory, to a probability distribution p1 , · · · , pn , one can associate the Shannon entropy: P H(p1 , · · · , pn ) = − i pi log pi , where the logarithm is taken in base 2.

Quantum entanglement

419 Since one can think of a mixed state ρ as a probability distribution over an ensemble, this leads naturally to the definition of the von Neumann entropy:

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S(ρ) = −Tr (ρ log ρ) , where the logarithm is again taken in base 2. In general, to calculate log ρ, one would use the Borel functional calculus. If ρ acts on a finite dimensional Hilbert space and has eigenvalues λ1 , · · · , λn , then we recover the Shannon entropy: P S(ρ) = −Tr (ρ log ρ) = i λi log λi .

Since an event of probability 0 should not contribute to the entropy, we adopt the convention that 0 log 0 = 0. This extends to the infinite dimensional case R as well: if ρ has spectral resolution ρ = λdPλ , then we assume the same convention when calculating R ρ log ρ = λ log λdPλ .

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Entropy provides one tool which can be used to quantify entanglement (although other entanglement measures exist). As in statistical mechanics, one can say that the more uncertainty (number of microstates) possessed by the system, the larger the entropy. For example, the entropy of any pure state is zero, which is unsurprising since there is no uncertainty about a system in a pure state. The entropy of any of the two subsystems of the entangled state discussed above is ln 2 (which can be shown to be the maximum entropy for 2 × 2 mixed states). If the overall system is pure, the entropy of one subsystem can be used to measure its degree of entanglement with the other subsystems. It turns out that, for pure states, the von Neumann entropy of reduced states is the unique measure of entanglement. On the other hand, uniqueness does not hold for mixed states. Physically speaking, this is because the uncertainty in the mixed state gives us entropy in itself, irrespective of whether or not the state is entangled.

As an aside, the information-theoretic definition is closely related to entropy in the sense of statistical mechanics (comparing the two definitions, we note that, in the present context, it is customary to set the Boltzmann constant k = 1). For example, by properties of the Borel functional calculus, we see that for any unitary operator U, S(ρ) = S(U ρU ∗ ).

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Indeed, without the above property, the von Neumann entropy would not be well-defined. In particular, U could be the time evolution operator of the system, i.e.   U (t) = exp −iHt ~

where H is the Hamiltonian of the system. This associates the reversibility of a process with its resulting entropy change, i.e. a process is reversible if and only if it leaves the entropy of the system invariant. This provides a connection between quantum information theory and thermodynamics.

Applications of entanglement

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Entanglement has many applications in quantum information theory. Mixed state entanglement can be viewed as a resource for quantum communication. With the aid of entanglement, otherwise impossible tasks may be achieved. Among the most well known such applications of entanglement are superdense coding and quantum state teleportation. Efforts to quantify this resource are often termed entanglement theory. See for example Entanglement Theory Tutorials 222. The Reeh-Schlieder theorem of quantum field theory is sometimes seen as the QFT analogue of quantum entanglement.

See also • • •

→Entanglement witness Fubini-Study metric →Separable states

•

→Squashed entanglement

References •

M. Horodecki, P. Horodecki, R. Horodecki, "Separability of Mixed States: Necessary and Sufficient Conditions", Physics Letters A 210, 1996.

•

L. Gurvits, "Classical deterministic complexity of Edmonds’ Problem and quantum entanglement", Proceedings of the thirty-fifth annual ACM symposium on Theory of computing, 2003.

222 http://www.imperial.ac.uk/quantuminformation

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External links •

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An Interview With Brian Clegg, Author of "The God Effect : Quantum Entanglement, Science’s Strangest Phenomenon" 223 California Literary Review • Multiple entanglement and quantum repeating 224 • How to entangle photons experimentally 225

Source: http://en.wikipedia.org/wiki/Quantum_entanglement

Principal Authors: Mct mht, CYD, Roadrunner, CSTAR, Caroline Thompson, Cortonin, Charles Matthews

Quantum field theory

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Quantum field theory (QFT) is the application of quantum mechanics to fields. It provides a theoretical framework, widely used in particle physics and condensed matter physics, in which to formulate consistent quantum theories of many-particle systems, especially in situations where particles may be created and destroyed. Non-relativistic quantum field theories are needed in condensed matter physics— for example in the BCS theory of superconductivity. Relativistic quantum field theories are indispensable in particle physics (see the standard model), although they are known to arise as effective field theories in condensed matter physics.

Origin

Quantum field theory originated in the problem of computing the energy radiated by an atom when it dropped from one quantum state to another of lower energy. This problem was first examined by Max Born and Pascual Jordan in 1925. In 1926, Max Born, Werner Heisenberg and Pascual Jordan wrote down the quantum theory of the electromagnetic field neglecting polarization and sources to obtain what would today be called a free field theory. In order to quantize this theory, they used the canonical quantization procedure. In 1927, Paul Dirac gave the first consistent treatment of this problem. Quantum field theory followed unavoidably from a quantum treatment of the only known classical field, ie, electromagnetism. The theory was required by the need to

223 http://calitreview.com/Interviews/clegg_8029.htm 224 http://www.physorg.com/news63037231.html

225 http://physicsweb.org/articles/world/11/3/9/1/world%2D11%2D3%2D9%2D3

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422 treat a situation where the number of particles changes. Here, one atom in the initial state becomes an atom and a photon in the final state.

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It was obvious from the beginning that the quantum treatment of the electromagnetic field required a proper treatment of relativity. Jordan and Wolfgang Pauli showed in 1928 that commutators of the field were actually Lorentz invariant. By 1933, Niels Bohr and Leon Rosenfeld had related these commutation relations to a limitation on the ability to measure fields at space-like separation. The development of the Dirac equation and the hole theory drove quantum field theory to explain these using the ideas of causality in relativity, work that was completed by Wendell Furry and Robert Oppenheimer using methods developed for this purpose by Vladimir Fock. This need to put together relativity and quantum mechanics was a second motivation which drove the development of quantum field theory. This thread was crucial to the eventual development of particle physics and the modern (partially) unified theory of forces called the standard model.

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In 1927 Jordan tried to extend the canonical quantization of fields to the wave function which appeared in the quantum mechanics of particles, giving rise to the equivalent name second quantization for this procedure. In 1928 Jordan and Eugene Wigner found that the Pauli exclusion principle demanded that the electron field be expanded using anti-commuting creation and annihilation operators. This was the third thread in the development of quantum field theory— the need to handle the statistics of multi-particle systems consistently and with ease. This thread of development was incorporated into many-body theory, and strongly influenced condensed matter physics and nuclear physics.

What QFT is

Just as quantum mechanics deals with operators acting upon a (separable) →Hilbert space, QFT also deals with operators acting upon a →Hilbert space. However, in the case of QFT, the operators are generated by what is known as operator-valued fields, that is, operators which are parametrized by a spacetime point. Intuitively, this means that operators can be localized. This definition applies even to the cases of theories which aren’t quantizations, and as such, is pretty general. This is sometimes stated as "position is an operator in QM but is a parameter in QFT" but this statement, while accurate, can be very misleading. QM deals with particles and one of the properties of a particle is its position as a function of time and in QM, this becomes the position operator as a function of time (it’s constant in the Schrödinger picture and varying in the Heisenberg picture). QFT, on the other hand, deals with fields on a fundamental level and particles only emerge as localized excitations (aka quanta aka quasiparticles) Quantum field theory

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ˆ(t) and p ˆ (t), x

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of the ground state (aka the vacuum) and it’s precisely these quantum fields which correspond to the operator valued functions. Put more simply, instead of looking at the operators generated by

we now look at operators generated by ˆ t) φ(x,

And just as in QM, we may work in the →Schrödinger picture, the →Heisenberg picture or the interaction picture (in the context of perturbation theory). Only the Heisenberg picture is manifestly Lorentz covariant.

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The energy is given by the Hamiltonian operator, which can be generated from the quantum fields, and corresponds to the generator of infinitesimal time translations. (the condition that the generator of infinitesimal time translations can be generated by the quantum fields rules out many unphysical theories, which is a good thing) We further assume that this Hamiltonian is bounded from below and has a lowest energy eigenstate (this rules out theories which are unstable and have no stable solutions, which is also a good thing), which may or may not be degenerate. (although there are physical QFTs which have a lower bound to the Hamiltonian but don’t have a lowest energy eigenstate, like N=1 super QCD theories with too few quarks...) This lowest energy eigenstate is called the vacuum in particle physics and the ground state in condensed matter physics. (QFT appears in the continuum limit of condensed matter systems) This simple explanation of what QFT really is, is often obscured in treatments which jump straight to the path integral approach, which is a good computational technique but often obscures the underlying ideas. QFT most definitely isn’t the same thing as classical field theory or classical field theory with some "minor" quantum corrections, which is a mistake many high energy physicists are prone to making at times, especially when working in the semiclassical approximation.

Technical statement

Quantum field theory corrects several limitations of ordinary quantum mechanics, which we will briefly discuss now. The →Schrödinger equation, in its most commonly encountered form, is h 2 i |p| ∂ 2m + V (r) |ψ(t)i = i~ ∂t |ψ(t)i

Quantum field theory

424 where |ψi denotes the quantum state (notation) of a particle with mass m, in the presence of a potential V .

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The first problem occurs when we seek to extend the equation to large numbers of particles. As described in the article on identical particles, quantum mechanical particles of the same species are indistinguishable, in the sense that the state of the entire system must be symmetric (bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged. These multi-particle states are extremely complicated to write. For example, the general quantum state of a system of N bosons is written as rQ Nj ! P j |φ1 · · · φN i = p∈Sn |φp(1) i · · · |φp(N ) i N!

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where |φi i are the single-particle states, Nj is the number of particles occupying state j, and the sum is taken over all possible permutations p acting on N elements. In general, this is a sum of N ! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. Large numbers of particles are needed in condensed matter physics where typically the number of particles is on the order of Avogadro’s number, approximately 10 23. The second problem arises when trying to reconcile the Schrödinger equation with special relativity. It is possible to modify the Schrödinger equation to include the rest energy of a particle, resulting in the →Klein-Gordon equation or the Dirac equation. However, these equations have many unsatisfactory qualities; for instance, they possess energy eigenvalues which extend to –∞, so that there seems to be no easy definition of a ground state. Such inconsistencies occur because these equations neglect the possibility of dynamically creating or destroying particles, which is a crucial aspect of relativity. Einstein’s famous mass-energy relation predicts that sufficiently massive particles can decay into several lighter particles, and sufficiently energetic particles can combine to form massive particles. For example, an electron and a positron can annihilate each other to create photons. Such processes must be accounted for in a truly relativistic quantum theory. This problem brings to the fore the notion that a consistent relativistic quantum theory, even of a single particle, must be a many particle theory.

Quantizing a classical field theory Canonical quantization

Quantum field theory solves these problems by consistently quantizing a field. By interpreting the physical observables of the field appropriately, one can create a (rather successful) theory of many particles. Here is how it is: Quantum field theory

425 1. Each normal mode oscillation of the field is interpreted as a particle with frequency f.

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2. The quantum number n of each normal mode (which can be thought of as a harmonic oscillator) is interpreted as the number of particles. The energy associated with the mode of excitation is therefore = (n + 1/2)~ω which directly follows from the energy eigenvalues of a one dimensional harmonic oscillator in quantum mechanics. With some thought, one may similarly associate momenta and position of particles with observables of the field.

Having cleared up the correspondence between fields and particles (which is different from non-relativistic QM), we can proceed to define how a quantum field behaves. Two caveats should be made before proceeding further: •

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Each of these "particles" obeys the usual uncertainty principle of quantum mechanics. The "field" is an operator defined at each point of spacetime. Quantum field theory is not a wildly new theory. Classical field theory is the same as classical mechanics of an infinite number of dynamical quantities (say, tiny elements of rubber on a rubber sheet). Quantum field theory is the quantum mechanics of this infinite system.

The first method used to quantize field theory was the method now called canonical quantization (earlier known as second quantization). This method uses a Hamiltonian formulation of the classical problem. The later technique of Feynman path integrals uses a Lagrangian formulation. Many more methods are now in use; for an overview see the article on quantization.

Canonical quantization for bosons

Suppose we have a system of N bosons which can occupy mutually orthogonal single-particle states |φ1 i, |φ2 i, |φ3 i, and so on. The usual method of writing a multi-particle state is to assign a state to each particle and then impose exchange symmetry. As we have seen, the resulting wavefunction is an unwieldy sum of N ! terms. In contrast, in the second quantized approach we will simply list the number of particles in each of the single-particle states, with the understanding that the multi-particle wavefunction is symmetric. To be specific, suppose that N = 3, with one particle in state |φ1 i and two in state|φ2 i. The normal way of writing the wavefunction is √1 3

[|φ1 i|φ2 i|φ2 i + |φ2 i|φ1 i|φ2 i + |φ2 i|φ2 i|φ1 i]

In second quantized form, we write this as

Quantum field theory

426 |1, 2, 0, 0, 0, · · ·i

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which means "one particle in state 1, two particles in state 2, and zero particles in all the other states."

Though the difference is entirely notational, the latter form makes it easy for us to define creation and annihilation operators, which add and subtract particles from multi-particle states. These creation and annihilation operators are very similar to those defined for the quantum harmonic oscillator, which added and subtracted energy quanta. However, these operators literally create and annihilate particles with a given quantum state. The bosonic annihilation operator a2 and creation operator a†2 have the following effects: √ a2 |N1 , N2 , N3 , · · ·i = N2 | N1 , (N2 − 1), N3 , · · ·i a†2 |N1 , N2 , N3 , · · ·i =

√

N2 + 1 | N1 , (N2 + 1), N3 , · · ·i

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We may well ask whether these are operators in the usual quantum mechanical sense, i.e. linear operators acting on an abstract →Hilbert space. In fact, the answer is yes: they are operators acting on a kind of expanded Hilbert space, known as a →Fock space, composed of the space of a system with no particles (the so-called vacuum state), plus the space of a 1-particle system, plus the space of a 2-particle system, and so forth. Furthermore, the creation and annihilation operators are indeed Hermitian conjugates, which justifies the way we have written them. The bosonic creation and annihilation operators obey the commutation relation h h i i   ai , a†j = δij ai , aj = 0 , a†i , a†j = 0 ,

where δ stands for the Kronecker delta. These are precisely the relations obeyed by the "ladder operators" for an infinite set of independent quantum harmonic oscillators, one for each single-particle state. Adding or removing bosons from each state is therefore analogous to exciting or de-exciting a quantum of energy in a harmonic oscillator. The final step toward obtaining a quantum field theory is to re-write our original N -particle Hamiltonian in terms of creation and annihilation operators acting on a Fock space. For instance, the Hamiltonian of a field of free (noninteracting) bosons is P H = k Ek a†k ak

where Ek is the energy of the k-th single-particle energy eigenstate. Note that Quantum field theory

427 a†k ak | · · · , Nk , · · ·i = Nk | · · · , Nk , · · ·i.

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Canonical quantization for fermions It turns out that the creation and annihilation operators for fermions must be defined differently, in order to satisfy the Pauli exclusion principle. For fermions, the occupation numbers Ni can only take on the value 0 or 1, since particles cannot share quantum states. We then define the fermionic annihilation operators c and creation operators c† by cj |N1 , N2 , · · · , Nj = 0, · · ·i = 0

cj |N1 , N2 , · · · , Nj = 1, · · ·i = (−1)(N1 +···+Nj−1 ) |N1 , N2 , · · · , Nj = 0, · · ·i

c†j |N1 , N2 , · · · , Nj = 0, · · ·i = (−1)(N1 +···+Nj−1 ) |N1 , N2 , · · · , Nj = 1, · · ·i

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c†j |N1 , N2 , · · · , Nj = 1, · · ·i = 0

The fermionic creation and annihilation operators obey an anticommutation relation, n n o o  ci , cj = 0 , c†i , c†j = 0 , ci , c†j = δij

One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible for the particles to share single-particle states, in accordance with the exclusion principle.

Significance of creation and annihilation operators When we re-write a Hamiltonian using a Fock space and creation and annihilation operators, as in the previous example, the symbol N , which stands for the total number of particles, drops out. This means that the Hamiltonian is applicable to systems with any number of particles. Of course, in many common situations N is a physically important and perfectly well-defined quantity. For instance, if we are describing a gas of atoms sealed in a box, the number of atoms had better remain a constant at all times. This is certainly true for the above Hamiltonian. Viewing the Hamiltonian as the generator of time evolution, we see that whenever an annihilation operator ak destroys a particle during an infinitesimal time step, the creation operator a†k to the left of it instantly puts it back. Therefore, if we start with a state of N non-interacting particles then we will always have N particles at a later time.

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On the other hand, it is often useful to consider quantum states where the particle number is ill-defined, i.e. linear superpositions of vectors from the Fock space that possess different values of N . For instance, it may happen that our bosonic particles can be created or destroyed by interactions with a field of fermions. Denoting the fermionic creation and annihilation operators by c†k and ck , we could add a "potential energy" term to our Hamiltonian such as: P V = k,q Vq (aq + a†−q )c†k+q ck

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This describes processes in which a fermion in state k either absorbs or emits a boson, thereby being kicked into a different eigenstate k + q. In fact, this is the expression for the interaction between phonons and conduction electrons in a solid. The interaction between photons and electrons is treated in a similar way; it is a little more complicated, because the role of spin must be taken into account. One thing to notice here is that even if we start out with a fixed number of bosons, we will generally end up with a superposition of states with different numbers of bosons at later times. On the other hand, the number of fermions is conserved in this case. In condensed matter physics, states with ill-defined particle numbers are also very important for describing the various superfluids. Many of the defining characteristics of a superfluid arise from the notion that its quantum state is a superposition of states with different particle numbers.

Field operators

We can now define field operators that create or destroy a particle at a particular point in space. In particle physics, these are often more convenient to work with than the creation and annihilation operators, because they make it easier to formulate theories that satisfy the demands of relativity. Single-particle states are usually enumerated in terms of their momenta (as in the particle in a box problem.) We can construct field operators by applying the Fourier transform to the creation and annihilation operators for these states. For example, the bosonic field annihilation operator φ(r) is P φ(r) ≡ i eiki ·r ai

The bosonic field operators obey the commutation relation  †    [φ(r), φ(r0 )] = 0 , φ (r), φ† (r0 ) = 0 , φ(r), φ† (r0 ) = δ 3 (r − r0 ) where δ(x) stands for the Dirac delta function. As before, the fermionic relations are the same, with the commutators replaced by anticommutators.

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It should be emphasized that the field operator is not the same thing as a singleparticle wavefunction. The former is an operator acting on the Fock space, and the latter is just a scalar field. However, they are closely related, and are indeed commonly denoted with the same symbol. If we have a Hamiltonian with a space representation, say P ~2 P 2 H = − 2m i ∇i + i<j U (|ri − rj |)

where the indices i and j run over all particles, then the field theory Hamiltonian is R 3 R R ~2 H = − 2m d r φ(r)† ∇2 φ(r) + d3r d3r0 φ(r)† φ(r0 )† U (|r − r0 |)φ(r0 )φ(r) This looks remarkably like an expression for the expectation value of the energy, with φ playing the role of the wavefunction. This relationship between the field operators and wavefunctions makes it very easy to formulate field theories starting from space-projected Hamiltonians.

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Quantization of classical fields

So far, we have shown how one goes from an ordinary quantum theory to a quantum field theory. There are certain systems for which no ordinary quantum theory exists. These are the "classical" fields, such as the electromagnetic field. There is no such thing as a wavefunction for a single photon in classical electromagnetisim, so a quantum field theory must be formulated right from the start. The essential difference between an ordinary system of particles and the electromagnetic field is the number of dynamical degrees of freedom. For a system of N particles, there are 3N coordinate variables corresponding to the position of each particle, and 3N conjugate momentum variables. One formulates a classical Hamiltonian using these variables, and obtains a quantum theory by turning the coordinate and position variables into quantum operators, and postulating commutation relations between them such as   qi , pj = δij

For an electromagnetic field, the analogue of the coordinate variables are the values of the electrical potential φ(x) and the vector potential A(x) at every point x. This is an uncountable set of variables, because x is continuous. This prevents us from postulating the same commutation relation as before. The way out is to replace the Kronecker delta with a Dirac delta function. This ends up giving us a commutation relation exactly like the one for field operators! We therefore end up treating "fields" and "particles" in the same way, using Quantum field theory

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Path integral methods The axiomatic approach

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the apparatus of quantum field theory. Only by accident electrons were not regarded as de Broglie waves and photons governed by geometrical optics were not the dominant theory when QFT was developed.

There have been many attempts to put quantum field theory on a firm mathematical footing by formulating a set of axioms for it. These attempts fall into two broad classes.

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The first class of axioms (most notably the Wightman, Osterwalder-Schrader, and Haag-Kastler systems) tried to formalize the physicists’ notion of an "operator-valued field" within the context of functional analysis. These axioms enjoyed limited success. It was possible to prove that any QFT satisfying these axioms satisfied certain general theorems, such as the spin-statistics theorem and the PCT theorems. Unfortunately, it proved extraordinarily difficult to show that any realistic field theory (e.g. quantum chromodynamics) satisfied these axioms. Most of the theories which could be treated with these analytic axioms were physically trivial: restricted to low-dimensions and lacking in interesting dynamics. Constructive quantum field theory is the construction of theories which satisfy one of these sets of axioms. Important work was done in this area in the 1970s by Segal, Glimm, Jaffe and others. In the 1980s, a second wave of axioms were proposed. These axioms (associated most closely with Atiyah and Segal, and notably expanded upon by Witten, Borcherds, and Kontsevich) are more geometric in nature, and more closely resemble the path integrals of physics. They have not been exceptionally useful to physicists, as it is still extraordinarily difficult to show that any realistic QFTs satisfy these axioms, but have found many applications in mathematics, particularly in representation theory, algebraic topology, and geometry. Finding the proper axioms for quantum field theory is still an open and difficult problem in mathematics. In fact, one of the Clay Millennium Prizes offers $1,000,000 to anyone who proves the existence of a mass gap in Yang-Mills theory. It seems likely that we have not yet understood the underlying structures which permit the Feynman path integrals to exist.

Renormalization

Some of the problems and phenomena eventually addressed by renormalization actually appeared earlier in the classical electrodynamics of point particles in the 19th and early 20th century. The basic problem is that the observable properties of an interacting particle cannot be entirely separated from the field Quantum field theory

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Figure 39

Edward Witten

that mediates the interaction. The standard classical example is the energy of a charged particle. To cram a finite amount of charge into a single point requires an infinite amount of energy; this manifests itself as the infinite energy of the particle’s electric field. The energy density grows to infinity as one gets close to the charge. A single particle state in quantum field theory incorporates within it multiparticle states. This is most simply demonstrated by examining the evolution of a single particle state in the interaction picture—   |ψ(t)i = eiHI t |ψ(0)i = 1 + iHI t − 12 HI2 t2 − 3!i HI3 t3 + 4!1 HI4 t4 + · · · |ψ(0)i.

Taking the overlap with the initial state, one retains the even powers of H I. These terms are responsible for changing the number of particles during propagation, and are therefore quintessentially a product of quantum field theory. Corrections such as these are incorporated into wave-function renormalization

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Gauge theories

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and mass renormalization. Similar corrections to the interaction Hamiltonian, H I, include vertex renormalization, or, in modern language, effective field theory.

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A gauge theory is a theory which admits a symmetry with a local parameter. For example, in every quantum theory the global phase of the wave function is arbitrary and does not represent something physical, so the theory is invariant under a global change of phases (adding a constant to the phase of all wave functions, everywhere); this is a global symmetry. In quantum electrodynamics, the theory is also invariant under a local change of phase, that is - one may shift the phase of all wave functions so that in every point in space-time the shift is different. This is a local symmetry. However, in order for a well-defined derivative operator to exist, one must introduce a new field, the gauge field, which also transforms in order for the local change of variables (the phase in our example) not to effect the derivative. In quantum electrodynamics this gauge field is the electromagnetic field. The change of local change of variables is termed gauge transformation. In quantum field theory the excitations of fields represent particles. The particle associated with excitations of the gauge field is the gauge boson, which is the photon in the case of quantum electrodynamics. The degrees of freedom in quantum field theory are local fluctuations of the fields. The existence of a gauge symmetry reduces the number of degrees of freedom, simply because some fluctuations of the fields can be transformed to zero by gauge transformations, so they are equivalent to having no fluctuations at all, and they therefore have no physical meaning. Such fluctuations are usually called "non-physical degrees of freedom" or gauge artifacts; Usually some of them have a negative norm, making them inadequate for a consistent theory. Therefore, if a classical field theory has a gauge symmetry, then its quantized version (i.e. the corresponding quantum field theory) will have this symmetry as well. In other words, a gauge symmetry cannot have a quantum anomaly. If a gauge symmetry is anomalous (i.e. not kept in the quantum theory) then the theory is non-consistent: for example, in quantum electrodynamics, had there been a gauge anomaly, this would require the appearance of photons with longitudinal polarization and polarization in the time direction, the latter having a negative norm, rendering the theory inconsistent; another possibility would be for these photons to appear only in intermediate processes but not in the final products of any intercation, making the theory non unitary and again inconsistent (see optical theorem).

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In general, the gauge transformations of a theory consist several different transformations, which may not be commutative. These transformations are together described by a mathematical object known as a gauge group. Infinitesimal gauge transformations are the gauge group generators. Therefore the number of gauge bosons is the group rank (i.e. number of generators forming an orthogonal basis). All the fundamental interactions in nature are described by gauge theories. These are: •

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Quantum electrodynamics, whose gauge transformation is a local change of phase, so that the gauge group is U(1). The gauge boson is the photon. • Quantum chromodynamics, whose gauge group is SU(3). The gauge bosons are eight gluons. • The electroweak Theory, whose gauge group is U (1)×SU (2) (a direct product of U(1) and SU(2)). • Gravity, whose classical theory is general relativity, admits the equivalence principle which is a form of gauge symmetry.

Supersymmetry

Supersymmetry assumes that every fundamental fermion has a superpartner which is a boson and vice versa. It was introduced in order to solve the socalled Hierarchy Problem, that is, to explain why particles not protected by any symmetry (like the Higgs boson) do not receive radiative corrections to its mass driving it to the larger scales (GUT, Planck...). It was soon realized that supersymmetry has other interesting properties: its gauged version is an extension of general relativity (Supergravity), and it is a key ingredient for the consistency of string theory. The way supersymmetry protects the hierarchies is the following: since for every particle there is a superpartner with the same mass, any loop in a radiative correction is cancelled by the loop corresponding to its superpartner, rendering the theory UV finite. Since no superpartners have yet been observed, if supersymmetry exists it must be broken (through a so-called soft term, which breaks supersymmetry without ruinning its helpful features). The simplest models of this breaking require that the energy of the superpartners not be too high; in these cases, supersymmetry is expected to be observed by experiments at the Large Hadron Collider.

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Beyond local field theory

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History More details can be found in the article on the history of quantum field theory.

Quantum field theory was created by Dirac when he attempted to quantize the electromagnetic field in the late 1920s. The early development of the field involved Fock, Jordan, Pauli, Heisenberg, Bethe, Tomonaga, Schwinger, Feynman, and Dyson. This phase of development culminated with the construction of the theory of quantum electrodynamics in the 1950s.

Gauge theory was formulated and quantized, leading to the unification of forces embodied in the standard model of particle physics. This effort started in the 1950s with the work of Yang and Mills, was carried on by Martinus Veltman and a host of others during the 1960s and completed during the 1970s by the work of Gerard ’t Hooft, Frank Wilczek, David Gross and David Politzer.

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Parallel developments in the understanding of phase transitions in condensed matter physics led to the study of the renormalization group. This in turn led to the grand synthesis of theoretical physics which unified theories of particle and condensed matter physics through quantum field theory. This involved the work of Michael Fisher and Leo Kadanoff in the 1970s which led to the seminal reformulation of quantum field theory by Kenneth Wilson. The study of quantum field theory is alive and flourishing, as are applications of this method to many physical problems. It remains one of the most vital areas of theoretical physics today, providing a common language to many branches of physics.

See also • • • • • • •

List of quantum field theories Feynman path integral Quantum chromodynamics Quantum electrodynamics Schwinger-Dyson equation Relationship between string theory and quantum field theory Abraham-Lorentz force

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Suggested reading Wilczek, Frank ; Quantum Field Theory, Review of Modern Physics 71 (1999) S85-S95. Review article written by a master of Q.C.D., Nobel laureate 2003 226. Full text available at : hep-th/9803075 227

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Ryder, Lewis H. ; Quantum Field Theory (Cambridge University Press, 1985), [ISBN 0-521-33859-X] Highly readable textbook, certainly the best introduction to relativistic Q.F.T. for particle physics.

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Zee, Anthony ; Quantum Field Theory in a Nutshell, Princeton University Press (2003) [ISBN 0-691-01019-6]. • Peskin, M and Schroeder, D. ;An Introduction to Quantum Field Theory (Westview Press, 1995) [ISBN 0201503972]

Weinberg, Steven ; The Quantum Theory of Fields (3 volumes) Cambridge University Press (1995). A monumental treatise on Q.F.T. written by a leading expert, Nobel laureate 1979 228.

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Loudon, Rodney ; The Quantum Theory of Light (Oxford University Press, 1983), [ISBN 0198511558]

•

D.A. Bromley (2000). Gauge Theory of Weak Interactions. Springer. ISBN 3540676724.

•

Gordon L. Kane (1987). Modern Elementary Particle Physics. Perseus Books. ISBN 0201117495.

External links • •

Siegel, Warren ; Fields 229 (also available from arXiv:hep-th/9912205) ’t Hooft, Gerard ; The Conceptual Basis of Quantum Field Theory, Handbook of the Philosophy of Science, Elsevier (to be published). Review article written by a master of gauge theories, [http://nobelprize.org/physics/laureates/1999/thooft-autobio.html’’Nobel laureate 1999]. Full text available in 230.

226 http://nobelprize.org/physics/laureates/2004/wilczek-autobio.html 227 http://fr.arxiv.org/abs/hep-th/9803075 228 http://nobelprize.org/physics/laureates/1979/weinberg-lecture.html 229 http://insti.physics.sunysb.edu/%7Esiegel/errata.html 230 http://www.phys.uu.nl/~thooft/lectures/basisqft.pdf’’pdf’’

Quantum field theory

436 Srednicki, Mark ; Quantum Field Theory 231 Kuhlmann, Meinard ; Quantum Field Theory 232, Stanford Encylopedia of Philosophy

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• •

Source: http://en.wikipedia.org/wiki/Quantum_field_theory

Principal Authors: CYD, Bambaiah, Stupidmoron, Phys, Odddmonster, Lethe, Arnero, Charles Matthews, Itinerant1, AmarChandra

Quantum fluctuation

In quantum physics, a quantum fluctuation is the temporary change in the amount of energy in a point in space, arising from Werner Heisenberg’s uncertainty principle.

DR A

According to one formulation of the principle, energy and time can be related by the relation ∆E∆t ≈

h 2π

That means that conservation of energy can appear to be violated, but only for small times. This allows the creation of particle-antiparticle pairs of virtual particles. The effects of these particles are measurable, for example, in the effective charge of the electron, different from its "naked" charge. In the modern view, energy is always conserved, but the eigenstates of the Hamiltonian (energy observable) aren’t the same as (don’t commute with) the particle number operators. Quantum fluctuations may have been very important in the origin of the structure of the universe: according to the model of inflation the ones that existed when inflation began were amplified and formed the seed of all current observed structure.

Quantum fluctuations of a field

A reasonably clear distinction can be made between quantum fluctuations and thermal fluctuations of a quantum field (at least for a free field; for interacting

231 http://gabriel.physics.ucsb.edu/~mark/qft.html 232 http://plato.stanford.edu/entries/quantum-field-theory/

Quantum fluctuation

437

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fields, renormalization complicates matters a lot). For the quantized KleinGordon field, we can calculate the probability density that we would observe a configuration ϕt (x) at a time t in terms of its fourier transform ϕ˜t (k) to be h i R d3 k ∗ p ρ0 [ϕt ] = exp − ~1 (2π) ˜t (k) |k|2 + m2 ϕ˜t (k) . 3ϕ

In contrast, for the classical Klein-Gordon field at non-zero temperature, the Gibbs probability density that we would observe a configuration ϕt (x) at a time t is h i R d3 k ∗ 1 2 + m2 ) ϕ 1 ˜ (k) (|k| ˜ (k) . ρE [ϕt ] = exp [−H[ϕt ]/kT ] = exp − kT t 3ϕ 2 (2π) t

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The amplitude of quantum fluctuations is controlled by the amplitude of Planck’s constant ~, just as the amplitude of thermal fluctuations is controlled by kT . Note that the following three points are closely related: (1) Planck’s constant has units p of action instead of units of energy, (2) the quantum kernel is |k|2 + m2 instead of 21 (|k|2 + m2 ) (the quantum kernel is nonlocal from a classical heat kernel viewpoint, but it is local in the sense that it does not allow signals to be transmitted), (3) the quantum vacuum state is Lorentz invariant (although not manifestly in the above), whereas the classical thermal state is not (the classical dynamics is Lorentz invariant, but the Gibbs probability density is not a Lorentz invariant initial condition). We can construct a classical continuous random field that has the same probability density as the quantum vacuum state, so that the principal difference from quantum field theory is the measurement theory (measurement in quantum theory is different from measurement for a classical continuous random field, in that classical measurements are always mutually compatible — in quantum mechanical terms they always commute). Quantum effects that are consequences only of quantum fluctuations, not of subtleties of measurement incompatibility, can alternatively be modelled by classical continuous random fields.

Quantum annealing: A novel utilization of quantum fluctuations Quantum fluctuations are recently being used to anneal glassy systems (physical glass, or equivalently, hard combinatorial optimization problem with ragged energy/cost landscape) to their minimal/ground states. Thus it provides a general algorithmic scheme for classical/quantum computers.

Quantum fluctuation

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• • •

Casimir effect Virtual particle Quantum annealing

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See also

Source: http://en.wikipedia.org/wiki/Quantum_fluctuation

Principal Authors: Arnab das, AstroNomer, Eequor, Michael Hardy, RoboDick, Phys, Nowhither, Andre Engels

Quantum foam

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Quantum foam, also referred to as spacetime foam, is a concept in quantum mechanics, devised by John Wheeler in 1955. It is sometimes likened to the old concept of the ether/aether.

The foam is a qualitative description of the turbulence that the phenomenon creates at extremely small distances of the order of the Planck length. At such small scales of time and space the uncertainty principle allows particles and energy to briefly come into existence, and then annihilate, without violating conservation laws. As the scale of time and space being discussed shrinks, the energy of the virtual particles increases. At sufficiently small scale space is not smooth as would be expected from observations at larger scales.

Foaming through the universe

Quantum foam is theorized to create masses of virtual particles. They are particle-antiparticle pairs, and prior to their annihilation, exist for a short period of time, on the order of the Planck time. They are created randomly from photons; the higher the energy of the photon from which they are created, the longer the time they will exist prior to annihilation. These virtual particles make their existence known by the Casimir effect. It is thought that there are constant quantum fluctuations in "empty" space, even at the energetic homogeneity referred to as absolute zero. Due to this, quantum fluctuations are often described using the term "zero-point energy". The "foamy" spacetime would look like a complex turbulent storm-tossed sea. Some physicists theorize the formation of wormholes therein; speculation arising from this includes the possibility of hyperspatial links to other universes. As far as realistic phenomena are concerned, it’s thought that the hyperspatial Quantum foam

439 nature of the quantum foam may account for such diverse physical principles as inertia, propagation of light, and time flow.

FT

Reginald Cahill has developed a theory called Process physics, which describes space as a quantum foam system in which gravity is an inhomogeneous flow of the quantum foam into matter. According to this theory, the so-called spiral galaxy rotation-velocity anomaly may be explained without the need for dark matter. Various scientists have theorized that quantum foam is an incredibly powerful source of zero-point energy. It has been estimated that one cubic centimeter of space contains enough zero point energy to boil all the world’s oceans. However, estimates of this energy vary widely due to the huge disparity in the calculations of the quantum foam density, which vary more than 1:10 100. Physicist Michio Kaku thinks that this enormous uncertainty in the estimation of quantum-foam density would represent the largest disparity for any quantity in all of physics.

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See also • • • • • • •

Dirac sea Hawking radiation Hyperspace theory Planck time "Rolling ball" topology Vacuum energy Wormhole

References •

• •

John Archibald Wheeler with Kenneth Ford. Geons, Black Holes, and Quantum Foam. 1995. Reginald T. Cahill. Gravity as Quantum Foam In-Flow. June 2003. 233 Process Physics 234 Resource Index

Source: http://en.wikipedia.org/wiki/Quantum_foam

Principal Authors: GregorB, Peak, ErkDemon, Stevertigo, Platypus222

233 http://www.scieng.flinders.edu.au/cpes/people/cahill_r/HPS15.pdf 234 http://www.mountainman.com.au/process_physics/

Quantum foam

440

Quantum Hall effect

σ=ν

e2 h,

FT

The quantum Hall effect is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductance σ takes on the quantized values

where e is the elementary charge and h is Planck’s constant. In the "ordinary" quantum Hall effect, known as the integer quantum Hall effect, ν takes on integer values (ν = 1, 2, 3, etc.). There is another type of quantum Hall effect, known as the fractional quantum Hall effect, in which ν can occur as a vulgar fraction (ν = 2/7, 1/3, 2/5, 3/5, 5/2 etc.)

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The quantization of the Hall conductance has the important property of being incredibly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of e 2/h to nearly one part in a billion. This phenomenon, referred to as "exact quantization", has been shown to be a subtle manifestation of the principle of gauge invariance. It has allowed for the definition of a new practical standard for electrical resistance: the resistance unit h/e 2, roughly equal to 25 812.8 ohms, is referred to as the von Klitzing constant R K 235 (after Klaus von Klitzing, the discoverer of exact quantization) and since 1990, a fixed conventional value R K-90 236 is used in resistance calibrations worldwide. The quantum Hall effect also provides an extremely precise independent determination of the fine structure constant, a quantity of fundamental importance in quantum electrodynamics.

The integer quantization of the Hall conductance was originally predicted by Ando, Matsumoto, and Uemura in 1975, on the basis of an approximate calculation. Several workers subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs. It was only in 1980 that von Klitzing, working with samples developed by Michael Pepper and Gerhard Dorda, made the totally unexpected discovery that the Hall conductivity was exactly quantized. For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics. The link between exact quantization and gauge invariance was subsequently found by Robert Laughlin. The fractional effect is due to completely different physics, and was experimentally discovered in 1982 by Daniel Tsui and Horst Störmer, in experiments

235 http://physics.nist.gov/cgi-bin/cuu/Value?rk|search_for=RK 236 http://physics.nist.gov/cgi-bin/cuu/Value?rk90|search_for=RK

Quantum Hall effect

441

References • •

T. Ando, Y. Matsumoto, and Y. Uemura, J. Phys. Soc. Jpn. 39, 279 (1975) DOI: 10.1143/JPSJ.39.279 237 K. von Klitzing, G. Dorda, and M. Pepper, Phys. Rev. Lett. 45, 494 (1980) DOI: 10.1103/PhysRevLett.45.494 238 R.B. Laughlin, Phys. Rev. B. 23, 5632 (1981) DOI: 10.1103/PhysRevB.23.5632 239 D.C. Tsui, H.L. Stormer, and A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1982) DOI: 10.1103/PhysRevLett.48.1559 240 R.B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983) DOI: 10.1103/PhysRevLett.50.1395 241 R. de-Picciotto, M. Reznikov, M. Heiblum, V. Umansky, G. Bunin and D. Mahalu, Nature 389, 162-164 (1997)

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•

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performed on gallium arsenide heterostructures developed by Arthur Gossard. The effect was explained by Robert B. Laughlin in 1983, using a novel quantum liquid phase that accounts for the effects of interactions between electrons. Tsui, Störmer, and Laughlin were awarded the 1998 Nobel Prize for their work. Although it was generally assumed that the discrete resistivity jumps found in the Tsui experiment were due to the presence of fractional charges, it was not until 1997 that R. de-Picciotto, et. al., indirectly observed fractional charges through measurements of quantum shot noise. The fractional quantum hall effect continues to be influential in theories about topological order.

•

•

•

Source: http://en.wikipedia.org/wiki/Quantum_Hall_effect

Principal Authors: CYD, Michael Hardy, Jaraalbe, Shaddack, Glenn

237 http://dx.doi.org/10.1143/JPSJ.39.279 238 http://dx.doi.org/10.1103/PhysRevLett.45.494

239 http://dx.doi.org/10.1103/PhysRevB.23.5632 240 http://dx.doi.org/10.1103/PhysRevLett.48.1559 241 http://dx.doi.org/10.1103/PhysRevLett.50.1395

Quantum Hall effect

442

Quantum harmonic oscillator

FT

The quantum harmonic oscillator is the quantum mechanical analogue of the classical harmonic oscillator. It is one of the most important model systems in quantum mechanics because, as in classical mechanics, a wide variety of physical situations can be reduced to it either exactly or approximately. In particular, a system near an equilibrium configuration can often be described in terms of one or more harmonic oscillators. Furthermore, it is one of the few quantum mechanical systems for which a simple exact solution is known. The following discussion of the quantum harmonic oscillator relies on the article mathematical formulation of quantum mechanics.

One-dimensional harmonic oscillator Diatomic molecules

DR A

In diatomic molecules, the natural frequency can be found by: q ω = mkr

242

where

ω = 2πf is the angular frequency, k is the bond force constant, and mr is the reduced mass.

Hamiltonian and energy eigenstates

In the one-dimensional harmonic oscillator problem, a particle of mass m is subject to a potential V (x) = (1/2)mω 2 x 2. The Hamiltonian of the particle is: H=

p2 2m

+ 21 mω 2 x2

where x is the position operator, and p is the momentum operator (p = ∂ −i~ ∂x ). The first term represents the kinetic energy of the particle, and the second term represents the potential energy in which it resides. In order to

242 http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html

Quantum harmonic oscillator

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443

Figure 40 Wavefunction representations for the first six bound eigenstates, n = 0 to 5. The horizontal axis shows the position x. The graphs are not normalised

find the energy levels and the corresponding energy eigenstates, we must solve the time-independent →Schrödinger equation, H |ψi = E |ψi.

We can solve the differential equation in the coordinate basis, using a power series method. It turns out that there is a family of solutions,  
1/4 p mω
mωx2 · exp − · Hn hx|ψn i = √ 1n · mω π~ 2~ ~ x 2 n!

n = 0, 1, 2, . . .

The first six solutions (n = 0 to 5) are shown on the right. The functions Hn are the Hermite polynomials:

Quantum harmonic oscillator

FT

444

Figure 41 Probability densities |ψ n(x)| 2 for the bound eigenstates, beginning with the ground state (n = 0) at the bottom and increasing in energy toward the top. The horizontal axis shows the position x, and brighter colors represent higher probability densities. 2

dn −x2 dxn e

DR A

Hn (x) = (−1)n ex

They should not be confused with the Hamiltonian, which is also denoted by H. The corresponding energy levels are
En = ~ω n + 21 .

This energy spectrum is noteworthy for two reasons. Firstly, the energies are "quantized", and may only take the discrete values of ~ω times 1/2, 3/2, 5/2, and so forth. This is a feature of many quantum mechanical systems. In the following section on ladder operators, we will engage in a more detailed examination of this phenomenon. Secondly, the lowest achievable energy is not zero, but ~ω/2, which is called the "ground state energy" or zero-point energy. It is not obvious that this is significant, because normally the zero of energy is not a physically meaningful quantity, only differences in energies. Nevertheless, the ground state energy has many implications, particularly in quantum gravity. Note that the ground state probability density is concentrated at the origin. This means the particle spends most of its time at the bottom of the potential well, as we would expect for a state with little energy. As the energy increases, the probability density becomes concentrated at the "classical turning points", where the state’s energy coincides with the potential energy. This is consistent with the classical harmonic oscillator, in which the particle spends most of its time (and is therefore most likely to be found) at the turning points, where it is the slowest. The correspondence principle is thus satisfied. Quantum harmonic oscillator

445

Ladder operator method

a a†

p mω 2~ x + p mω 2~ x −

= =

i mω p
i mω p




FT

The power series solution, though straightforward, is rather tedious. The "ladder operator" method, due to Paul Dirac, allows us to extract the energy eigenvalues without directly solving the differential equation. Furthermore, it is readily generalizable to more complicated problems, notably in quantum field theory. Following this approach, we define the operators a and its adjoint a †

The operator a is not Hermitian since it and its adjoint a † are not equal. In deriving the form of a †, we have used the fact that the operators x and p, which represent observables, are Hermitian. These observable operators can be expressed as a linear combination of the ladder operators as

x =

q

p

i

~mω 2

a† + a




a† − a




DR A

=

~ q2mω

The x and p operators obey the following identity, known as the canonical commutation relation: [x, p] = i~.

The square brackets in this equation are a commonly-used notational device, known as the commutator, defined as [A, B] ≡ AB − BA.

Using the above, we can prove the identities
H = ~ω a† a + 1/2   a, a† = 1.

Now, let |ψE i denote an energy eigenstate with energy E. The inner product of any ket with itself must be non-negative, so (a |ψE i , a |ψE i) = hψE | a† a |ψE i ≥ 0.

Expressing a †a in terms of the Hamiltonian:

Quantum harmonic oscillator

446 H hψE | ~ω −

1 2

|ψE i =



E ~ω

−

1 2



≥ 0,

FT

so that E ≥ ~ω/2. Note that when (a |ψE i) is the zero ket (i.e. a ket with length zero), the inequality is saturated, so that E = ~ω/2. It is straightforward to check that there exists a state satisfying this condition; it is the ground (n = 0) state given in the preceding section. Using the above identities, we can now show that the commutation relations of a and a † with H are:

[H, a]†  = H, a =

−~ωa ~ωa†

.

Thus, provided (a |ψE i) is not the zero ket,

H(a |ψE i) = = =

([H, a] + aH) |ψE i (−~ωa + aE) |ψE i (E − ~ω)(a |ψE i)

DR A

.

Similarly, we can show that

H(a† |ψE i) = (E + ~ω)(a† |ψE i).

In other words, a acts on an eigenstate of energy E to produce, up to a multiplicative constant, another eigenstate of energy E − ~ω, and a † acts on an eigenstate of energy E to produce an eigenstate of energy E + ~ω. For this reason, a is called a "lowering operator", and a † a "raising operator". The two operators together are called "ladder operators". In quantum field theory, a and a † are alternatively called "annihilation" and "creation" operators because they destroy and create particles, which correspond to our quanta of energy. Given any energy eigenstate, we can act on it with the lowering operator, a, to produce another eigenstate with ~ω, less energy. By repeated application of the lowering operator, it seems that we can produce energy eigenstates down to E = -∞. However, this would contradict our earlier requirement that E ≥ ~ω/2. Therefore, there must be a ground-state energy eigenstate, which we label |0i (not to be confused with the zero ket), such that a |0i = 0(zero ket).

Quantum harmonic oscillator

447

H |0i = (~ω/2) |0i

FT

In this case, subsequent applications of the lowering operator will just produce zero kets, instead of additional energy eigenstate. Furthermore, we have shown above that

Finally, by acting on |0i with the raising operator and multiplying by suitable normalization factors, we can produce an infinite set of energy eigenstates {|0i , |1i , |2i , ..., |ni , ...}, such that H |ni = ~ω(n + 1/2) |ni

which matches the energy spectrum which we gave in the preceding section.

Natural length and energy scales

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The quantum harmonic oscillator possesses natural scales for length and energy, which can be used to simplify the problem. These can be found by nondimensionalization. The result is that if we measure energy in units of ~ω and distance in units of (~/ (mω))1/2 , then the Schrödinger equation becomes: 2

∂ 1 2 H = − 21 ∂u 2 + 2u ,

and the energy eigenfunctions and eigenvalues become hx|ψn i =

√ 1 π −1/4 exp(−u2 /2)Hn (u) 2n n!

En = n + 21 .

To avoid confusion, we will not adopt these natural units in this article. However, they frequently come in handy when performing calculations.

N -dimensional harmonic oscillator

The one-dimensional harmonic oscillator is readily generalizable to N dimensions, where N = 1, 2, 3, ... . In one dimension, the position of the particle was specified by a single coordinate, x. In N dimensions, this is replaced by N position coordinates, which we label x 1, ..., x N . Corresponding to each position coordinate is a momentum; we label these p 1, ..., p N . The canonical commutation relations between these operators are

[xi , pj ] = [xi , xj ] = [pi , pj ] =

i~δi,j 0 0

Quantum harmonic oscillator

448

The Hamiltonian for this system is  P  p2i 1 2 2 H= N i=1 2m + 2 mω xi .

FT

.

As the form of this Hamiltonian makes clear, the N -dimensional harmonic oscillator is exactly analogous to N independent one-dimensional harmonic oscillators with the same mass and spring constant. In this case, the quantities x 1, ..., x N would refer to the positions of each of the N particles. This is a happy property of the r 2 potential, which allows the potential energy to be separated into terms depending on one coordinate each.

DR A

This observation makes the solution straightforward. For a particular set of quantum numbers {n} the energy eigenfunctions for the N -dimensional oscillator are expressed in terms of the 1-dimensional eigenfunctions as: Q hx|ψ{n} i = N i=1 hxi |ψni i In the ladder operator method,
we define N sets of ladder operators,

= =

ai a†i

p mω 2~ xi + p mω 2~ xi −

i mω pi
i mω pi

.

By a procedure analogous to the one-dimensional case, we can then show that each of the a i and a † i operators lower and raise the energy by ω respectively. The energy levels of the system are h i E = ~ω (n1 + · · · + nN ) + N2 . ni = 0, 1, 2, . . .

As in the one-dimensional case, the energy is quantized. The ground state energy is N times the one-dimensional energy, as we would expect using the analogy to N independent one-dimensional oscillators. There is one further difference: in the one-dimensional case, each energy level corresponds to a unique quantum state. In N -dimensions, except for the ground state, the energy levels are degenerate, meaning there are several states with the same energy. The degeneracy can be calculated relatively easily, as an example, consider the 3-dimensional case: Define n = n 1 + n 2 + n 3. All states with the same n Quantum harmonic oscillator

449

(n+1)(n+2) 2

Related problems

FT

will have the same energy. For a given n, we choose a particular n 1. Then n 2 + n 3 = n - n 1. There are n - n 1 + 1 possible groups {n 2, n 3}. n 2 can take on the values 0 to n - n 1, and for each n 2 the value of n 3 is fixed. The degree of degeneracy therefore is: P P P n(n+1) gn = nn1 =0 n−n1 +1 = nn1 =0 n+1− nn1 =0 n1 = (n+1)(n+1)− 2 =

The quantum harmonic oscillator can be extended in many interesting ways. We will briefly discuss two of the more important extensions, the anharmonic oscillator and coupled harmonic oscillators.

Anharmonic oscillator

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As mentioned in the introduction, a system residing "near" the minimum of some potential may be treated as a harmonic oscillator. In this approximation, we Taylor-expand the potential energy around the minimum and discard terms of third or higher order, resulting in an approximate quadratic potential. Once we have studied the system in this approximation, we may wish to investigate the corrections due to the discarded higher-order terms, particularly the thirdorder term.

The anharmonic oscillator Hamiltonian is the harmonic oscillator Hamiltonian with an additional x 3 potential: H=

p2 2m

+ 12 mω 2 x2 + λx3

If the harmonic approximation is valid, the coefficient is small compared to the quadratic term. We may therefore use perturbation theory to determine the corrections to the states and energy levels imposed by the anharmonic term. This task may be simplified by using the ladder operators to rewrite the anharmonic term as  3 2 ~ λ 2mω (a + a† )3 . It turns out that the correction to the energies vanish to first-order in . The second-order corrections are given by the usual formula in perturbation theory: 1 ∆E (2) = λ2 hψE | x3 E−H x3 |ψE i . 0

Quantum harmonic oscillator

450 This is straightforward, though tedious, to evaluate. One failing of this method, however, is that it does not take into account the possibility of the particle tunnelling out, since it is no longer bound on both sides.

FT

Coupled harmonic oscillators

In this problem, we consider N equal masses which are connected to their neighbors by springs, in the limit of large N. The masses form a linear chain in one dimension, or a regular lattice in two or three dimensions.

As in the previous section, we denote the positions of the masses by x 1, x 2, ..., as measured from their equilibrium positions (i.e. x k = 0 if particle k is at its equilibrium position.) In two or more dimensions, the xs are vector quantities. The Hamiltonian of the total system is P p2i 1 2P 2 H= N i=1 2m + 2 mω {ij}(nn) (xi − xj ) The potential energy is summed over "nearest-neighbor" pairs, so there is one term for each spring.

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Remarkably, there exists a coordinate transformation to turn this problem into a set of independent harmonic oscillators, each of which corresponds to a particular collective distortion of the lattice. These distortions display some particle-like properties, and are called phonons. Phonons occur in the ionic lattices of many solids, and are extremely important for understanding many of the phenomena studied in solid state physics.

See also • • • •

Gas in a harmonic trap →Creation and annihilation operators →Coherent state Morse potential

References •

Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X. • Liboff, Richard L. (2002). Introductory Quantum Mechanics. AddisonWesley. ISBN 0805387145.

Quantum harmonic oscillator

451

External links Quantum Harmonic Oscillator 243

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Source: http://en.wikipedia.org/wiki/Quantum_harmonic_oscillator Principal Authors: CYD, Michael Hardy, HappyCamper, PAR, Dmn

Quantum indeterminacy

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Quantum indeterminacy is the apparent necessary incompleteness in the description of a physical system, that has become one of the characteristics of the standard description of quantum physics. Prior to quantum physics, it was thought that (a) a physical system had a determinate state which uniquely determined all the values of its measurable properties, and conversely (b) the values of its measurable properties uniquely determined the state. Albert Einstein may have been the first person to carefully point out the radical effect the new quantum physics would have on our notion of physical state. Quantum indeterminacy can be quantitatively characterized by a probability distribution on the set of outcomes of measurements of an observable. The distribution is uniquely determined by the system state, and moreover quantum mechanics provides a recipe for calculating this probability distribution. Indeterminacy in measurement was not an innovation of quantum mechanics, since it had established early on by experimentalists that errors in measurement may lead to indeterminate outcomes. However, by the latter half of the eighteenth century, measurement errors were well understood and it was known that they could either be reduced by better equipment or accounted for by statistical error models. In quantum mechanics, however, indeterminacy is of a much more fundamental nature, having nothing to do with errors or disturbance.

Measurement

An adequate account of quantum indeterminacy requires a theory of measurement. Many theories have been proposed since the beginning of quantum mechanics and quantum measurement continues to be an active research area

243 http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/hosc.html

Quantum indeterminacy

452

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in both theoretical and experimental physics (Braginski and Khalili 1992.) Possibly the first systematic attempt at a mathematical theory was developed by John von Neumann. The kind of measurements he investigated in (von Neumann, 1955) are now called projective measurements. That theory was based in turn on the theory of projection-valued measures for self-adjoint operators which had been recently developed (by von Neumann and independently by Marshall Stone) and the Hilbert space formulation of quantum mechanics (attributed by von Neumann to Paul Dirac).

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In this formulation, the state of a physical system corresponds to a vector of length 1 in a →Hilbert space H over the complex numbers. An observable is represented by a self-adjoint operator A on H. If H is finite dimensional, by the spectral theorem, A has an orthonormal basis of eigenvectors. If the system is in state ψ, then immediately after measurement the system will occupy a state which is an eigenvector e of A and the observed value will be the corresponding eigenvalue of the equation A e = e. It is immediate from this that measurement in general will be non-deterministic. Quantum mechanics, moreover, gives a recipe for computing a probability distribution Pr on the possible outcomes given the initial system state is ψ. The probability is

Pr(λ) = hE(λ)ψ | ψi

where E() is the projection onto the space of eigenvectors of A with eigenvalue .

Quantum indeterminacy

453

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FT

Example

Bloch sphere showing eigenvectors for Pauli Spin matrices. The Bloch sphere is a two-dimensional surface the points of which correspond to the state space of a spin 1/2 particle. At the state ψ the values of σ 1 are +1 whereas the values of σ 2 and σ 3 take the values +1, -1 with probability 1/2. In this example, we consider a single spin 1/2 particle (such as an electron) in which we only consider the spin degree of freedom. The corresponding Hilbert space is the two-dimensional Hilbert space C 2, with each quantum state corresponding to a unit vector in C 2 (unique up to phase). In this case, the state space can be geometrically represented as the surface of a sphere, as shown in the figure on the right. The Pauli  spinmatrices 

σ1 =

0 1

1 , 0

σ2 =

0 i

 −i , 0

σ3 =



 1 0 0 −1

are self-adjoint and correspond to spin-measurements along the 3 coordinate axes. Quantum indeterminacy

454 The Pauli matrices all have the eigenvalues +1, -1. For σ 1, these eigenvalues correspond to the eigenvectors √1 (1, 1), √1 (1, −1) 2 2

•

FT

•

For σ 3, they correspond to the eigenvectors (1, 0), (0, 1)

Thus in the state ψ=

√1 (1, 1), 2

σ 1 has the determinate value +1, while measurement of σ 3 can produce either +1, -1 each with probability 1/2. In fact, there is no state in which measurement of both σ 1 and σ 3 have determinate values.

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There are various questions that can be asked about the above indeterminacy assertion. •

Can the indeterminacy be understood as similar to an error in measurement explainable by an error parameter? More precisely, is there a hidden parameter that could account for the statistical indeterminacy in a completely classical way? • Can the indeterminacy be understood as a disturbance of the system being measured?

Von Neumann formulated the question 1) and provided an argument why the answer had to be no, if one accepted the formalism he was proposing, although his argument contained a flaw. The definitive negative answer to 1) has been established by experiment that Bell’s inequalities are violated (see Bell test experiments.) The answer to 2) depends on how disturbance is understood (particularly since measurement is disturbance), but in the most natural interpretation the answer is also no. To see this, consider two sequences of measurements: (A) which measures exclusively σ 1 and (B) which measures only σ 3 of a spin system in the state ψ. The measurement outcomes of (A) are all +1, while the statistical distribution of the measurements (B) is still divided between +1, -1 with probability 1/2.

Other examples of indeterminacy

Quantum indeterminacy can also be illustrated in terms of a particle with a definitely measured momentum for which there must be a fundamental limit Quantum indeterminacy

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to how precisely its location can be specified. This quantum uncertainty principle can be expressed in terms of other variables, for example, a particle with a definitely measured energy has a fundamental limit to how precisely one can specify how long it will have that energy. The units involved in quantum uncertainty are on the order of Planck’s constant (found experimentally to be 6.6 x 10 -34 J·s).

Indeterminacy and incompleteness

Quantum indeterminacy is the assertion that the state of a system does not determine a unique collection of values for all its measurable properties. Indeed in the quantum mechanical formalism, for a given quantum state, each one of these measurable values will be obtained non-deterministically in accordance with a probability distribution which is uniquely determined by the system state. Note that the state is destroyed by measurement, so when we refer to a collection of values, each measured value in this collection must be obtained using a freshly prepared state.

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This indeterminacy might be regarded as a kind of essential incompleteness in our description of a physical system. Notice however, that the indeterminacy as stated above only applies to values of measurements not to the quantum state. For example, in the spin 1/2 example discussed above, the system can be prepared in the state ψ by using measurement of σ 1 as a filter which retains only those particles such that σ 1 yields +1. By the von Neumann (so-called) postulates, immediately after the measurement the system is assuredly in the state ψ. However, Einstein did believe that quantum state cannot be a complete description of a physical system and, it is commonly thought, never came to terms with quantum mechanics. In fact, Einstein, Boris Podolsky and Nathan Rosen did show that if quantum mechanics is correct, then the classical view of how the real world works (at least after special relativity) is no longer tenable. This view included the following two ideas: •

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A measurable property of a physical system whose value can be predicted with certainty is actually an element of reality (this was the terminology used by EPR). Effects of local actions have a finite propagation speed.

This failure of the classical view was one of the conclusions of the EPR thought experiment in which two remotely located observers, now commonly referred to as Alice and Bob, perform independent measurements of spin on a pair of electrons, prepared at a source in a special state called a spin singlet state. It was a conclusion of EPR, using the formal apparatus of quantum theory, Quantum indeterminacy

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that once Alice measured spin in the x direction, Bob’s measurement in the x direction was determined with certainty, whereas immediately before Alice’s measurement Bob’s outcome was only statistically determined. From this it follows that either value of spin in the x direction is not an element of reality or that the effect of Alice’s measurement has infinite speed of propagation.

Indeterminacy for mixed states

We have described indeterminacy for a quantum system which is in a pure state. Mixed states are a more general kind of state obtained by a statistical mixture of pure states. For mixed states the "quantum recipe" for determining the probability distribution of a measurement is determined as follows:

Let A be an observable of a quantum mechanical system. A is given by a densely defined self-adjoint operator on H. The spectral measure of A is a projectionvalued measure defined by the condition R

EA (U ) =

U

λd E(λ),

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for every Borel subset U of R. Given a mixed state S, we introduce the distribution of A under S as follows:

DA (U ) = Tr(EA (U )S).

This is a probability measure defined on the Borel subsets of R which is the probability distribution obtained by measuring A in S.

See also • • • • • • • •

Quantum mind just about any of the quantum mechanics articles, including →Quantum entanglement →Complementarity (physics) Interpretations of quantum mechanics Quantum measurement Counterfactual definiteness EPR paradox

References

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A. Aspect, Bell’s inequality test: more ideal than ever, Nature 398 189 (1999). 244 V. Braginski and F. Khalili, Quantum Measurements, Cambridge University Press, 1992. G. Bergmann, The Logic of Quanta, American Journal of Physics, 1947. Reprinted in Readings in the Philosophy of Science, Ed. H. Feigl and M. Brodbeck, Appleton-Century-Crofts, 1953. Discusses measurement, accuracy and determinism. J.S. Bell, On the Einstein-Poldolsky-Rosen paradox, Physics 1 195 (1964). A. Einstein, B. Podolsky, and N. Rosen, Can quantum-mechanical description of physical reality be considered complete? 245 Phys. Rev. 47 777 (1935). 246 G. Mackey, Mathematical Foundations of Quantum Mechanics, W. A. Benjamin, 1963 (paperback reprint by Dover 2004). J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. Reprinted in paperback form. Originally published in German in 1932. R. Omnès, Understanding Quantum Mechanics, Princeton University Press, 1999.

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External links •

Common Misconceptions Regarding Quantum Mechanics 247 See especially part III "Misconceptions regarding measurement".

Source: http://en.wikipedia.org/wiki/Quantum_indeterminacy

Principal Authors: CarlHewitt, CSTAR, JWSchmidt, DV8 2XL, William M. Connolley

244 http://www-ece.rice.edu/~kono/ELEC565/Aspect_Nature.pdf 245 http://www.drchinese.com/David/EPR.pdf

246 http://prola.aps.org/abstract/PR/v47/i10/p777_1 247 http://www.oberlin.edu/physics/dstyer/TeachQM/misconnzz.pdf

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In physics, a quantum leap or quantum jump is a change of an electron within an atom from one energy state to another. This is a discontinuous change in which the electron goes from one energy level to another without passing through any intermediate levels. This phenomenon contradicted expectations set by classical theories, that the electron’s energy should be able to vary continuously. Quantum leaps of electrons cause the emission of electromagnetic radiation in quantized units called photons. All emission of light occurs as a result of quantum leaps.

More generally, a quantum leap is the smallest possible change, as when one’s bank account balance goes from $500.00 (five hundred dollars) to $500.01 (five hundred dollars and one cent). There are no possible amounts intermediate between those.

Vernacular usage

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In the vernacular, the term quantum leap has come to mean an abrupt change, especially an advance or augmentation. The term dates back to early-to-mid20th century. The vernacular usage is not always in accord with the original meaning, in that a large abrupt change is often implied. A quantum leap in quantum mechanics is by definition the smallest change possible. The usages agree, however, in that both describe an advance that happens all at once, rather than gradually over time. A ’quantum leap in technology’ is thus a revolutionary advance, rather than an evolutionary one.

See also • • • •

Fluorescence Phosphorescence Stimulated emission Sinclair QL

External links •

Are there quantum jumps? 248

Source: http://en.wikipedia.org/wiki/Quantum_leap

248 http://www.mikomma.de/schroe/quantumjumps.htm

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Principal Authors: Michael Hardy, Srleffler, Maveric149, Hinakana, Djinn112

Quantum levels are fixed levels with a logarithmic, descending quantum pattern in the visible spectrum of light that can be observed through a spectrometer while looking at intense flows of electricity through the various halides on the periodic table in a vacuum tube. They also have some use in chemistry when dealing with the movement of electrons to different orbital levels around the atom and the energy levels involved in such actions.

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• →Quantum mechanics • Absorption spectrum • Emission spectrum

Source: http://en.wikipedia.org/wiki/Quantum_level

Principal Authors: Enochlau, JYOuyang, Michael Hardy, Borofkin, Charles Matthews

Quantum mechanics

For a non-technical introduction to the topic, please see Introduction to Quantum mechanics.

Quantum mechanics is a fundamental branch of theoretical physics that replaces classical mechanics and classical electromagnetism at the atomic and subatomic levels. It is the underlying mathematical framework of many fields of physics and chemistry, including condensed matter physics, atomic physics, molecular physics, computational chemistry, quantum chemistry, particle physics, and nuclear physics. Along with general relativity, quantum mechanics is one of the pillars of modern physics.

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Figure 42 Fig. 1: The wavefunctions of an electron in a hydrogen atom possessing definite energy (increasing downward: n=1,2,3,...) and angular momentum (increasing across: s, p, d,...). Brighter areas correspond to higher probability density for a position measurement. Wavefunctions like these are directly comparable to Chladni’s figures of acoustic modes of vibration in classical physics and are indeed modes of oscillation as well: they possess a sharp energy and thus a sharp frequency. The angular momentum and energy are quantized, and only take on discrete values like those shown (as is the case for resonant frequencies in acoustics).

Introduction

The term quantum (Latin, "how much ") refers to discrete units that the theory assigns to certain physical quantities, such as the energy of an atom at rest (see Figure 1, at right). The discovery that waves could be measured in particle-like small packets of energy called quanta led to the branch of physics that deals with atomic and subatomic systems which we today call Quantum Mechanics. The foundations of quantum mechanics were established during the first half of the 20th century by Max Planck, Albert Einstein, Niels Bohr, Werner Heisenberg, Erwin Schrödinger, Max Born, John von Neumann, Paul Dirac, Wolfgang Pauli and others. Some fundamental aspects of the theory are still actively studied.

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Quantum mechanics is a more fundamental theory than Newtonian mechanics and classical electromagnetism, in the sense that it provides accurate and precise descriptions for many phenomena that these "classical" theories simply cannot explain on the atomic and subatomic level. It is necessary to use quantum mechanics to understand the behavior of systems at atomic length scales and smaller. For example, if Newtonian mechanics governed the workings of an atom, electrons would rapidly travel towards and collide with the nucleus. However, in the natural world the electron normally remains in a stable orbit around a nucleus – seemingly defying classical electromagnetism. Quantum mechanics was initially developed to explain the atom, especially the spectra of light emitted by different atomic species. The quantum theory of the atom developed as an explanation for the electron’s staying in its orbital, which could not be explained by Newton’s laws of motion and by classical electromagnetism.

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In the formalism of quantum mechanics, the state of a system at a given time is described by a complex number wave functions (sometimes referred to as orbitals in the case of atomic electrons), and more generally, elements of a complex vector space. This abstract mathematical object allows for the calculation of probabilities of outcomes of concrete experiments. For example, it allows to compute the probability of finding an electron in a particular region around the nucleus at a particular time. Contrary to classical mechanics, one cannot in general make predictions of arbitrary accuracy. For instance electrons cannot in general be pictured as localized particles in space but rather should be thought of as "clouds" of negative charge spread out over the entire orbit. These clouds represent the regions around the nucleus where the probability of "finding" an electron is the largest. The Heisenberg’s Uncertainty Principle quantifies the inability to precisely locate the particle. The other exemplar that led to quantum mechanics was the study of electromagnetic waves such as light. When it was found in 1900 by Max Planck that the energy of waves could be described as consisting of small packets or quanta, Albert Einstein exploited this idea to show that an electromagnetic wave such as light could be described by a particle called the photon with a discrete energy dependent on its frequency. This led to a theory of unity between subatomic particles and electromagnetic waves called wave-particle duality in which particles and waves were neither one nor the other, but had certain properties of both. While quantum mechanics describes the world of the very small, it also is needed to explain certain "macroscopic quantum systems" such as superconductors and superfluids. Broadly speaking, quantum mechanics incorporates four classes of phenomena that classical physics cannot account for: (i) the quantization (discretization) Quantum mechanics

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Since the early days of quantum theory, physicists have made many attempts to combine it with the other highly successful theory of the twentieth century, Albert Einstein’s General Theory of Relativity. While quantum mechanics is entirely consistent with special relativity, serious problems emerge when one tries to join the quantum laws with general relativity, the more elaborate description of nature which includes gravity. Resolving these inconsistencies has been a major goal of twentieth- and twenty-first-century physics. Despite the proposal of many novel ideas, the unification of quantum mechanics—which reigns in the domain of the very small—and general relativity—a superb description of the very large—remains a tantalizing future possibility. (See quantum gravity, string theory.)

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Because everything is composed of quantum-mechanical particles, the laws of classical physics must approximate the laws of quantum mechanics in the appropriate limit. This is often expressed by saying that in case of large quantum numbers quantum mechanics "reduces" to classical mechanics and classical electromagnetism . This requirement is called the correspondence, or classical limit.

Quantum mechanics can be formulated in either a relativistic or non-relativistic manner. Relativistic quantum mechanics (quantum field theory) provides the framework for some of the most accurate physical theories known. Still, nonrelativistic quantum mechanics is also used due to its simplicity and when relativistic effects are negligible. We will use the terms quantum mechanics, quantum physics, and quantum theory synonymously, to refer to both relativistic and non-relativistic quantum mechanics. It should be noted, however, that certain authors refer to "quantum mechanics" in the more restricted sense of non-relativistic quantum mechanics. Also, in quantum mechanics, the use of the term particle typically refers to an elementary or subatomic particle.

Description of the theory

There are a number of mathematically equivalent formulations of quantum mechanics. One of the oldest and most commonly used formulations is the transformation theory invented by Cambridge theoretical physicist Paul Dirac, which unifies and generalizes the two earliest formulations of quantum mechanics, matrix mechanics (invented by Werner Heisenberg) and wave mechanics (invented by Erwin Schrödinger).

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In this formulation, the instantaneous state of a quantum system encodes the probabilities of its measurable properties, or "observables". Examples of observables include energy, position, momentum, and angular momentum. Observables can be either continuous (e.g., the position of a particle) or discrete (e.g., the energy of an electron bound to a hydrogen atom).

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Generally, quantum mechanics does not assign definite values to observables. Instead, it makes predictions about probability distributions; that is, the probability of obtaining each of the possible outcomes from measuring an observable. Naturally, these probabilities will depend on the quantum state at the instant of the measurement. There are, however, certain states that are associated with a definite value of a particular observable. These are known as "eigenstates" of the observable ("eigen" meaning "own" in German). In the everyday world, it is natural and intuitive to think of everything being in an eigenstate of every observable. Everything appears to have a definite position, a definite momentum, and a definite time of occurrence. However, Quantum Mechanics does not pinpoint the exact values for the position or momentum of a certain particle in a given space in a finite time, but, rather, it only provides a range of probabilities of where that particle might be. Therefore, it became necessary to use different words for a) the state of something having an uncertainty relation and b) a state that has a definite value. The latter is called the "eigenstate" of the property being measured.

A concrete example will be useful here. Let us consider a free particle. In quantum mechanics, there is wave-particle duality so the properties of the particle can be described as a wave. Therefore, its quantum state can be represented as a wave, of arbitrary shape and extending over all of space, called a wavefunction. The position and momentum of the particle are observables. The Uncertainty Principle of quantum mechanics states that both the position and the momentum cannot simultaneously be known with infinite precision at the same time. However, we can measure just the position alone of a moving free particle creating an eigenstate of position with a wavefunction that is very large at a particular position x, and zero everywhere else. If we perform a position measurement on such a wavefunction, we will obtain the result x with 100% probability. In other words, we will know the position of the free particle. This is called an eigenstate of position. If the particle is in an eigenstate of position then its momentum is completely unknown. An eigenstate of momentum, on the other hand, has the form of a plane wave. It can be shown that the wavelength is equal to h/p, where h is Planck’s constant and p is the momentum of the eigenstate. If the particle is in an eigenstate of momentum then its position is completely blurred out.

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Usually, a system will not be in an eigenstate of whatever observable we are interested in. However, if we measure the observable, the wavefunction will immediately become an eigenstate of that observable. This process is known as wavefunction collapse. If we know the wavefunction at the instant before the measurement, we will be able to compute the probability of collapsing into each of the possible eigenstates. For example, the free particle in our previous example will usually have a wavefunction that is a wave packet centered around some mean position x 0, neither an eigenstate of position nor of momentum. When we measure the position of the particle, it is impossible for us to predict with certainty the result that we will obtain. It is probable, but not certain, that it will be near x 0, where the amplitude of the wavefunction is large. After we perform the measurement, obtaining some result x, the wavefunction collapses into a position eigenstate centered at x.

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Wave functions can change as time progresses. An equation known as the →Schrödinger equation describes how wave functions change in time, a role similar to Newton’s second law in classical mechanics. The Schrödinger equation, applied to our free particle, predicts that the center of a wave packet will move through space at a constant velocity, like a classical particle with no forces acting on it. However, the wave packet will also spread out as time progresses, which means that the position becomes more uncertain. This also has the effect of turning position eigenstates (which can be thought of as infinitely sharp wave packets) into broadened wave packets that are no longer position eigenstates.

Some wave functions produce probability distributions that are constant in time. Many systems that are treated dynamically in classical mechanics are described by such "static" wave functions. For example, a single electron in an unexcited atom is pictured classically as a particle moving in a circular trajectory around the atomic nucleus, whereas in quantum mechanics it is described by a static, spherically symmetric wavefunction surrounding the nucle-

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us ( ). (Note that only the lowest angular momentum states, labeled s, are spherically symmetric). The time evolution of wave functions is deterministic in the sense that, given a wavefunction at an initial time, it makes a definite prediction of what the wavefunction will be at any later time. During a measurement, the change of the wavefunction into another one is not deterministic, but rather unpredictable, i.e., random. The probabilistic nature of quantum mechanics thus stems from the act of measurement. This is one of the most difficult aspects of quantum systems to understand. It was the central topic in the famous Bohr-Einstein debates, in which the two scientists attempted to clarify these fundamental principles by way of thought experiments. In the decades after the formulation of quantum mechanics, the question of what constitutes a "measurement" has been extensively studied. Interpretations of quantum mechanics have been formulated to do away with the concept of "wavefunction collapse"; see, for example, the relative state interpretation. The basic idea is that when a quantum system interacts with a measuring apparatus, their respective wavefunctions become entangled, so that the original quantum system ceases to exist as an independent entity. For details, see the article on measurement in quantum mechanics. Quantum mechanics

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As mentioned in the introduction, there are several classes of phenomena that appear under quantum mechanics which have no analogue in classical physics. These are sometimes referred to as "quantum effects".

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The first type of quantum effect is the quantization of certain physical quantities. Quantization first arose in the mathematical formulae of Max Planck in 1900 as discussed in the introduction. Max Planck was analyzing how the radiation emitted from a body was related to its temperature, in other words, he was analyzing the energy of a wave. The energy of a wave could not be infinite, so Planck used the property of the wave we designate as the frequency to define energy. Max Planck discovered a constant that when multiplied by the frequency of any wave gives the energy of the wave. This constant is referred to by the letter h in mathematical formulae. It is a cornerstone of physics. By measuring the energy in a discrete non-continuous portion of the wave, the wave took on the appearance of chunks or packets of energy. These chunks of energy resembled particles. So energy is said to be quantized because it only comes in discrete chunks instead of a continuous range of energies.

In the example we have given, of a free particle in empty space, both the position and the momentum are continuous observables. However, if we restrict the particle to a region of space (the so-called "particle in a box" problem), the momentum observable will become discrete; it will only take on the valh ues n 2L , where L is the length of the box, h is Planck’s constant, and n is an arbitrary nonnegative integer number. Such observables are said to be quantized, and they play an important role in many physical systems. Examples of quantized observables include angular momentum, the total energy of a bound system, and the energy contained in an electromagnetic wave of a given frequency. Another quantum effect is the uncertainty principle, which is the phenomenon that consecutive measurements of two or more observables may possess a fundamental limitation on accuracy. In our free particle example, it turns out that it is impossible to find a wavefunction that is an eigenstate of both position and momentum. This implies that position and momentum can never be simultaneously measured with arbitrary precision, even in principle: as the precision of the position measurement improves, the maximum precision of the momentum measurement decreases, and vice versa. Those variables for which it holds (e.g., momentum and position, or energy and time) are canonically conjugate variables in classical physics.

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Another quantum effect is the wave-particle duality. It has been shown that, under certain experimental conditions, microscopic objects like atoms or electrons exhibit particle-like behavior, such as scattering. ("Particle-like" in the sense of an object that can be localized to a particular region of space.) Under other conditions, the same type of objects exhibit wave-like behavior, such as interference. We can observe only one type of property at a time, never both at the same time. Another quantum effect is quantum entanglement. In some cases, the wave function of a system composed of many particles cannot be separated into independent wave functions, one for each particle. In that case, the particles are said to be "entangled". If quantum mechanics is correct, entangled particles can display remarkable and counter-intuitive properties. For example, a measurement made on one particle can produce, through the collapse of the total wavefunction, an instantaneous effect on other particles with which it is entangled, even if they are far apart. (This does not conflict with special relativity because information cannot be transmitted in this way.)

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Mathematical formulation

Main article: →Mathematical formulation of quantum mechanics. See also the discussion in Quantum logic. In the mathematically rigorous formulation of quantum mechanics, developed by Paul Dirac and John von Neumann, the possible states of a quantum mechanical system are represented by unit vectors (called "state vectors") residing in a complex separable →Hilbert space (variously called the "state space" or the "associated Hilbert space" of the system) well defined upto a complex number of norm 1 (the phase factor). In other words, the possible states are points in the projectivization of a Hilbert space. The exact nature of this Hilbert space is dependent on the system; for example, the state space for position and momentum states is the space of square-integrable functions, while the state space for the spin of a single proton is just the product of two complex planes. Each observable is represented by a densely defined Hermitian (or self-adjoint) linear operator acting on the state space. Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate. If the operator’s spectrum is discrete, the observable can only attain those discrete eigenvalues. The time evolution of a quantum state is described by the →Schrödinger equation, in which the Hamiltonian, the operator corresponding to the total energy of the system, generates time evolution.

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The inner product between two state vectors is a complex number known as a probability amplitude. During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states. The possible results of a measurement are the eigenvalues of the operator - which explains the choice of Hermitian operators, for which all the eigenvalues are real. We can find the probability distribution of an observable in a given state by computing the spectral decomposition of the corresponding operator. Heisenberg’s uncertainty principle is represented by the statement that the operators corresponding to certain observables do not commute. The Schrödinger equation acts on the entire probability amplitude, not merely its absolute value. Whereas the absolute value of the probability amplitude encodes information about probabilities, its phase encodes information about the interference between quantum states. This gives rise to the wave-like behavior of quantum states.

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It turns out that analytic solutions of Schrödinger’s equation are only available for a small number of model Hamiltonians, of which the quantum harmonic oscillator, the particle in a box, the hydrogen-molecular ion and the hydrogen atom are the most important representatives. Even the helium atom, which contains just one more electron than hydrogen, defies all attempts at a fully analytic treatment. There exist several techniques for generating approximate solutions. For instance, in the method known as perturbation theory one uses the analytic results for a simple quantum mechanical model to generate results for a more complicated model related to the simple model by, for example, the addition of a weak potential energy. Another method is the "semi-classical equation of motion" approach, which applies to systems for which quantum mechanics produces weak deviations from classical behavior. The deviations can be calculated based on the classical motion. This approach is important for the field of quantum chaos. An alternative formulation of quantum mechanics is Feynman’s path integral formulation, in which a quantum-mechanical amplitude is considered as a sum over histories between initial and final states; this is the quantum-mechanical counterpart of action principles in classical mechanics.

Interactions with other scientific theories

The fundamental rules of quantum mechanics are very broad. They state that the state space of a system is a Hilbert space and the observables are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators. These must be chosen appropriately in order to obtain a quantitative Quantum mechanics

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description of a quantum system. An important guide for making these choices is the correspondence principle, which states that the predictions of quantum mechanics reduce to those of classical physics when a system becomes large. This "large system" limit is known as the classical or correspondence limit. One can therefore start from an established classical model of a particular system, and attempt to guess the underlying quantum model that gives rise to the classical model in the correspondence limit. Unsolved problems in physics: In the correspondence limit of quantum mechanics: Is there a preferred interpretation of quantum mechanics? How does the quantum description of reality, which includes elements such as the superposition of states and wavefunction collapse, give rise to the reality we perceive?

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When quantum mechanics was originally formulated, it was applied to models whose correspondence limit was non-relativistic classical mechanics. For instance, the well-known model of the quantum harmonic oscillator uses an explicitly non-relativistic expression for the kinetic energy of the oscillator, and is thus a quantum version of the classical harmonic oscillator.

Early attempts to merge quantum mechanics with special relativity involved the replacement of the Schrödinger equation with a covariant equation such as the →Klein-Gordon equation or the Dirac equation. While these theories were successful in explaining many experimental results, they had certain unsatisfactory qualities stemming from their neglect of the relativistic creation and annihilation of particles. A fully relativistic quantum theory required the development of quantum field theory, which applies quantization to a field rather than a fixed set of particles. The first complete quantum field theory, quantum electrodynamics, provides a fully quantum description of the electromagnetic interaction. The full apparatus of quantum field theory is often unnecessary for describing electrodynamic systems. A simpler approach, one employed since the inception of quantum mechanics, is to treat charged particles as quantum mechanical objects being acted on by a classical electromagnetic field. For example, the elementary quantum model of the hydrogen atom describes the electric field of the hydrogen atom using a classical 1/r Coulomb potential. This "semiclassical" approach fails if quantum fluctuations in the electromagnetic field play an important role, such as in the emission of photons by charged particles. Quantum field theories for the strong nuclear force and the weak nuclear force have been developed. The quantum field theory of the strong nuclear force is

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called quantum chromodynamics, and describes the interactions of the subnuclear particles: quarks and gluons. The weak nuclear force and the electromagnetic force were unified, in their quantized forms, into a single quantum field theory known as electroweak theory. It has proven difficult to construct quantum models of gravity, the remaining fundamental force. Semi-classical approximations are workable, and have led to predictions such as Hawking radiation. However, the formulation of a complete theory of quantum gravity is hindered by apparent incompatibilities between general relativity, the most accurate theory of gravity currently known, and some of the fundamental assumptions of quantum theory. The resolution of these incompatibilities is an area of active research, and theories such as string theory are among the possible candidates for a future theory of quantum gravity.

Applications of quantum theory

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Quantum mechanics has had enormous success in explaining many of the features of our world. The individual behavior of the subatomic particles that make up all forms of matter - electrons, protons, neutrons, and so forth - can often only be satisfactorily described using quantum mechanics.Quantum mechanics has strongly influenced string theory, a candidate for a theory of everything (see Reductionism). It is also related to statistical mechanics.

Quantum mechanics is important for understanding how individual atoms combine covalently to form chemicals or molecules. The application of quantum mechanics to chemistry is known as quantum chemistry. (Relativistic) quantum mechanics can in principle mathematically describe most of chemistry. Quantum mechanics can provide quantitative insight into ionic and covalent bonding processes by explicitly showing which molecules are energetically favorable to which others, and by approximately how much. Most of the calculations performed in computational chemistry rely on quantum mechanics. Much of modern technology operates at a scale where quantum effects are significant. Examples include the laser, the transistor, the electron microscope, and magnetic resonance imaging. The study of semiconductors led to the invention of the diode and the transistor, which are indispensable for modern electronics. Researchers are currently seeking robust methods of directly manipulating quantum states. Efforts are being made to develop quantum cryptography, which will allow guaranteed secure transmission of information. A more distant goal is the development of quantum computers, which are expected to

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perform certain computational tasks exponentially faster than classical computers. Another active research topic is quantum teleportation, which deals with techniques to transmit quantum states over arbitrary distances.

Philosophical consequences

Main article: Interpretations of quantum mechanics

Since its inception, the many counter-intuitive results of quantum mechanics have provoked strong philosophical debate and many interpretations. Even fundamental issues such as Max Born’s basic rules concerning probability amplitudes and probability distributions took decades to be appreciated.

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The Copenhagen interpretation, due largely to the Danish theoretical physicist Niels Bohr, is the interpretation of quantum mechanics most widely accepted amongst physicists. According to it, the probabilistic nature of quantum mechanics predictions cannot be explained in terms of some other deterministic theory, and does not simply reflect our limited knowledge. Quantum mechanics provides probabilistic results because the physical universe is itself probabilistic rather than deterministic.

Albert Einstein, himself one of the founders of quantum theory, disliked this loss of determinism in measurement. He held that there should be a local hidden variable theory underlying quantum mechanics and consequently the present theory was incomplete. He produced a series of objections to the theory, the most famous of which has become known as the EPR paradox. John Bell showed that the EPR paradox led to experimentally testable differences between quantum mechanics and local hidden variable theories. Experiments have been taken as confirming that quantum mechanics is correct and the real world cannot be described in terms of such hidden variables. "Loopholes" in the experiments, however, mean that the question is still not quite settled. See the Bohr-Einstein debates

The Everett many-worlds interpretation, formulated in 1956, holds that all the possibilities described by quantum theory simultaneously occur in a "multiverse" composed of mostly independent parallel universes. This is not accomplished by introducing some new axiom to quantum mechanics, but on the contrary by removing the axiom of the collapse of the wave packet: All the possible consistent states of the measured system and the measuring apparatus (including the observer) are present in a real physical (not just formally mathematical, as in other interpretations) quantum superposition. (Such a superposition of consistent state combinations of different systems is called an entangled state.) While the multiverse is deterministic, we perceive non-deterministic behavior governed by probabilities, because we can observe only the universe, i.e. the Quantum mechanics

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consistent state contribution to the mentioned superposition, we inhabit. Everett’s interpretation is perfectly consistent with John Bell’s experiments and makes them intuitively understandable. However, according to the theory of quantum decoherence, the parallel universes will never be accessible for us, making them physically meaningless. This inaccessiblity can be understood as follows: once a measurement is done, the measured system becomes entangled with both the physicist who measured it and a huge number of other particles, some of which are photons flying away towards the other end of the universe; in order to prove that the wave function did not collapse one would have to bring all these particles back and measure them again, together with the system that was measured originally. This is completely impractical, but even if one can theoretically do this, it would destroy any evidence that the original measurement took place (including the physicist’s memory).

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In 1900, the German physicist Max Planck introduced the idea that energy is quantized, in order to derive a formula for the observed frequency dependence of the energy emitted by a black body. In 1905, Einstein explained the photoelectric effect by postulating that light energy comes in quanta called photons. The idea that each photon had to consist of energy in terms of quanta was a remarkable achievement as it effectively removed the possibility of black body radiation attaining infinite energy if it were to be explained in terms of wave forms only. In 1913, Bohr explained the spectral lines of the hydrogen atom, again by using quantization, in his paper of July 1913 On the Constitution of Atoms and Molecules. In 1924, the French physicist Louis de Broglie put forward his theory of matter waves by stating that particles can exhibit wave characteristics and vice versa.

These theories, though successful, were strictly phenomenological: there was no rigorous justification for quantization (aside, perhaps, for Henri Poincaré’s discussion of Planck’s theory in his 1912 paper Sur la théorie des quanta). They are collectively known as the old quantum theory. The phrase "quantum physics" was first used in Johnston’s Planck’s Universe in Light of Modern Physics. Modern quantum mechanics was born in 1925, when the German physicist Heisenberg developed matrix mechanics and the Austrian physicist Schrödinger invented wave mechanics and the non-relativistic Schrödinger equation. Schrödinger subsequently showed that the two approaches were equivalent. Heisenberg formulated his uncertainty principle in 1927, and the Copenhagen interpretation took shape at about the same time. Starting around 1927, Paul Quantum mechanics

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Dirac began the process of unifying quantum mechanics with special relativity by discovering the Dirac equation for the electron. He also pioneered the use of operator theory, including the influential bra-ket notation, as described in his famous 1930 textbook. During the same period, Hungarian polymath John von Neumann formulated the rigorous mathematical basis for quantum mechanics as the theory of linear operators on Hilbert spaces, as described in his likewise famous 1932 textbook. These, like many other works from the founding period still stand, and remain widely used.

The field of quantum chemistry was pioneered by physicists Walter Heitler and Fritz London, who published a study of the covalent bond of the hydrogen molecule in 1927. Quantum chemistry was subsequently developed by a large number of workers, including the American theoretical chemist Linus Pauling at Cal Tech, and John Slater into various theories such as Molecular Orbital Theory or Valence Theory.

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Beginning in 1927, attempts were made to apply quantum mechanics to fields rather than single particles, resulting in what are known as quantum field theories. Early workers in this area included Dirac, Pauli, Weisskopf, and Jordan. This area of research culminated in the formulation of quantum electrodynamics by Feynman, Dyson, Schwinger, and Tomonaga during the 1940s. Quantum electrodynamics is a quantum theory of electrons, positrons, and the electromagnetic field, and served as a role model for subsequent quantum field theories. The theory of quantum chromodynamics was formulated beginning in the early 1960s. The theory as we know it today was formulated by Politzer, Gross and Wilzcek in 1975. Building on pioneering work by Schwinger, Higgs, Goldstone, Glashow, Weinberg and Salam independently showed how the weak nuclear force and quantum electrodynamics could be merged into a single electroweak force.

Founding experiments •

Thomas Young’s double-slit experiment demonstrating the wave nature of light (c1805) • Henri Becquerel discovers radioactivity (1896) • Joseph John Thomson’s cathode ray tube experiments (discovers the electron and its negative charge) (1897) • The study of black body radiation between 1850 and 1900, which could not be explained without quantum concepts. • The photoelectric effect: Einstein explained this in 1905 (and later received a Nobel prize for it) using the concept of photons, particles of light with quantized energy Quantum mechanics

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Robert Millikan’s oil-drop experiment, which showed that electric charge occurs as quanta (whole units), (1909) Ernest Rutherford’s gold foil experiment disproved the plum pudding model of the atom which suggested that the mass and positive charge of the atom are almost uniformly distributed. (1911) Professor Walter Ernhart-Plank’s Proton Collapse experiment disproved the Rutherford model and temporarily cast doubt on the distribution of protons throughout an atom. Otto Stern and Walter Gerlach conduct the →Stern-Gerlach experiment, which demonstrates the quantized nature of particle spin (1920) Clinton Davisson and Lester Germer demonstrate the wave nature of the electron 1 in the Electron diffraction experiment (1927) Clyde L. Cowan and Frederick Reines confirm the existence of the neutrino in the neutrino experiment (1955) Claus Jönsson‘s double-slit experiment with electrons (1961)

See also Basics of quantum mechanics →Measurement in quantum mechanics Quantum electrochemistry Quantum chemistry Quantum computers

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Quantum information →Quantum field theory Quantum thermodynamics Theoretical chemistry

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References •

P. A. M. Dirac, The Principles of Quantum Mechanics (1930) – the beginning chapters provide a very clear and comprehensible introduction • David Griffiths, Introduction to Quantum Mechanics, Prentice Hall, 1995. ISBN 0-13-111892-7 – A standard undergraduate level text written in an accessible style. • Richard P. Feynman, Robert B. Leighton and Matthew Sands (1965). The Feynman Lectures on Physics, Addison-Wesley. Richard Feynman’s original lectures (given at CALTECH in early 1962) can also be downloaded as an MP3 file from www.audible.com 249 • Hugh Everett, Relative State Formulation of Quantum Mechanics, Reviews of Modern Physics vol 29, (1957) pp 454-462. • Bryce DeWitt, R. Neill Graham, eds, The Many-Worlds Interpretation of Quantum Mechanics, Princeton Series in Physics, Princeton University Press (1973), ISBN 069108131X

249 http://www.audible.com

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Albert Messiah, Quantum Mechanics, English translation by G. M. Temmer of Mécanique Quantique, 1966, John Wiley and Sons, vol. I, chapter IV, section III. Richard P. Feynman, QED: The Strange Theory of Light and Matter – a popular science book about quantum mechanics and quantum field theory that contains many enlightening insights that are interesting for the expert as well Marvin Chester, Primer of Quantum Mechanics, 1987, John Wiley, N.Y. ISBN 0486428788 Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3th edition, World Scientific (Singapore, 2004) 250(also available online here 251) George Mackey (2004). The mathematical foundations of quantum mechanics. Dover Publications. ISBN 0486435172. Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X. Omnes, Roland (1999). Understanding Quantum Mechanics. Princeton University Press. ISBN 0691004358. J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955. H. Weyl, The Theory of Groups and Quantum Mechanics, Dover Publications 1950.

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Note 1: The Davisson-Germer experiment, which demonstrates the wave nature of the electron 252

External links General: • • •

A history of quantum mechanics 253 A Lazy Layman’s Guide to Quantum Physics 254 Introduction to Quantum Theory at Quantiki 255

250 http://www.worldscibooks.com/physics/5057.html 251 http://www.physik.fu-berlin.de/~kleinert/b5

252 http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/davger2.html 253 http://www-history.mcs.st-andrews.ac.uk/history/HistTopics/The_Quantum_age_begins.html 254 http://higgo.com/quantum/laymans.htm 255 http://cam.qubit.org/wiki/index.php/Introduction_to_Quantum_Theory

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Quantum Physics Made Relatively Simple 256: three video lectures by Hans Bethe Decoherence 257 by Erich Joos

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Course material:

• MIT OpenCourseWare: Chemistry 258. See 5.61 259, 5.73 260, and 5.74 261 • MIT OpenCourseWare: Physics 262. See 8.04 263, 8.05 264, and 8.06 265. • Imperial College Quantum Mechanics Course to Download 266 • A set of downloadable tutorials on Quantum Mechanics, Imperial College 267 • Spark Notes - Quantum Physics 268 FAQs: • • •

Many-worlds or relative-state interpretation 269 Measurement in Quantum mechanics 270 A short FAQ on quantum resonances 271

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Everything you wanted to know about the quantum world 272 — archive of articles from New Scientist magazine. • " Quantum Trickery: Testing Einstein’s Strangest Theory 273", The New York Times, December 27, 2005. •

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256 http://bethe.cornell.edu/ 257 http://www.decoherence.de/

258 http://ocw.mit.edu/OcwWeb/Chemistry/index.htm 259 http://ocw.mit.edu/OcwWeb/Chemistry/5-61Fall-2004/CourseHome/index.htm 260 http://ocw.mit.edu/OcwWeb/Chemistry/5-73Fall-2005/CourseHome/index.htm 261 http://ocw.mit.edu/OcwWeb/Chemistry/5-74Spring-2005/CourseHome/index.htm 262 http://ocw.mit.edu/OcwWeb/Physics/index.htm 263 http://ocw.mit.edu/OcwWeb/Physics/8-04Quantum-Physics-ISpring2003/CourseHome/index.htm 264 http://ocw.mit.edu/OcwWeb/Physics/8-05Fall-2004/CourseHome/index.htm 265 http://ocw.mit.edu/OcwWeb/Physics/8-06Spring-2005/CourseHome/index.htm 266 http://www.imperial.ac.uk/quantuminformation/qi/tutorials 267 http://www3.imperial.ac.uk/portal/page?_pageid=161,482073&_dad=portallive&_schema=PORTALLIVE

#Quantum%20Information%20Tutorials

268 http://www.sparknotes.com/testprep/books/sat2/physics/chapter19section3.rhtml 269 http://www.hedweb.com/manworld.htm 270 http://www.mtnmath.com/faq/meas-qm.html 271 http://www.thch.uni-bonn.de/tc/people/brems.vincent/vincent/faq.html 272 http://www.newscientist.com/channel/fundamentals/quantum-world 273 http://www.nytimes.com/2005/12/27/science/27eins.html?ex=1293339600&en=caf5d835203c3500

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477 Quantum Mechanics (Stanford Encyclopedia of Philosophy) 274 David Mermin on the future directions of physics 275 "Quantum Physics Quackery" 276 by Victor Stenger, Skeptical Inquirer (January/February 1997). • Crank Dot Net’s quantum physics page 277 — "cranks, crackpots, cooks & loons on the net" • Hinduism & Quantum Physics 278

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Source: http://en.wikipedia.org/wiki/Quantum_mechanics

Principal Authors: CYD, Lethe, David R. Ingham, Ancheta Wis, Voyajer, Laurascudder, Andris, Bensaccount, El C, Anville

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Quantum mechanics, philosophy and controversy Quantum mechanics has had many detractors including Albert Einstein and Erwin Schroedinger. Quantum mechanics has had a profound affect on philosophy. Determinism is a philosophical view that the universe is governed by determinism if given a specific state of the universe at a specific time, the future state of the universe is fixed as a matter of natural law. The philosophy of determinism was derived from science, from Newton’s laws, and pre-Newtonian physics, in that the ability to predict future outcomes in the universe (such as future position of planets) was made possible by science. Quantum mechanics took away predictability and therefore was a blow to philosophy. However, the main founder of quantum mechanics, Niels Bohr, is said to have a philosophy of determinism similar to the rationalization by Immanuel Kant. This article will attempt, without going into religious implications which are personal matters, to explain the position of many physicists on quantum mechanics and the profound effect that quantum mechanics has had on philosophy.

274 http://plato.stanford.edu/entries/qm/ 275 http://www.physicstoday.org/pt/vol-54/iss-2/p11.html

276 http://www.csicop.org/si/9701/quantum-quackery.html 277 http://www.crank.net/quantum.html 278 http://www.hinduism.co.za/hinduism.htm

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Philosophical determinism

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The 18th century saw many advances in the domain of science. After Newton, most scientists agreed on the presupposition that the universe is governed by (natural) laws that can be discovered and formalized by means of scientific observation and experiment. This position is known as determinism. However, while determinism was the fundamental presupposition of post-Newtonian physics, it quickly lead philosophers to a tremendous problem: if the universe, and thus the entire world is governed by natural law, then that means that human beings are also governed by natural law in their own actions. In other words, it means that there is no such thing as human freedom. If it is accepted that everything in the world is governed by natural law, then we must also accept that it is not possible for us to will our own actions as free individuals; rather, they must be determined by universal laws of nature. Conversely, if it is accepted that human beings do have free will, then we must accept that the world is not entirely governed by natural law. However, if the world is not entirely governed by natural law, then the task of science is rendered impossible: if the task of science is to discover and formalize the laws of nature, then what task is left for science if it has been decided that nature is not entirely governed by laws? Thus, there are extremely compelling reasons to want to accept both free will and determinism. However, the two seem totally irreconcilable. Immanuel Kant whose work dates towards the end of the 18th Century, attempted to reconcile the seemingly incompatible schools of thought known as empiricism (e.g., David Hume) and rationalism (e.g., René Descartes). According to the empiricists, the only possible knowledge of the world is the knowledge that can be obtained by means of perception (inductive reasoning). Thus, for the empiricists concepts are abstractions that we derive by mentally comparing several different perceptions and noting some quality shared by all of them: for example, we see a fire engine, a rubber ball, and a dress, we perceive some quality that is shared by these different objects, and we abstract this quality from the objects themselves in order to arrive at the concept of the color red. For Hume and the empiricists, this means that our concepts, such as the concept of cause and effect, are not actually legitimate properties of the world, but are rather mental constructs that we produce from repeated observation. Since we can never actually perceive cause and effect (because it is not an object, but rather a relation), we can never obtain certain knowledge of whether it actually exists. In other words, since we can’t perceive it, we can never be totally sure that we are not just imagining it. For the rationalists, on the other hand, the situation is entirely the reverse, and the only certain knowledge is the knowledge that we derive by means of pure logic (deductive reasoning). The privileged model of certainty for the rationalists is mathematics. For the Quantum mechanics, philosophy and controversy

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rationalists, we can never be certain of any knowledge derived from perception, since we are capable of perceiving objects that are almost certainly false in dreams. In other words, since there is no difference between an object perceived while we are awake and the same object perceived during a dream, we can never derive certainty from perception. Mathematics does not require any perceivable object in order to arrive at its proofs, because it works in purely logical relations between concepts. Kant approached this problem most famously in his major work of epistemology, The Critique of Pure Reason (Kritik der reinen Vernunft, 1781). In order to reconcile these disparate views, Kant found it necessary to split the world into two completely separate aspects: • •

1. The world as appearance – that is, as it appears to us in our perceptions. 2. The world as a thing-in-itself – in other words, independent of all human perception.

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For Kant, all scientific knowledge (which at that time included philosophy too) refers to the world of/as appearance. The world as a thing-in-itself is, according to Kant, not a "possible object of experience", and all human (conceptual) knowledge refers to the world as we experience it. By splitting the world in the world in this way, Kant was able to offer compelling solutions to some of the most historically difficult questions faced by philosophy. Most importanly, it allowed him to offer a solution to the question of free will versus determinism. Kant argued that, in the world of appearances, determinism is the rule. In other words, according to Kant, in the world of appearances, there is no object that is not governed by the laws of nature. However, this doesn’t preclude the possibility that human freedom exists, with the proviso that it exists as a thingin-itself. In other words, for Kant, human freedom is not a possible object of experience, but that doesn’t make it any less real. Even though we can never perceive human freedom, the mere fact that we can will actions for which we can find no cause in the world of appearances is enough to make human freedom a reasonable assumption. It must remain an assumption, since we cannot have knowledge of something that is not an object of experience, but it is an assumption worth making, since it is what makes morality possible. Thus, Kant was able to offer a coherent answer to the question of how it is possible for both free will and determinism to apply to the same world, but in order to do so, he found it necessary to split the world into these two totally separate aspects. This method made possible tremendous advances in philosophical thought throughout the late eighteenth and early nineteenth centuries. However, it also set strict limits on human knowledge. For Kant, we cannot ’know’

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freedom or any other thing-in-itself in a rigorously scientific way. Rather, freedom is more like a necessary assumption. We can only ’know’ objects as they appear to us – our knowledge is only knowledge of the world of appearances. Any claim to have knowledge of objects as they are in themselves is an illegitimate use of the faculties of reason and understanding. This is why, for Kant, it is impossible to prove the existence of God, of the soul, or of human freedom: none of these are possible objects of experience. This is not just a historically specific problem that might be overcome as science advances and we learn more and more; it is constitutive of all human knowledge. In other words, even while Kant enabled great leaps forward in philosophical thought, he did so by introducing a concept of human knowledge as essentially limited, and essentially fallible.

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Although twentieth century scientists left the question of human will to the philosophers, scientists themselves felt very firmly grounded in the idea that science could make predictions according to Newton’s laws with regard to objects in nature. Therefore, determinism was still one of the fundamental axioms of scientific thought. Even with Einstein’s theory of relativity, determinism was not seriously challenged. This was all about to change.

Consequences of the uncertainty principle The Uncertainty Principle is a main theory in the physical science of quantum mechanics that explains the universe at atomic and subatomic scales. The Uncertainty Principle was developed as an answer to the question: How does one measure the location of an electron around a nucleus? In March 1926, working in Niels Bohr’s institute, Werner Heisenberg formulated the principle of uncertainty thereby laying the foundation of what became known as the Copenhagen interpretation of quantum mechanics. Heisenberg had been studying the papers of Paul Dirac and Jordan. Heisenberg discovered a problem with measurement of basic variables in the equations. His analysis showed that uncertainties, or imprecisions, always turned up if one tried to measure the position and the momentum of a particle at the same time. Heisenberg concluded that these uncertainties or imprecisions in the measurements were not the fault of the experimenter, but fundamental in nature and inherent in quantum mechanics. The term Copenhagen interpretation of quantum mechanics was often used interchangeably with and as a synonym for Heisenberg’s Uncertainty Principle by detractors who believed in fate and determinism and saw the common features of the Bohr-Heisenberg theories as a threat. Within the widely but not universally accepted Copenhagen interpretation of quantum mechanics (i.e. it Quantum mechanics, philosophy and controversy

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was not accepted by Einstein or other physicists such as Alfred Lande), the uncertainty principle is taken to mean that on an elementary level, the physical universe does not exist in a deterministic form, but rather as a collection of probabilities, or potentials. For example, the pattern (probability distribution) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. The Copenhagen interpretation holds that it cannot be predicted by any method, not even with theoretically infinitely precise measurements. Albert Einstein was not happy with the uncertainty principle, and he challenged Niels Bohr and Werner Heisenberg with a famous thought experiment (See the Bohr-Einstein debates for more details).

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It is this interpretation that Einstein was questioning when he said "I cannot believe that God would choose to play dice with the universe." Bohr, who was one of the authors of the Copenhagen interpretation responded, "Einstein, don’t tell God what to do." Niels Bohr himself acknowledged that quantum mechanics and the uncertainty principle were counter-intuitive when he stated, "Anyone who is not shocked by quantum theory has not understood a single word." The basic debate between Einstein and Bohr (including Heisenberg’s Uncertainty Principle) was that Einstein was in essence saying: "Of course, we can know where something is; we can know the position of a moving particle if we know every possible detail, and thereby by extension, we can predict where it will go." Bohr and Heisenberg were saying the opposite: "There is no way to know where a moving particle is ever even given every possible detail, and thereby by extension, we can never predict where it will go." Einstein was convinced that this interpretation was in error. His reasoning was that all previously known probability distributions arose from deterministic events. The distribution of a flipped coin or a rolled dice can be described with a probability distribution (50% heads, 50% tails). But this does not mean that their physical motions are unpredictable. Ordinary mechanics can be used to calculate exactly how each coin will land, if the forces acting on it are known. And the heads/tails distribution will still line up with the probability distribution (given random initial forces). Einstein was not adverse to quantum mechanics as a whole, but specifically with the uncertainty principle itself. As to other basic principles of quantum mechanics, Einstein whose own general relativity was firmly rooted in field theory said: "The de Broglie-Schrödinger method, which has in a certain sense the character of a field theory, does indeed deduce the existence of only discrete states, Quantum mechanics, philosophy and controversy

482 in surprising agreement with empirical facts. It does so on the basis of differential equations applying a kind of resonance argument." (Albert Einstein, On Quantum Physics, 1954)

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Niels Bohr himself appears to have taken Kant’s view that there are two aspects of reality, what we can say about and what it is, when Bohr said: "There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature."

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In other words, there may be no quantum leaping of electrons as predicted by quantum mechanics, there may be a definite position of a particle contrary to the uncertainty principle, but the only way we mere humans can describe mathematically in a useful way what we see in the real world is to use Quantum Mechanics. This is because theories are simple models of complex systems. The universe is too complex to describe without simple models. Because quantum mechanics is useful and continues to provide sound mathematics when tested, it is a mainstream theory of the universe at the quantum level. Niels Bohr’s comment was saying that he himself believed that in all probability the natural world was different than the explanation given by quantum mechanics which is similar to Kant’s view. Heisenberg wrote a conversation between himself and Einstein further debating their different viewpoints as follows: •

Heisenberg: "One cannot observe the electron orbits inside the atom. [...] but since it is reasonable to consider only those quantities in a theory that can be measured, it seemed natural to me to introduce them only as entities, as representatives of electron orbits, so to speak." • Einstein: "But you don’t seriously believe that only observable quantities should be considered in a physical theory?" • "I thought this was the very idea that your relativity theory is based on?" Heisenberg asked in surprise. • "Perhaps I used this kind of reasoning," replied Einstein, "but it is nonsense nevertheless. [...] In reality the opposite is true: only the theory decides what can be observed."– (translated from "Der Teil und das Ganze" by W. Heisenberg)

Werner Heisenberg himself said, "‘I myself . . . only came to believe in the uncertainty relations after many pangs of conscience. . . ." He knew what he was saying didn’t make sense, but it helped measurements at quantum levels so much, he did it anyway. Richard Feynman, another major contributor to quantum theory said, "We have always had a great deal of difficulty understanding the world view that quantum mechanics represents. At least I do, Quantum mechanics, philosophy and controversy

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because I’m an old enough man that I haven’t got to the point that this stuff is obvious to me. Okay, I still get nervous with it.... You know how it always is, every new idea, it takes a generation or two until it becomes obvious that there’s no real problem. I cannot define the real problem, therefore I suspect there’s no real problem, but I’m not sure there’s no real problem." He meant that he understood quantum mechanics very well, but that in 1982 some 50 years later, he still couldn’t reconcile himself to it. That is why Einstein spent the entire rest of his life trying to disprove the Uncertainty Principle however there is no other theory to replace quantum mechanics that is so successful at the quantum level.

Erwin Schrödinger controversy

Later in life, the inventor of wave mechanics of quantum theory, Erwin Schrödinger began a campaign against the generally accepted quantum description of wave-particle duality and tried to propose a theory in terms of waves only. This led him into controversy with other leading physicists since he rejected mainstream quantum mechanical theory.

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Sometimes Schrödinger’s wave equation is erroneously said to give the exact location of the electron and doesn’t need the uncertainty principle. Actually Schrödinger’s wave equation explains the exact location of a wave. A wave not being a point particle has a natural integral probability distribution as a widespread disturbance. So Schrödinger’s equation does in a sense give the exact location of the electron, however, only in its state of being a widespread disturbance, a wave. Schrödinger later in life tried to develop a theory that would show the electron is only a wave and not a particle at all and that fundamentally the atom is only a wave, thus making the uncertainty principle obsolete as it was only needed to show the uncertainty of the particle-like position of the electron and other subatomic particles. This was not a new theory. The idea that the atom could be explained mathematically as a wave was introduced in 1922 by Charles Galton Darwin, a physicist, in his paper at 279 . However, the consequences would be that planets, galaxies, human beings and atoms are completely described as waves of physical disturbance, some waves being massless as in the case of light and some waves being massive as the case of the subatomic particles of the atom. Einstein rejected such a theory when Schrödinger proposed it to him. Einstein followed intuitive lines of thinking which is why he rejected Heisenberg’s uncertainty principle. There are difficulties in describing a single wave as having two polarities if the atom were a single wave and the idea of waves producing spin and magnetic moment seem hard to overcome, but a solution was proposed in 1927 by Arthur Ruark

279 http://www.pubmedcentral.gov/picrender.fcgi?tool=pmcentrez&blobtype=pdf&artid=1085216

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in THE IMPULSE MOMENT OF THE LIGHT QUANTUM at 280. However, it is accepted that quantum mechanics teaches that solid objects only appear solid due to forces. The atom is mostly space and it is the negative charge of the electrons that keep atoms from collapsing into each other. Schrödinger became so disenchanted with the idea of wave-particle duality that he was known to have said concerning it:

"Let me say at the outset, that in this discourse, I am opposing not a few special statements of quantum physics held today (1950s), I am opposing as it were the whole of it, I am opposing its basic views that have been shaped 25 years ago, when Max Born put forward his probability interpretation, which was accepted by almost everybody." (Schrödinger Erwin, The Interpretation of Quantum Physics. Ox Bow Press, Woodbridge, CN, 1995). "I don’t like it, and I’m sorry I ever had anything to do with it." (Erwin Schrodinger talking about Quantum Physics)

Comments by other quantum physicists

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"This is the third of four lectures on a rather difficult subject – the theory of quantum electrodynamics – and since there are obviously more people here tonight than there were before, some of you haven’t heard the other two lectures and will find this lecture incomprehensible. Those of you who have heard the other two lectures will also find this lecture incomprehensible, but you know that that’s all right: as I explained in the first lecture, the way we have to describe Nature is generally incomprehensible to us." Richard P. Feynman, QED, The Strange Theory of Light and Matter, p. 77 [Princeton University Press, 1985]

"The discomfort that I feel is associated with the fact that the observed perfect quantum correlations seem to demand something like the "genetic" hypothesis. For me, it is so reasonable to assume that the photons in those experiments carry with them programs, which have been correlated in advance, telling them how to behave. This is so rational that I think that when Einstein saw that, and the others refused to see it, he was the rational man. The other people, although history has justified them, were burying their heads in the sand. I feel that Einstein’s intellectual superiority over Bohr, in this instance, was enormous; a vast gulf between the man who saw clearly what was needed, and the obscurantist. So for me, it is a pity that Einstein’s idea doesn’t work. The reasonable thing just doesn’t work." John Stewart Bell (1928-1990), author of "Bell’s Theorem" (or "Bell’s Inequality"), quoted in Quantum Profiles, by Jeremy Bernstein [Princeton University Press, 1991, p. 84]

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External links

The Center of Quantum Philosophy 281

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"Thus the last and most successful creation of theoretical physics, namely quantum mechanics (QM), differs fundamentally from both Newton’s mechanics, and Maxwell’s e-m field. For the quantities which figure in Quantum Physics’ laws make no claim to describe physical reality itself, but only probabilities of the occurrence of a physical reality that we have in view. · · · I cannot but confess that I attach only a transitory importance to this interpretation. I still believe in the possibility of a model of reality - that is to say, of a theory which represents things themselves and not merely the probability of their occurrence. On the other hand, it seems to me certain that we must give up the idea of complete localization of the particle in a theoretical model. This seems to me the permanent upshot of Heisenberg’s principle of uncertainty." (Albert Einstein, On Quantum Physics, 1954)

Source: http://en.wikipedia.org/wiki/Quantum_mechanics%2C_philosophy_and_controversy

Quantum mineralogy

Quantum mineralogy is the branch of physics and chemistry that uses fundamental (quantum-level) properties of particular elements to describe the macroscopic properties of minerals containing those elements.

References •

Quantum mineralogy by Bryan C. Chakoumakos in McGraw-Hill Encyclopedia of Science & Technology Online 282

Source: http://en.wikipedia.org/wiki/Quantum_mineralogy

281 http://www.quantumphil.org/ 282 http://www.accessscience.com/Encyclopedia/5/56/Est_562950_frameset.html?doi

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In physics, a quantum phase transition (QPT) is a phase transition between different quantum phases (phases of matter at zero temperature). Contrary to classical phase transitions, quantum phase transitions can be only be accessed by varying a physical parameter - such as magnetic field or pressure - at absolute zero temperature. The transition describes an abrupt change in the ground state of a many-body system due to its quantum fluctuations. Such quantum phase transitions can be first-order phase transition or continuous.

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To understand quantum phase transitions, it is useful to contrast them to classical phase transitions (CPT) (also called thermal phase transitions). A CPT describes a discontinuity in the thermodynamic properties of a system. It signals a reorganization of the particles; A canonical example is the freezing transition of water describing the transition between liquid and ice. The classical phase transitions are driven by a competition between the energy of a system and the entropy of its thermal fluctuations. A classical system does not have entropy at zero temperature and therefore no phase transition can occur. In contrast, even at zero temperature a quantum-mechanical system has quantum fluctuations and therefore can still support phase transitions. As a physical parameter is varied, quantum fluctuations can drive a phase transition into a different phase of matter. A canonical quantum phase transition is the wellstudied superconductor/insulator transition in disordered thin films which separates two quantum phases having different symmetries. Quantum magnets provide another example of QPT.

Source: http://en.wikipedia.org/wiki/Quantum_phase_transition

Principal Authors: Mikkalai, Taxman, Lankiveil, Commander Keane, Folajimi

Quantum solid

In physics, a quantum solid is a type of solid that is "intrinsically restless", in the sense that atoms continuously vibrate about their position and exchange places even at the absolute zero of temperature. The archetypal quantum solid is low density solid helium.

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References E.Polturak and N.Gov, Inside a quantum solid, Contemporary Physics 44, No.2, 145-151, (2003).

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•

Source: http://en.wikipedia.org/wiki/Quantum_solid

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Quantum state

In quantum mechanics, a quantum state is any possible state in which a quantum mechanical system can be. A fully specified quantum state can be described by a state vector, a wavefunction, or a complete set of quantum numbers for a specific system. A partially known quantum state, such as an ensemble with some quantum numbers fixed, can be described by a density operator.

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488

Bra-ket notation

Basis states

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Paul Dirac invented a powerful and intuitive notation to describe quantum states, known as bra-ket notation. For instance, one can refer to an |excited atom> or to |↑i for a spin-up particle, hiding the underlying complexity of the mathematical description, which is revealed when the state is projected onto a coordinate basis. For instance, the simple notation |1s> describes the first hydrogen atom bound state, but becomes a complicated function in terms of Laguerre polynomials and spherical harmonics when projected onto the basis of position vectors |r>. The resulting expression Ψ(r)=, which is known as the wavefunction, is a special representation of the quantum state, namely, its projection into position space. Other representations, such as projection into momentum space, are possible. The various representations are simply different expressions of a single physical quantum state.

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Any quantum state |ψi can be expressed in terms of a sum of basis states (also called basis kets) |ki i in the form P |ψi = i ci |ki i

where ci are the coefficients representing the probability amplitude, such that the absolute square of the probability amplitude, |ci |2 is the probability of a measurement in terms of the basis states yielding the state |ki i. The normalization condition mandates that the total sum of probabilities is equal to one, P 2 i |ci | = 1. The simplest understanding of basis states is obtained by examining the quantum harmonic oscillator. In this system, each basis state |ni has an energy
En = ~ω n + 12 . The set of basis states can be extracted using a construction operator a ˆ† and a destruction operator a ˆ in what is called the ladder operator method.

Superposition of states

If a quantum mechanical state |ψi can be reached by more than one path, then |ψi is said to be a linear superposition of states. In the case of two paths, if the states after passing through path α and path β are

|αi =

√1 |0i 2

+

√1 |1i, 2

√1 |0i 2

−

√1 |1i, 2

and

|βi =

Quantum state

489

then |ψi is defined as the normalized linear sum of these two states. If the two paths are equally likely, this yields √1 ( √1 |0i 2 2

+

√1 |1i) 2

+

FT

|ψi = √12 |αi + √12 |βi = √1 ( √1 |0i − √1 |1i) = |0i. 2 2 2

Note that in the states |αi and |βi, the two states |0i and |1i each have a prob1 ability of 2 , as obtained by the absolute square of the probability amplitudes, 1

1

which are √2 and ± √2 . In a superposition, it is the probability amplitudes which add, and not the probabilities themselves. The pattern which results from a superposition is often called an interference pattern. In the above case, |0i is said to constructively interfere, and |1i is said to destructively interfere. For more about superposition of states, see the double-slit experiment.

Pure and mixed states

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A pure quantum state is a state which can be described by a single ket vector, or as a sum of basis states. A mixed quantum state is a statistical distribution of pure states. The expectation value hai of a measurement A on a pure quantum state is given by P P P hai = hψ|A|ψi = i ai hψ|αi ihαi |ψi = i ai |hαi |ψi|2 = i ai P (αi )

where |αi i are basis kets for the operator A, and P (αi ) is the probability of |ψi being measured in state |αi i. In order to describe a statistical distribution of pure states, or mixed state, the density operator (or density matrix), ρ, is used. This extends quantum mechanics to quantum statistical mechanics. The density operator is defined as P ρ = s ps |ψs ihψs |

where ps is the fraction of each ensemble in pure state |ψs i. The ensemble average of a measurement A on a mixed state is given by P P P [A] = hAi = s ps hψs |A|ψs i = s i ps ai |hαi |ψs i|2 = tr(ρA) where it is important to note that two types of averaging are occurring, one being a quantum average over the basis kets of the pure states, and the other being a statistical average over the ensemble of pure states. Quantum state

490

Mathematical formulation

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For a mathematical discussion on states as functionals, see GNS construction. There, the same objects are described in a C*-algebraic context.

See also • • • • • • •

→Quantum harmonic oscillator →Bra-ket notation Orthonormal basis →Wavefunction →Probability amplitude Density operator Qubit

Source: http://en.wikipedia.org/wiki/Quantum_state

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Principal Authors: Cortonin, Fresheneesz, Laussy, Bkalafut, CSTAR

Quantum statistical mechanics

Quantum statistical mechanics is the study of statistical ensembles of quantum mechanical systems. A statistical ensemble is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the →Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. One such formalism is provided by quantum logic.

Expectation

From classical probability theory we know that the expectation of a random variable X is completely determined by its distribution D X by

Exp(X) =

R

R

λ d DX (λ)

assuming, of course that the random variable is integrable or the random variable is non-negative. Similarly, let A be an observable of a quantum mechanical system. A is given by a densely defined self-adjoint operator on H. The spectral measure ofRA defined by

EA (U ) =

U

λd E(λ),

Quantum statistical mechanics

491

DA (U ) = Tr(EA (U )S).

FT

uniquely determines A and conversely, is uniquely determined by A. E A is a boolean homomorphism from the Borel subsets of R into the lattice Q of selfadjoint projections of H. In analogy with probability theory, given a state S, we introduce the distribution of A under S which is the probability measure defined on the Borel subsets of R by

Similarly, the expected value of A is defined in terms of the probability distribution D A by

Exp(A) =

R

R

λ d DA (λ).

Note that this expectation is relative to the mixed state S which is used in the definition of D A .

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Remark. For technical reasons, one needs to consider separately the positive and negative parts of A defined by the Borel functional calculus for unbounded operators. One can easily show:

Exp(A) = Tr(AS) = Tr(SA).

Note that if S is a pure state corresponding to the vector ψ,

Exp(A) = hψ|A|ψi.

Von Neumann entropy

Of particular significance for describing randomness of a state is the von Neumann entropy of S formally defined by

H(S) = − Tr(S log2 S) .

Actually the operator S log 2 S is not necessarily trace-class. However, if S is a non-negative self-adjoint operator not of trace class we define Tr(S ) = +∞. Also note that any density operator S can be diagonalized, that it can be represented in some orthonormal basis by a (possibly infinite) matrix of the form

Quantum statistical mechanics

0 λ2

0 0

··· ··· ··· ··· ···

0

and we define P

H(S) = −

i

λn ···

 ··· · · ·    · · ·

λi log2 λi .

FT

492

 λ1 0    0

The convention is that 0 log2 0 = 0, since an event with probability zero should not contribute to the entropy. This value is an extended real number (that is in [0, ∞]) and this is clearly a unitary invariant of S.

Remark. It is indeed possible that H(S ) = +∞ for some density operator S. In fact Tbe the diagonal matrix  1 2(log2 2)2

0

1 3(log2 3)2

0

··· ··· ··· ··· ···

0 0

··· · · ·     · · ·

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   T =  

0

0

1 n(log2 n)2

···

T is non-negative trace class and one can show T log 2 T is not trace-class. Theorem. Entropy is a unitary invariant.

In analogy with classical entropy (notice the similarity in the definitions), H(S ) measures the amount of randomness in the state S. The more dispersed the eigenvalues are, the larger the system entropy. For a system in which the space H is finite-dimensional, entropy is maximized for the states S which in diagonal form   have the representation 1 n

0

0   0

0

1 n

··· ... ··· ···

0 0   1 n

For such an S, H(S ) = log 2 n. The state S is called the maximally mixed state.

Recall that a pure state is one the form S = |ψihψ|,

for ψ a vector of norm 1.

Quantum statistical mechanics

493 Theorem. H(S ) = 0 if and only if S is a pure state. For S is a pure state if and only if its diagonal form has exactly one non-zero entry which is a 1.

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Entropy can be used as a measure of quantum entanglement.

Gibbs canonical ensemble

Consider an ensemble of systems described by a Hamiltonian H with average energy E. If H has pure-point spectrum and the eigenvalues En of H go to + ∞ sufficiently fast, e -r H will be a non-negative trace-class operator for ever positive r. The Gibbs canonical ensemble is described by the state

S=

e−βH Tr(e−βH )

where β is such that the ensemble average of energy satisfies

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Tr(SH) = E ,and

Tr(e−βH ) =

P

n

e−βEn

is the quantum mechanical version of the canonical partition function. The probability that a system chosen at random from the ensemble will be in a state corresponding to energy eigenvalue Em is m Pe−βE−βE . n n

e

Under certain conditions the Gibbs canonical ensemble maximizes the von Neumann entropy of the state subject to the energy conservation requirement.

References •

J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press, 1955.

•

F. Reif, Statistical and Thermal Physics, McGraw-Hill, 1985.

Source: http://en.wikipedia.org/wiki/Quantum_statistical_mechanics Principal Authors: CSTAR, Mct mht, Sietse Snel, Phys, E2mb0t

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Quantum superposition

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Quantum superposition is the application of the superposition principle to quantum mechanics. The superposition principle is the addition of the amplitudes of waves from interference. In quantum mechanics it is the amplitudes of wavefunctions, or state vectors, that add. It occurs when an object simultaneously "possesses" two or more values for an observable quantity (e.g. the position or energy of a particle).

More specifically, in quantum mechanics, any observable quantity corresponds to an eigenstate of a Hermitian linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the projection postulate states that the state will be randomly collapsed onto one of the values in the superposition (with a probability proportional to the square of the amplitude of that eigenstate in the linear combination).

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The question naturally arose as to why "real" (macroscopic, Newtonian) objects and events do not seem to display quantum mechanical features such as superposition. In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as →Schrödinger’s cat, which highlighted the dissonance between quantum mechanics and Newtonian physics. In fact, quantum superposition does result in many directly observable effects, such as interference peaks from an electron wave in a double-slit experiment. If two observables correspond to noncommutative operators, they obey an uncertainty principle and a distinct state of one observable corresponds to a superposition of many states for the other observable.

See also • • •

→Wave packet Quantum computation →Penrose Interpretation

Source: http://en.wikipedia.org/wiki/Quantum_superposition

Principal Authors: Reddi, Stevenj, Stevertigo, PopUpPirate, D.328, El C, John187, Retodon8

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Quantum Theory Parallels to Consciousness Can parallels to certain generic concepts and logical relationships of quantum theory: •

help theoretically characterize general features of conscious experience, in particular, nonlocal consciousness?

•

transcend the scope of analogy and metaphor to enhance quantitative prediction and provide guidelines for experimental design?

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Ordering in random physical processes may be attached to "mindful" meditation via lucid empathy and participatory interest making otherwise stochastic processes vulnerable to anomalous statistical behavior. (See, for example, the experiments on "Field Consciousness" discussed in Radin, 1997, Chapter 10, pp. 157-174).

During so-called "unusual", "expanded" or "transcendental" states of consciousness, certain macromolecules or entire cell ensembles in the brain may be mentally decoupled from their thermodynamic environment, thus enabling them to exist in quantum states. By their inherent "long-wavelength" nature, quantum states are "spread out" beyond their immediate locus. Accordingly, quantum mind-brain states would be capable of entanglement with similar states in the mind-brains of other individuals or with labile (stochastic) states of matter in the remote environment (Schmid, 2000a), (Schmid, 2000c). Labile states of matter are inherent to the stochastic behavior of inanimate matter, the psychomotorical lability of animate beings, the neuropsychoemotional ambivalence of cognitive beings as well as the ill/pathological/disturbed states of living tissue in general. Such nonlocal states of consciousness may include prayer, meditation, trance, and dreaming, as well as mystical, out-of-body, and near-death experiences and, especially, pathological mental states induced by drugs or psychosis. In fact, "faith", in the sense of an open ("mindful") trusting belief in an inner connection to other people as well as to the world in general is the core of spirituality: meaningfulness and belonging. Intention and decision involve precise cognitive-emotional processes which may correspond to short-wavelength wavefunctions which are sharply localized. By contrast, "attachment" may reduce potential barriers around certain macromolecules in the mind-brain, thus relaxing tightly centered (classical) Quantum Theory Parallels to Consciousness

496

FT

molecular couplings to their biophysical surroundings and leading to the mental quality of "clarity". Subjective "clarity" (lucid empathy, participatory interest and vivid mental imagery void of expectation) may enable such thermodynamically decoupled states to selectively entangle with others. The quantum physicist Niels Bohr once said that mental clarity is complementary to mental precision. How these ideas allow for such phenomena as precognition, telepathy - encompassing clairvoyance, distant anticipation, remote perception, synchronicity, and the like - psychic healing, and psychokinesis between the members of a fortune teller/event-, percipient/agent-, healer/patient-, or influencer/object-pair is discussed.

References

Braud, W. G., & Schlitz, M. J. (1991). Consciousness Interactions with remote biological systems: Anomalous intentionality effects. Subtle Energies, 2(1), 1-46.

•

Jahn, R. G., & Dunne, B. J. (1986). On the quantum mechanics of consciousness, with application to anomalous phenomena. Foundations of Physics, 16(8), 721-772.

•

Radin, D. I. (1997). The Conscious Universe: The Scientific Truth of Psychic Phenomena. San Francisco: HarperEdge.

•

Schmid, G. B. (2000a). Das Geheimnis psychogener Todesfälle. intra Psychologie und Gesellschaft, 45(September), 14-23.

•

Schmid, G. B. (2000b). Tod durch Vorstellungskraft: Das Geheimnis psychogener Todesfälle (1. ed.). Wien-New York: Springer-Verlag.

•

Schmid, G. B. (2000c). Tod durch Vorstellungskraft? Die geheimnisvolle Macht der Gedanken. An der Urania 17 / D-10787 Berlin: URANIA.

•

Tittel, W., Brendel, J., Gisin, B., Herzog, T., Zbinden, H., & Gisin, N. (1998). Experimental demonstration of quantum-correlations over more than 10 kilometers. Physical Review A, 57(5), 3229-3232.

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•

Source: http://en.wikipedia.org/wiki/Quantum_Theory_Parallels_to_Consciousness Principal Authors: Cholmes75, Omphaloscope, Harlanpaine

Quantum Theory Parallels to Consciousness

497

Quantum tomography

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Quantum tomography or quantum state tomography is the process of reconstructing the quantum state (density matrix) for a source of quantum systems by measurements on the systems coming from the source. To be able to uniquely identify the state, the measurements must be tomographically complete, that is the measured operators must form an operator basis on the →Hilbert space of the system. In quantum process tomography on the other hand, known quantum states are used to probe a quantum process to find out how the process can be described.

Source: http://en.wikipedia.org/wiki/Quantum_tomography Principal Authors: V79, Remuel, Conscious

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Quantum tunnelling

Quantum tunnelling (or tunneling) is the quantum-mechanical effect of transitioning through a classically-forbidden energy state. It can be generalized to other types of classically-forbidden transitions as well. Consider rolling a ball up a hill. If the ball is not given enough velocity, then it will not roll over the hill. This scenario makes sense from the standpoint of classical mechanics, but is an inapplicable restriction in quantum mechanics simply because quantum mechanical objects do not behave like classical objects such as balls. On a quantum scale, objects exhibit wavelike behavior. For a quantum particle moving against a potential energy "hill", the wave function describing the particle can extend to the other side of the hill. This wave represents the probability of finding the particle in a certain location, meaning that the particle has the possibility of being detected on the other side of the hill. This behavior is called tunnelling; it is as if the particle has ’dug’ through the potential hill. As this is a quantum and non-classical effect, it can generally only be seen in nanoscopic phenomena — where the wave nature of particles is more pronounced. It should be noted that availability of states is necessary for tunneling to occur. In the above example, the quantum mechanical ball will not appear inside the hill because there is no available "space" for it to exist, but it can tunnel to Quantum tunnelling

498

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the other side of the hill, where there is free space. Analogously, a particle can tunnel through the barrier, but unless there are states available within the barrier, the particle can only tunnel to the other side of the barrier. The wavefunction describing a particle only expresses the probability of finding the particle at a location assuming a free state exists.

History and consequences

In the early 1900s, radioactive materials were known to have characteristic exponential decay rates or half lives. At the same time, radiation emissions were known to have certain characteristic energies. By 1928, George Gamow had solved the theory of the alpha decay of a nucleus via tunnelling. Classically, the particle is confined to the nucleus because of the high energy requirement to escape the very strong potential. Classically, it takes an enormous amount of energy to pull apart the nucleus. In quantum mechanics, however, there is a probability the particle can tunnel through the potential and escape. Gamow solved a model potential for the nucleus and derived a relationship between the half-life of the particle and the energy of the emission.

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Alpha decay via tunnelling was also solved concurrently by Ronald Gurney and Edward Condon. Shortly thereafter, both groups considered whether particles could also tunnel into the nucleus.

After attending a seminar by Gamow, Max Born recognized the generality of quantum-mechanical tunnelling. He realised that tunnelling phenomena was not restricted to nuclear physics, but was a general result of quantum mechanics that applies to many different systems. Today the theory of tunnelling is even applied to the early cosmology of the universe. Quantum tunnelling was later applied to other situations, such as the cold emission of electrons, and perhaps most importantly semiconductor and superconductor physics. Phenomena such as field emission, important to flash memory, are explained by quantum tunnelling. Tunnelling is a source of major current leakage in Very-large-scale integration (VLSI) electronics, and results in the substantial power drain and heating effects that plague high-speed and mobile technology. Another major application is in electron-tunnelling microscopes (see scanning tunnelling microscope) which can resolve objects that are too small to see using conventional microscopes. Electron tunnelling microscopes overcome the limiting effects of conventional microscopes (optical aberrations, wavelength limitations) by scanning the surface of an object with tunnelling electrons.

Quantum tunnelling

499

Semiclassical calculation Let us consider the time-independent →Schrödinger equation for one particle, in one dimension, under the influence of a hill potential V (x). 2

2

d2 Ψ(x) dx2

=

2m ~2

FT

~ d − 2m Ψ(x) + V (x)Ψ(x) = EΨ(x) dx2

(V (x) − E) Ψ(x)

Now let us recast the wave function Ψ(x) as the exponential of a function. Ψ(x) = eΦ(x) Φ00 (x) + Φ0 (x)2 =

2m ~2

(V (x) − E)

Now let us separate Φ0 (x) into real and imaginary parts. Φ0 (x) = A(x) + ıB(x)

2m ~2

(V (x) − E)

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A0 (x) + A(x)2 − B(x)2 = B 0 (x) − 2A(x)B(x) = 0

Next we want to take the semiclassical approximation to solve this. That means we expand each function as a power series in ~. From the equations we can already see that the power series must start with at least an order of ~−1 to satisfy the real part of the equation. But as we want a good classical limit, we also want to start with as high a power of Planck’s constant as possible. P i A(x) = ~1 ∞ i=0 ~ Ai (x) B(x) =

1 ~

P∞

i=0 ~

i B (x) i

The constraints on the lowest order terms are as follows. A0 (x)2 − B0 (x)2 = 2m (V (x) − E)

A0 (x)B0 (x) = 0

If the amplitude varies slowly as compared to the phase, we set A0 (x) = 0 and get

Quantum tunnelling

500 p B0 (x) = ± 2m (E − V (x))

ı

Ψ(x) ≈ C e √

dx

(E−V (x))+θ ~2 2m [4] 2 (E−V (x)) ~

FT

Which is obviously only valid when you have more energy than potential classical motion. After the same procedure on the next order of the expansion we get R p 2m

On the other hand, if the phase varies slowly as compared to the amplitude, we set B0 (x) = 0 and get p A0 (x) = ± 2m (V (x) − E)

Which is obviously only valid when you have more potential than energy tunnelling motion. Grinding out the next order of the expansion yields R p 2m R p 2m +

C+ e

dx

~2

(V (x)−E)

√

−

dx

+C− e (x)−E)

(V [4] 2m ~2

~2

(V (x)−E)

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Ψ(x) ≈

It is apparent from the denominator, that both these approximate solutions are bad near the classical turning point E = V (x). What we have are the approximate solutions away from the potential hill and beneath the potential hill. Away from the potential hill, the particle acts similarly to a free wave - the phase is oscillating. Beneath the potential hill, the particle undergoes exponential changes in amplitude. In a specific tunnelling problem, Rwe p might already suspect that the transition − dx 2m (V (x)−E) ~2 amplitude be proportional to e and thus the tunnelling be exponentially dampened by large deviations from classically permitable motion. But to be complete we must find the approximate solutions everywhere and match coefficients to make a global approximate solution. We have yet to approximate the solution near the classical turning points E = V (x).

Let us label a classical turning point x1 . Now because we are near E = V (x1 ), we can easily expand 2m (V (x) − E) in a power series. ~2 2m ~2

(V (x) − E) = U1 (x − x1 ) + U2 (x − x1 )2 + · · ·

Let us only approximate to linear order 2

d Ψ(x) dx2

2m ~2

(V (x) − E) = U1 (x − x1 )

= U1 (x − x1 )Ψ(x)

Quantum tunnelling

501 This differential equation looks deceptively simple. It takes some trickery to transform this into a Bessel equation. The solution is as follows.   √   √  √ 1 1 Ψ(x) = x − x1 C+ 1 J+ 1 23 U1 (x − x1 ) 3 + C− 1 J− 1 23 U1 (x − x1 ) 3 3

3

3

FT

3

Hopefully this solution should connect the far away and beneath solutions. Given the 2 coefficients on one side of the classical turning point, we should be able to determine the 2 coefficients on the other side of the classical turning point by using this local solution to connect them. We should be able to find a relationship between C, θ and C+ , C− . Fortunately the Bessel function solutions will asymptote into sine, cosine and exponential functions in the proper limits. The relationship can be found as follows.
C+ = 21 C cos θ − π4 π 4




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C− = −C sin θ −

Now we can easily construct global solutions and solve tunnelling problems. C outgoing 2 , for a particle tunnelling through a The transmission coefficient, C incoming single potential barrier is found to be R x2 p 2m −2

T = 

e

x1

−2

1+ 14 e

dx

~2

(V (x)−E)

R x2 p 2m x1

dx

~2

(V (x)−E)

2

Where x1 , x2 are the 2 classical turning points for the potential barrier. If we take the classical limit of all other physical parameters much larger than Planck’s constant, abbreviated as ~ → 0, we see that the transmission coefficient correctly goes to zero. This classical limit would have failed in the unphysical, but much simpler to solve, situation of a square potential.

Quantum tunnelling

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FT

502

See also • • • • •

→Josephson effect →SQUID Tunnel diode WKB approximation Scanning tunnelling microscope

References •

Razavy, Mohsen (2003). Quantum Theory of Tunneling. World Scientific. ISBN 9812380191. • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X. • Liboff, Richard L. (2002). Introductory Quantum Mechanics. AddisonWesley. ISBN 0805387145.

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External links

FT

Source: http://en.wikipedia.org/wiki/Quantum_tunnelling Principal Authors: Light current, C h fleming, Yguff88, GregRM, Folajimi, Mako098765, Nobbie, Shaddack, Asr1, Oleg Alexandrov

Quantum vibration

A quantum vibration is a vibration of a chemical bond in a molecule that must be treated quantum mechanically. The low-lying vibration energy states can be described as states of the quantum harmonic oscillator, and as higher vibrational states, near the bond disassociation limit, as Morse oscillators.

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A molecule can vibrate in many ways and each of them we can call a vibrational mode. The vibrations can be seen with IR spectroscopy for example. As a help to calculate the number of vibrational modes, it’s convenient to determine the number of degrees of freedom available to vibration. As a generalization any molecule consisting of N atoms will have 3N freedoms for translational motion. 3 degrees are translational freedoms. There are also 3 degrees of rotational freedom for non-linear molecules and 2 degrees of rotational freedom for linear molecules. This leaves 3N-5 degrees of vibrational freedom for linear molecules and 3N-6 degrees of vibrational freedom for non-linear molecules.

As an example H 2O, a non-linear molecule, will have 3*3-6 = 3 degrees of vibrational freedoms, or modes.

Source: http://en.wikipedia.org/wiki/Quantum_vibration

Principal Authors: Wolf530, Salsb, Cypa, Grendelkhan, Conscious

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Quantum well

Fabrication

FT

A quantum well is a potential well that confines particles, which were originally free to move in three dimensions, to two dimensions, forcing them to occupy a planar region. The effects of quantum confinement take place when the quantum well thickness becomes comparable at the de Broglie wavelength of the carriers (generally electrons and holes), leading to energy levels called "energy subbands", i.e., the carriers can only have discrete energy values.

Quantum wells are formed in semiconductors by having a material, like gallium arsenide sandwiched between two layers of a material with a wider bandgap, like aluminium arsenide. These structures can be grown by molecular beam epitaxy or chemical vapor deposition with control of the layer thickness down to monolayers.

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Applications

Because of their quasi-two dimensional nature, electrons in quantum wells have a sharper density of states than bulk materials. As a result quantum wells are in wide use in diode lasers. They are also used to make HEMTs (High Electron Mobility Transistors), which are used in low-noise electronics By doping either the well itself, or preferably, the barrier of a quantum well with donor impurities, a two-dimensional electron gas (abbreviated 2DEG) can be formed. This quasi-two dimensional system has interesting properties at low temperature. One such property is the quantum Hall effect, seen at high magnetic fields. Acceptor dopants can lead to a two-dimensional hole gas.

See also • • • •

→Particle in a box Quantum wire Quantum dot Quantum-well intermixing (QWI)

Source: http://en.wikipedia.org/wiki/Quantum_well

Principal Authors: Jaraalbe, Tantalate, Rkuchta, DV8 2XL, Dobromila

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Quantum Zeno effect

FT

The quantum Zeno effect is a quantum mechanical phenomenon first described by George Sudarshan and Baidyanaith Misra of the University of Texas in 1977. It describes the situation that an unstable particle, if observed continuously, will never decay. This occurs because every measurement causes the wavefunction to "collapse" to a pure eigenstate of the measurement basis.

In general, the Zeno effect can be de ned as class of phenomena when a transition is suppressed by some interaction which allows the interpretation of the nal state in terms of "a transition has not yet occurred" or "a transition already occurred" ( 283). In quantum mechanics, such an interaction is called “measurement” because its result can be interpreted in terms of classical mechanics. Frequent measurement prohibits the transition. Various versions of the Zeno e ect fall into the de nition above.

DR A

Given a system in a state A, which is the eigenstate of some measurement operator. Say the system under free time evolution will decay with a certain probability into state B. If measurements are made periodically, with some finite interval between each one, at each measurement, the wavefunction collapses to an eigenstate of the measurement operator. Between the measurements, the system evolves away from this eigenstate into a superposition state of the states A and B. When the superposition state is measured, it will again collapse, either back into state A as in the first measurement, or away into state B. The probability that it will collapse back into the same state A is higher if the system has had less time to evolve away from it. In the limit as the time between measurements goes to zero, the probability of a collapse back to the original state A goes to one. Hence, the system doesn’t evolve from A to B. In reality, collapse of the wavefunction is not a discrete, instantaneous event. A measurement could be approximated by strongly coupling the quantum system to the noisy thermal environment for a brief period of time. The time it takes for the wavefunction to "collapse" is related to the decoherence time of the system when coupled to the environment. The stronger the coupling is, and the shorter the decoherence time, the faster it will collapse. So in the decoherence picture, the quantum Zeno effect corresponds to the limit where a quantum system is continuously coupled to the environment, and where that coupling is infinitely strong, and where the "environment" is an infinitely large source of thermal randomness.

283 http://annex.jsap.or.jp/OSJ/opticalreview/TOC-Lists/vol12/12e0363tx.htm

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506

FT

Experimentally, strong suppression of the evolution of a quantum system due to environmental coupling has been observed in a number of microscopic systems. One such experiment was performed in October 1989 by Itano, Heinzen, Bollinger and Wineland at NIST ( PDF 284). Approximately 5000 9Be + ions were stored in a cylindrical Penning trap and laser cooled to below 250mK. A resonant RF pulse was applied which, if applied alone, would cause the entire ground state population to migrate into an excited state. After the pulse was applied, the ions were monitored for photons emitted due to relaxation. The ion trap was then regularly "measured" by applying a sequence of ultraviolet pulses, during the RF pulse. As expected, the ultraviolet pulses suppressed the evolution of the system into the excited state. The results were in good agreement with theoretical models. The quantum Zeno effect takes its name from Zeno’s arrow paradox, which is the argument that since an arrow in flight does not move during any single instant, it couldn’t possibly be moving overall.

DR A

See also •

Interference

Source: http://en.wikipedia.org/wiki/Quantum_Zeno_effect

Principal Authors: Tim Starling, Seth Ilys, Domitori, Pjacobi, Gerd Breitenbach

Quasistability

Quasistability is the local stability of a system at a local minimum of a potential. The local minimum usually is not a global true minimum of the potential. The quasistable state may decay to a global minimum state via quantum mechanic effect.

Source: http://en.wikipedia.org/wiki/Quasistability

284 http://www.boulder.nist.gov/timefreq/general/pdf/858.pdf

Quasistability

507

QWiki

FT

Qwiki 285 is a quantum physics wiki devoted to the collective creation of content that is technical and useful to practicing scientists. The site is nominally centered around quantum physics, but all scientists are invited to contribute, including – but not limited to – computer scientists, control theorists, electrical engineers, and mathematicians. More specifically, this site is designed for people who post content to the arXiv and quant-ph.

Source: http://en.wikipedia.org/wiki/QWiki Principal Authors: Melaen, Mellery, JanusDC

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Range criterion

In quantum mechanics, in particular quantum information, the Range criterion is a necessary condition that a state must satisfy in order to be separable. In other words, it is a separability criterion.

The result

Consider a quantum mechanical system composed of n subsystems. The state space H of such a system is the tensor product of those of the subsystems, i.e. H = H1 ⊗ · · · ⊗ Hn . For simplicity we will assume throughout that all relevant state spaces are finite dimensional. The criterion reads as follows: If ρ is a separable mixed state acting on H, then the range of ρ is spanned by a set of product vectors.

Proof

P In general, if a matrix M is of the form M = i vi vi∗ , it is obvious that the range of M, Ran(M), is contained in the linear span of {vi }. On the other hand, we can also show vi lies in Ran(M), for all i. Assume w.l.o.g. i = 1. We can write M = v1 v1∗ + T , where T is Hermitian and positive semidefinite. There are two possibilities: 1) span{v1 } ⊂Ker(T). Clearly, in this case, v1 ∈ Ran(M).

285 http://’’’qwiki’’’.caltech.edu

Range criterion

508

FT

2) Notice 1) is true if and only if Ker(T) ⊥ ⊂ span{v1 }⊥ , where ⊥ denotes orthogonal compliment. By Hermiticity of T, this is the same as Ran(T) ⊂ span{v1 }⊥ . So if 1) does not hold, the intersection Ran(T) ∩ span{v1 } is nonempty, i.e. there exists some complex number α such that T w = αv1 . So M w = hw, v1 iv1 + T w = (hw, v1 i + α)v1 . Therefore v1 lies in Ran(M).

Thus Ran(M) coincides with the linear span of {vi }. The range criterion is a special case of this fact. A density matrix ρ acting on H is separable if and only if it can be written as P ∗ ⊗ · · · ⊗ ψ ψ∗ ρ = i ψ1,i ψ1,i n,i n,i

DR A

∗ is a (un-normalized) pure state on the j -th subsystem. This is where ψj,i ψj,i also P ∗ ⊗ · · · ⊗ ψ ∗ ). ρ = i (ψ1,i ⊗ · · · ⊗ ψn,i )(ψ1,i n,i

But this is exactly the same form as M from above, with the vectorial product state ψ1,i ⊗ · · · ⊗ ψn,i replacing vi . It then immediately follows that the range of ρ is the linear span of these product states. This proves the criterion.

References •

P. Horodecki, "Separability Criterion and Inseparable Mixed States with Positive Partial Transposition", Physics Letters A 232, (1997).

Source: http://en.wikipedia.org/wiki/Range_criterion

Relativistic particle

A relativistic particle is a particle moving with a speed close to the speed of light, such that effects of special relativity are important for the description of its behavior. Massless particles (e.g., photons) are always moving at the speed of light, therefore they are always relativistic. Massive particles are relativistic when their kinetic energy is comparable or greater than the energy mc2 corresponding to their rest mass. (This condition Relativistic particle

509

FT

implies that their speed is close to the speed of light.) Such relativistic particles are generated in particle accelerators, and are naturally occurring in cosmic radiation. In astrophysics, jets of relativistic plasma are produced by the centers of active galaxies and quasars. A charged relativistic particle crossing the interface of two media with different dielectric constants emits transition radiation. This is exploited in the transition radiation detectors of high-velocity particles. See also:

Special relativity Relativistic wave equations Lorentz factor Relativistic mass Relativistic plasma Relativistic jet Relativistic beaming

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• • • • • • •

Source: http://en.wikipedia.org/wiki/Relativistic_particle

Resonance

This article is about resonance in physics. For other senses of this term, see resonance (disambiguation).

In physics, resonance is the tendency of a system to oscillate with higher amplitude when the frequency of its oscillations matches the system’s natural frequency of vibration (its resonant frequency) than it does at other frequencies.

Examples

Examples are the acoustic resonances of musical instruments, the tidal resonance of the Bay of Fundy, orbital resonance as exemplified by some moons of the solar system’s gas giants, the resonance of the basilar membrane in the biological transduction of auditory input, and resonance in electrical circuits. A resonant object, whether mechanical, acoustic, or electrical, will probably have more than one resonant frequency (especially harmonics of the strongest resonance). It will be easy to vibrate at those frequencies, and more difficult to vibrate at other frequencies. It will "pick out" its resonant frequency from Resonance

510 a complex excitation, such as an impulse or a wideband noise excitation. In effect, it is filtering out all frequencies other than its resonance.

Theory

FT

See also: center frequency

For a linear oscillator with a resonant frequency Ω, the intensity of oscillations I when the system is driven with a driving frequency ω is given by: I(ω) ∝

Γ 2 2

2

(ω−Ω) +( Γ2 )

.

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The intensity is defined as the square of the amplitude of the oscillations. This is a Lorentzian function, and this response is found in many physical situations involving resonant systems. Γ is a parameter dependent on the damping of the oscillator, and is known as the linewidth of the resonance. Heavily damped oscillators tend to have broad linewidths, and respond to a wider range of driving frequencies around the resonant frequency. The linewidth is inversely proportional to the Q factor, which is a measure of the sharpness of the resonance.

Quantum mechanics

A resonance is a quantum state whose mean energy lies above the fragmentation threshold of a system and is associated with: •

a pronounced variation of the cross sections if the fragmentation energy lies in the neighbourhood of the energy of the resonance (energy-dependent definition) - The width of this neighbourhood is called the width of the resonance.

•

an exponential decay of the system when the system has a mean energy close to the resonance energy (time-dependent definition, i.e. in timeresolved spectroscopy) - The lifetime (or inverse of the exponent of the exponential signal) of the resonance is proportional to the inverse of its width. Resonances are usually classified into shape and Feshbach resonances or into Breit-Wigner and →Fano resonances.

Quantum field theory

In quantum field theory, resonance is an unstable particle/bound state. It is characterized by a complex pole off the real line in the S-matrix (which happens to be analytic). A sharp resonance is a resonance with a sharp peak in the S-matrix (which corresponds to a long lifetime compared to the reciprocal of its mass) while a broad resonance is a resonance with a spread out peak (which corresponds to a short lifetime relative to the reciprocal of its mass). If Resonance

511 a resonance is too broad, it might not be considered as a particle at all even if it has a complex pole (far from the real line). See also relativistic Breit-Wigner distribution

FT

If the resonance happens to be a "fundamental particle" (i.e. described by a "fundamental field" of its own), it shows up as a complex pole off the real line in the 2-point connected correlation function (i.e. the propagator).

’Old Tacoma Narrows’ bridge failure

The Old Tacoma Narrows Bridge has been popularized in physics text books as a classical example of resonance, but this description is misleading. It is more correct to say that it failed due to the action of self-excited forces, by an aeroelastic phenomenon known as flutter. Robert H. Scanlan, father of the field of bridge aerodynamics, wrote an article about this misunderstanding 286.

See also Center frequency Driven harmonic motion Formant Harmonic oscillator Impedance Q factor Resonator Schumann resonance Simple harmonic motion Tuned circuit Wave Gluonic vacuum field

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• • • • • • • • • • • •

Reference

External links • •

Lectures in Physics 287 - Resonance from an energetic perspective RMCybernetics - Resonance 288 Resonance Research.

286 K. Billah and R. Scanlan (1991), Resonance, Tacoma Narrows Bridge Failure, and Undergraduate

Physics Textbooks, American Journal of Physics, 59(2), 118–124 (PDF)(http://www.ketchum.org /billah/Billah-Scanlan.pdf) 287 http://www.vias.org/physics/bk3_02_04.html 288 http://www.rmcybernetics.com/research/resonance/resonance.htm

Resonance

512 Greene, Brian, " Resonance in strings 289". The Elegant Universe, NOVA (PBS) • Hyperphysics section on resonance concepts 290 • A short FAQ on quantum resonances 291 • Resonance versus resonant 292 • YouTube 293 - Video of the effects of resonance on rice

FT

•

Source: http://en.wikipedia.org/wiki/Resonance

Principal Authors: Omegatron, Heron, Jitse Niesen, Michael Hardy, Hyacinth, DrBob, N.MacInnes, Phys

Ring wave guide

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In quantum mechanics, the ring wave guide starts from the one dimensional, time independent →Schrödinger equation: 2

~ − 2m ∇2 ψ = Eψ

Using polar coordinates on the 1 dimensional ring, the wave function depends only on the angular coordinate, and so ∇2 =

1 ∂2 r2 ∂θ2

Requiring that the wave function be periodic in θ with a period 2 π (from the demand that the wave functions be single-valued functions on the circle), and that they be normalized leads to the conditions R 2π 2 0 |ψ(θ)| dθ = 1 , and

ψ(θ) = ψ(θ + 2 π)

Under these conditions, the solution to the Schrodinger equation is given by

289 http://www.pbs.org/wgbh/nova/elegant/resonance.html 290 http://hyperphysics.phy-astr.gsu.edu/hbase/sound/rescon.html#c1

291 http://www.thch.uni-bonn.de/tc/people/brems.vincent/vincent/faq.html 292 http://users.ece.gatech.edu/~mleach/misc/resonance.html 293 http://www.youtube.com/watch?v=Zkox6niJ1Wc

Ring wave guide

513 ψ(θ) =

√ r

e±i ~

√1 2π

2mEθ

r

e±i ~

√

2mEθ

r

e±i2π ~

√

r

= e±i ~

2mE

√

FT

The energy eigenvalues E are quantized because of the periodic boundary conditions, and they are required to satisfy 2mE(θ+2π) ,

= 1 = ei2πn

or

This leads to the energy eigenvalues E=

n2 ~2 2mr2

where n = 0, 1, 2, 3, . . .

The full wave functions are, therefore ψ(θ) =

√1 2π

e±inθ

Quantum states found:

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n = 0:

ψ is a constant function, and E = 0. This represents a stationary particle (no angular momentum spinning around the ring).

n = 1:

E=

~2 2mr2

and

ψ(θ) =

√1 2π

e±iθ

This produces two independent states that have the same energy level (degeneracy) and can be linearly combined arbitrarily; instead of exp(± · · ·) one can choose the sine and cosine functions. These two states represent particles spinning around the ring in clockwise and counterclockwise directions. The angular momentum is ±~.

n = 2 (and higher):

the energy level is proportional to n2 , the angular momentum to n. There are always two (degenerate) quantum states.

Ring wave guide

514

Application

FT

Except for the case n = 0, there are two quantum states for every value of n (corresponding to e±inθ ). Therefore there are 2n+1 states with energies less than an energy indexed by the number n.

In organic chemistry, aromatic compounds contain atomic rings, such as benzene rings (the Kekulé structure) consisting of five or six, usually carbon, atoms. So does the surface of "buckyballs" (buckminsterfullerene). These molecules are exceptionally stable. The above explains why the ring behaves like a circular wave guide. The excess (valency) electrons spin around in both directions.

To fill all energy levels up to n requires 2 × (2n + 1) electrons, as electrons have additionally two possible orientations of their spins.

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The rule that 4n + 2 excess electrons in the ring produces an exceptionally stable ("aromatic") compound, is known as the Hückel’s rule.

Source: http://en.wikipedia.org/wiki/Ring_wave_guide

Principal Authors: Pfalstad, Linas, Oleg Alexandrov, Waveguy, Wik

Ritz method

In physics, the Ritz method is a variational method named after Walter Ritz. It can be applied in quantum mechanical problems to provide an upper-bound on the ground state energy. As with other variational methods, a trial wave function is tested on the system. This trial function contains one or more adjustable parameters, which are varied to find a lowest energy configuration. It can be shown that the ground state energy, E0 , satisfies an inequality: R ˆ dτ E0 < Ψ∗ HΨ

that is, the ground-state energy is less than this value. The trial wave-function will always give an expectation value larger than the ground-energy (or at least, equal to it). If the trial wave function is known to be orthogonal to the ground state, then it will provide a boundary for the energy of some excited state.

Ritz method

515

FT

The Ritz ansatz function (trial function - i.e., the assumed form of the eigenfunctions) is a linear combination of N known basis functions {Ψi }, parametrized by unknown coefficients: P Φ= N i=1 ci Ψi . With a known hamiltonian, we can write its expected value as 

PN PN PN ∗ PN ˆ c c H ci Ψi ci Ψi |H| j=1 i j ij A PNi=1  = Pi=1 ≡ B ε = Pi=1N . N PN ∗ cΨ| i=1 i i

cΨ i=1 i i

i=1

c c S j=1 i j ij

The basis functions are usually not orthogonal, so that the overlap matrix S  is has nonzero diagonal elements. Either {ci } or c∗i (the conjugation of the first) can be used to minimze the expectation value. For instance, by making  the partial derivatives of ε over c∗i zero, the following equality is obtained for every k = 1,2,...,N: PN c (Hkj −εSkj ) j=1 j ∂ε = 0, ∗ = B ∂c k

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which leads to a set of N secular equations:
PN for k = 1, 2, ..., N . j=1 cj Hkj − εSkj = 0

 In the above equations, energy ε and the coefficients cj are unknown. With respect to c, this is a homogeneous set of linear equations, which has a solution when the determinant of the coefficients to these unknowns is zero:
det Hkj − εSkj = 0,

which in turn is true only for N values of ε. Furthermore, since the hamiltonian is a hermitian operator. matrix H is also hermitian and the values of εi will be real. The lowest value among εi (i=1,2,..,N), ε0 , will be the best approximation to the ground state for the basis functions used. The remaining N-1 energies are estimates of excited state energies. An approximation for the  wave function of state i can be obtained by finding the coefficients cj from the corresponding secular equation.

Source: http://en.wikipedia.org/wiki/Ritz_method

Principal Authors: Karol Langner, Klemen Kocjancic, Nimur, Emersoni

Ritz method

516

Rutherford model

FT

The Rutherford model of the atom was devised by Ernest Rutherford around 1911 after he performed scattering experiments which showed that the →Plum pudding model of the atom was incorrect. In the Rutherford model, an atom is made up of a nucleus surrounded by a cloud of orbiting electrons. However, the Rutherford model did not attribute any structure to the orbiting electrons. The Rutherford model of the atom was soon superseded by the Bohr atom, which used some of the early quantum mechanical results to give structure to the orbiting electrons.

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The Rutherford model was very important because it proposed the concept of the nucleus. After the discovery of the Rutherford model, the study of the atom branched into two separate fields, nuclear physics which studies the nucleus of the atom, and atomic physics which studies the structure of the orbiting electrons. In the Rutherford model the nucleus consisted of protons and embedded electrons, this was however proven false later.

See also •

Atomic nucleus

External Links • • •

World Of Atoms - "Rutherford’s Model" 294 Rutherford’s Model 295 Rutherford’s Model 296

Source: http://en.wikipedia.org/wiki/Rutherford_model

294 http://library.thinkquest.org/C0122360/full/1-1_e.html 295 http://www.rwc.uc.edu/koehler/biophys/7a.html

296 http://www2.kutl.kyushu-u.ac.jp/seminar/MicroWorld1_E/Part2_E/P25_E/Rutherford_model_E.htm

Rutherford model

517

Rutherford scattering

FT

In physics, Rutherford scattering is a phenomenon that was explained by Ernest Rutherford in 1911, and led to the development of the orbital theory of the atom. It is now exploited by the materials analytical technique Rutherford backscattering. Rutherford scattering is also sometimes referred to as Coulomb scattering because it relies on static electric (Coulomb) forces. A similar process probed the insides of nuclei in the 1960s, called deep inelastic scattering.

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The discovery was made by Hans Geiger and Ernest Marsden in 1909 when they performed the gold foil experiment under the direction of Rutherford, in which they fired a beam of alpha particles (helium nuclei) at layers of gold leaf only a few atoms thick. At the time of the experiment, the atom was thought to be analogous to a plum pudding (as proposed by J.J. Thomson), with the negative charges (the plums) found throughout a positive sphere (the pudding). If the plum-pudding model were correct, the positive “pudding”, being more spread out than in the current model of a concentrated nucleus, would not be able to exert such large coulombic forces, and the alpha particles should only be deflected by small angles as they pass through.

However, the intriguing results showed that around 1 in 8000 alpha particles were deflected by very large angles (over 90◦ ). From this, Rutherford concluded that the majority of the mass was concentrated in a minute, positively charged region (the nucleus) surrounded by electrons. When a (positive) alpha particle approached sufficiently close to the nucleus, it was repelled strongly enough to rebound at high angles. The small size of the nucleus explained the small number of alpha particles that were repelled in this way. Rutherford showed, using the method below, that the size of the nucleus was about 10 -14 m.

Details of calculating nuclear size

For head on collisions between alpha particles and the nucleus, all the kinetic 1 energy ( 2 mv 2 ) of the alpha particle is turned into potential energy and the particle is at rest. The distance from the centre of the alpha particle to the centre of the nucleus (b ) at this point is a maximum value for the radius, if it is evident from the experiment that the particles have not hit the nucleus. Applying the inverse-square law between the charges on the electron and nucleus, one can write: 1 2 2 mv

=

1 4π0

·

q1 q2 b

Rearranging:

Rutherford scattering

518 b=

1 4π0

·

2q1 q2 mv 2

For an alpha particle: m (mass) = 6.7×10 -27 kg q 1 = 2×(1.6×10 -19) C q 2 (for gold) = 79×(1.6×10 -19) C v (initial velocity) = 2×10 7 m/s

FT

• • • •

Substituting these in gives the value of about 2.7×10 -14 m. (The true radius is about 7.3×10 -15 m.)

See also: •

Coulomb collision

References •

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E. Rutherford, The Scattering of α and β Particles by Matter and the Structure of the Atom 297, Philosophical Magazine. Series 6, vol. 21. May 1911 • H. Geiger and E. Marsden, On a Diffuse Reflection of the α-Particles 298, Proceedings of the Royal Society, 1909 A vol. 82, p. 495-500

Source: http://en.wikipedia.org/wiki/Rutherford_scattering

Principal Authors: Sodium, Linas, Michael Hardy, Jll, HenkvD, Art Carlson, Awolf002, Andre Engels

Rydberg formula

The Rydberg formula (Rydberg-Ritz formula) is used in atomic physics for determining the full spectrum of light emission from hydrogen, later extended to be useful with any element by use of the Rydberg-Ritz combination principle.

297 http://fisica.urbenalia.com/arts/structureatom.pdf 298 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/GM-1909.html

Rydberg formula

FT

519

The spectrum is the set of wavelengths of photons emitted when electrons jump between discrete energy levels, "shells" around the atom of a certain chemical element. This discovery was later to provide motivation for the creation of quantum physics.

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The formula was invented by the Swedish physicist Johannes Rydberg and presented on November 5, 1888.

Rydberg formula for hydrogen 1 λvac

= RH Z 2



1 n21

−

1 n22



Where

λvac is the wavelength of the light emitted in vacuum, RH is the Rydberg constant for hydrogen,

n1 and n2 are integers such that n1 < n2 ,

Z is the atomic number, which is 1 for hydrogen.

By setting n1 to 1 and letting n2 run from 2 to infinity, the spectral lines known as the Lyman series converging to 91nm are obtained, in the same manner: n1 n2 Name 1 2 → ∞ Lyman series

Converge toward

2 3 → ∞ Balmer series

365nm

3 4 → ∞ Paschen series

821nm

4 5 → ∞ Brackett series

1459nm

5 6 → ∞ Pfund series

2280nm

91nm

Rydberg formula

520 6 7 → ∞ Humphreys series 3283nm

FT

The Lyman series is in the ultraviolet while the Balmer series is in the visible and the Paschen, Brackett, Pfund, and Humphreys series are in the infrared.

Rydberg formula for any hydrogen-like element

The formula above can be extended for use with any hydrogen-like chemical elements.   1 2 1 − 1 = RZ 2 2 λvac n n 1

where

2

λvac is the wavelength of the light emitted in vacuum; R is the Rydberg constant for this element;

DR A

Z is the atomic number, i.e. the number of protons in the atomic nucleus of this element; n1 and n2 are integers such that n1 < n2 .

It’s important to notice that this formula can be applied only to hydrogen-like, also called hydrogenic atoms chemical elements, i.e. atoms with only one electron on external system of orbitals. Examples would include He +, Li 2+, Be 3+ etc.

History

By 1890, Rydberg had discovered a formula describing the relation between the wavelengths in lines of alkali metals and found that the Balmer equation was a special case. Although the Rydberg formula was later found to be imprecise with heavier atoms, it is still considered accurate for all the hydrogen series and for alkali metal atoms with a single valency electron orbiting well clear of the inner electron core. By 1906, Lyman had begun to analyze the hydrogen series of wavelengths in the ultraviolet spectrum named for him that were already known to fit the Rydberg formula. Rydberg simplified his calculations by using the ‘wavenumber’ (the number of waves occupying a set unit of length) as his unit of measurement. He plotted the wavenumbers of successive lines in each series against consecutive integers which represented the order of the lines in that particular series. Finding that

Rydberg formula

521

References

FT

the resulting curves were similarly shaped, he sought a single function which could generate all of them when appropriate constants were inserted.

Mike Sutton, “Getting the numbers right – the lonely struggle of Rydberg” Chemistry World, Vol. 1, No. 7, July 2004.

See also

Rydberg-Ritz combination principle

Source: http://en.wikipedia.org/wiki/Rydberg_formula

Principal Authors: Nixdorf, Voyajer, Laurascudder, Chris Roy, Xerxes314, Tantalate, W.marsh, Wibblywobbly

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Scattering channel

In scattering theory, a scattering channel is a quantum state of the colliding system before or after the collision (t → ±∞). The →Hilbert space spanned by the states before collision (in states) is equal to the ones spanned by the states after collision (out states) which are both →Fock spaces if there is a mass gap. This is the reason why the S matrix which maps the in states onto the out states must be unitary. The scattering channel are also called scattering asymptotes. The Møller operators are mapping the scattering channels onto the corresponding states which are solution of the →Schrödinger equation taking the interaction Hamiltonian into account. The Møller operators are isometric. See also: LSZ formalism

Source: http://en.wikipedia.org/wiki/Scattering_channel Principal Authors: Phys, OpenToppedBus

Scattering channel

522

Scattering theory

FT

In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a rainbow. Examples of particle scattering includes the motion of billiard balls or the →Rutherford scattering of alpha particles by gold nucleii. More precisely, scattering consists of the study of how solutions of partial differential equations, propagating freely "in the distant past", come together and interact with one another or with a boundary condition, and then propagate way "to the distant future".

The direct scattering problem is the problem determining the distribution of scattered radiation/particle flux basing on the characteristics of the scatterer.

DR A

The inverse scattering problem is the problem of determining the characteristics of an object (its shape, internal constitution, etc.) from measurement data of radiation or particles scattered from the object.

Since its early statement for radiolocation, the problem has found vast number of applications, such as echolocation, geophysical survey, nondestructive testing, medical imaging and quantum field theory, to name just a few.

In theoretical physics

In mathematical physics, scattering theory is a framework for studying and understanding the interaction or scattering of solutions to partial differential equations. In acoustics, the differential equation is the wave equation, and scattering studies how its solutions, the sound waves, scatter from solid objects or propagate through non-uniform media (such as sound waves, in sea water, coming from a submarine). In the case of classical electrodynamics, the differential equation is again the wave equation, and the scattering of light or radio waves is studied. In quantum mechanics and particle physics, the equations are those of QED, QCD and the Standard Model, the solutions of which correspond to fundamental particles. In quantum chemistry, the solutions correspond to atoms and molecules, governed by the Schroedinger equation.

Elastic and inelastic scattering

The example of scattering in quantum chemistry is particularly instructive, as the theory is reasonably complex while still having a good foundation on which

Scattering theory

523

Topics in physics

FT

to build an intuitive understanding. When two atoms are scattered off one another, one can understand them as being the bound state solutions of some differential equation. Thus, for example, the hydrogen atom corresponds to a solution to the Schroedinger equation with an inverse-square law central potential. The scattering of two hydrogen atoms will disturb the state of each atom, resulting in one or both becoming excited, or even ionized. Thus, collisions can be either elastic (the internal quantum states of the particles are not changed) or inelastic (the internal quantum states of the particles are changed). From the experimental viewpoint the observable quantity is the cross section. From the theoretical viewpoint the key quantity is the S matrix.

According to the optics classification of the Optical Society of America this field consists of the following topics: Aerosol and cloud effects Atmospheric scattering Backscattering Diffusion Extinction Index measurements Inverse scattering Linewidth

• • • • • • • •

Long-wave scattering Mie theory Multiple scattering Scattering measurements Brillouin scattering Molecular scattering Particle scattering Raman scattering

• • • • • • •

Rayleigh Scattering Scattering from rough surfaces Stimulated scattering Stimulated Brillouin scattering Stimulated Raman scattering Scintillation Turbid media

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• • • • • • • •

The mathematical framework

In mathematics, scattering theory deals with a more abstract formulation of the same set of concepts. For example, if a differential equation is known to have some simple, localized solutions, and the solutions are a function of a single parameter, that parameter can take the conceptual role of time. One then asks what might happen if two such solutions are set up far away from each other, in the "distant past", and are made to move towards each other, interact (under the constraint of the differential equation) and then move apart in the "future". The scattering matrix then pairs solutions in the "distant past" to those in the "distant future". Solutions to differential equations are often posed on manifolds. Frequently, the means to the solution requires the study of the spectrum of an operator on the manifold. As a result, the solutions often have a spectrum that can be identified with a →Hilbert space, and scattering is described by a certain map, the S matrix, on Hilbert spaces. Spaces with a discrete spectrum correspond to bound states in quantum mechanics, while a continuous spectrum is associated with scattering states. The study of inelastic scattering then asks how discrete and continuous spectra are mixed together.

Scattering theory

524

References •

FT

An important, notable development is the inverse scattering transform, central to the solution of many exactly solvable models.

Lectures of the European school on theoretical methods for electron and positron induced chemistry, Prague, Feb. 2005 299

Source: http://en.wikipedia.org/wiki/Scattering_theory

Principal Authors: Vb, Pflatau, Charles Matthews, David R. Ingham, Fuhghettaboutit

Schrödinger equation

DR A

In physics, the Schrödinger equation, proposed by the Austrian physicist Erwin Schrödinger in 1925, describes the space- and time-dependence of quantum mechanical systems. It is of central importance to the theory of quantum mechanics, playing a role analogous to Newton’s second law in classical mechanics.

In the mathematical formulation of quantum mechanics, each system is associated with a complex →Hilbert space such that each instantaneous state of the system is described by a unit vector in that space. This state vector encodes the probabilities for the outcomes of all possible measurements applied to the system. As the state of a system generally changes over time, the state vector is a function of time. The Schrödinger equation provides a quantitative description of the rate of change of the state vector. Using Dirac’s bra-ket notation, the definition of energy results in the time derivative operator: at time t by |ψ (t)i. The Schrödinger equation is ∂ H(t) |ψ (t)i = i~ ∂t |ψ (t)i

where i is the imaginary unit, t is time, ∂/∂t is the partial derivative with respect to t, ~ is the reduced Planck’s constant (Planck’s constant divided by 2π), ψ(t) is the wave function, and H (t) is the Hamiltonian (a self-adjoint operator acting on the state space). The Hamiltonian describes the total energy of the system. As with the force occurring in Newton’s second law, its exact form is not provided by the

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Schrödinger equation

525 Schrödinger equation, and must be independently determined based on the physical properties of the system.

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Time-independent Schrödinger equation

For many real-world problems the energy distribution does not change with time, and it is useful to determine how the stationary states vary with position x (independent of the time t). The Schrödinger equation is often introduced without bra-ket notation in the following ways: One dimensional time-independent 2 ~ d ψ(x) − 2m (dx)2 2

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:

+ U (x)ψ(x) = Eψ(x)

3-dimensional time-independent ~2 − 2m ∇2 ψ(r) + U (r)ψ(r)

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:

= Eψ(r)

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For every time-independent Hamiltonian, H, there exists a set of quantum states, |ψn i, known as energy eigenstates, and corresponding real numbers En satisfying the eigenvalue equation H |ψn (x)i = En |ψn (x)i .

Such a state possesses a definite total energy, whose value En is the eigenvalue of the state vector with the Hamiltonian. This eigenvalue equation is referred to as the time-independent Schrödinger equation. Self-adjoint operators such as the Hamiltonian have the property that their eigenvalues are always real numbers, as we would expect since the energy is a physically observable quantity. On inserting the time-independent Schrödinger equation into the full Schrödinger equation, we get ∂ i~ ∂t |ψn (t)i = En |ψn (t)i .

It is easy to solve this equation. One finds that the state vectors of the energy eigenstates change by only a complex phase: |ψ (t)i = e−iEt/~ |ψ (0)i .

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Energy eigenstates are convenient to work with because their time-dependence is so simple; that is why the time-independent Schrödinger equation is so useful. We can always choose a set of instantaneous energy eigenstates whose state vectors {|ni} form a basis for the state space. Then any state vector |ψ (t)i can be written as a linear superposition of energy eigenstates: P P 2 |ψ (t)i = n cn (t) |ni , H |ni = En |ni , n |cn (t)| = 1. (The last equation enforces the requirement that |ψ (t)i, like all state vectors, must be a unit vector.) Applying the Schrödinger equation to each side of the first equation, and using the fact that the energy basis vectors are by definition linearly independent, we readily obtain n i~ ∂c ∂t = En cn (t) .

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Therefore, if we know the decomposition of |ψ (t)i into the energy basis at time t = 0, its value at any subsequent time is given simply by P |ψ (t)i = n e−iEn t/~ cn (0) |ni .

Schrödinger wave equation

The state space of certain quantum systems can be spanned with a position basis. In this situation, the Schrödinger equation may be conveniently reformulated as a partial differential equation for a wavefunction, a complex scalar field that depends on position as well as time. This form of the Schrödinger equation is referred to as the Schrödinger wave equation. Elements of the position basis are called position eigenstates. We will consider only a single-particle system, for which each position eigenstate may be denoted by |ri, where the label r is a real vector. This is to be interpreted as a state in which the particle is localized at position r. In this case, the state space is the space of all square-integrable complex functions.

The wavefunction

We define the wavefunction as the projection of the state vector |ψ (t)i onto the position basis: ψ (r, t) ≡ hr|ψ (t)i .

Since the position eigenstates form a basis for the state space, the integral over all projection operators is the identity operator: R |ri hr| d3 r = I.

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This statement is called the resolution of the identity. With this, and the fact that kets have unit norm, we can show that where ψ (r, t)∗ denotes the complex conjugate of ψ (r, t). This important result tells us that the absolute square of the wavefunction, integrated over all space, must be equal to 1: R |ψ (r, t)|2 d3 r = 1.

We can thus interpret the absolute square of the wavefunction as the probability density for the particle to be found at each point in space. In other words, |ψ (r, t)|2 d3 r is the probability, at time t, of finding the particle in the infinitesimal region of volume d3 r surrounding the position r. We have previously shown that energy eigenstates vary only by a complex phase as time progresses. Therefore, the absolute square of their wavefunctions do not change with time. Energy eigenstates thus correspond to static probability distributions.

Operators in the position basis

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Any operator A acting on the wavefunction is defined in the position basis by Aψ (r, t) ≡ hr|A|ψ (t)i .

The operators A on the two sides of the equation are different things: the one on the right acts on kets, whereas the one of the left acts on scalar fields. It is common to use the same symbols to denote operators acting on kets and their projections onto a basis. Usually, the kind of operator to which one is referring is apparent from the context, but this is a possible source of confusion. Using the position-basis notation, the Schrödinger equation can be written as ∂ Hψ (r, t) = i~ ∂t ψ (r, t) .

This form of the Schrödinger equation is the Schrödinger wave equation. It may appear that this is an ordinary differential equation, but in fact the Hamiltonian operator typically includes partial derivatives with respect to the position variable r. This usually leaves us with a difficult linear partial differential equation to solve.

Non-relativistic Schrödinger wave equation In non-relativistic quantum mechanics, the Hamiltonian of a particle can be expressed as the sum of two operators, one corresponding to kinetic energy and the other to potential energy. The Hamiltonian of a particle with no electric charge and no spin in this case is:

Schrödinger equation

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i ~2 Hψ (r, t) = (T + V ) ψ (r, t) = − 2m ∇2 + V (r) ψ (r, t) = i~ ∂ψ ∂t (r, t)

T =

p2 2m

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where is the kinetic energy operator,

m is the mass of the particle,

p = ~i ∇ is the momentum operator,

V = V (r) is the potential energy operator,

V is a real scalar function of the position operator r,

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∇ is the gradient operator, and ∇2 is the Laplace operator.

This is a commonly encountered form of the Schrödinger wave equation, though not the most general one. The corresponding time-independent equation is i h 2 ~ ∇2 + V (r) ψ (r) = Eψ (r) . − 2m The relativistic generalisations of this wave equation are the Dirac equation, →Klein-Gordon equation, Proca equation, Maxwell equations etc, depending on spin and mass of the particle. See relativistic wave equations for details.

Probability currents

In order to describe how probability density changes with time, it is acceptable to define probability current or probability flux. The probability flux represents a flowing of probability across space.

For example, consider a Gaussian probability curve centered around x0 , imagine that x0 moving in a speed v toward the right. Then one may say that the probability is flowing toward right, i.e., there is a probability flux directed to the right. The probability flux j is defined as:

j=

~ m

·

1 2i

(ψ ∗ ∇ψ − ψ∇ψ ∗ ) =

~ m

Im (ψ ∗ ∇ψ)

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and measured in units of (probability)/(area × time) = r -2t -1. ∂ ∂t P

(x, t) + ∇ · j = 0

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The probability flux satisfies a quantum continuity equation, i.e.:

where P (x, t) is the probability density and measured in units of (probability)/(volume) = r -3. This equation is the mathematical equivalent of probability conservation law. It is easy to show that for a plane wave, |ψi = Aeikx e−iωt

the probability flux is given by j (x, t) = |A|2

k~ m.

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Solutions of the Schrödinger equation

Analytical solutions of the time-independent Schrödinger equation can be obtained for a variety of relatively simple conditions. These solutions provide insight into the nature of quantum phenomena and sometimes provide a reasonable approximation of the behavior of more complex systems (e.g., in statistical mechanics, molecular vibrations are often approximated as harmonic oscillators). Several of the more common analytical solutions include:

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The free particle The particle in a box The finite potential well The particle in a ring The particle in a spherically symmetric potential The quantum harmonic oscillator The hydrogen atom or hydrogen-like atom The ring wave guide The particle in a one-dimensional lattice (periodic potential)

For many systems, however, there is no analytic solution to the Schrödinger equation. In these cases, one must resort to approximate solutions. Some of the common techniques are:

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• • • •

Perturbation theory The variational principle underpins many approximate methods (like the popular Hartree-Fock method which is the basis of the post Hartree-Fock methods) Quantum Monte Carlo methods Density functional theory The WKB approximation discrete delta-potential method

See also

→Schrödinger picture Basic quantum mechanics Quantum number • Principal quantum number • Azimuthal quantum number • Magnetic quantum number • Spin quantum number • Dirac equation

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References

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E. Schrödinger, Ann. Phys. (Leipzig) 489 (1926) p.79 E. Schrödinger, Phys. Rev. 28 (1926) p. 1049

Modern reviews •

David J. Griffiths (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X.

External links •

Linear Schrödinger Equation 302 at EqWorld: The World of Mathematical Equations.

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Nonlinear Schrödinger Equation 303 at EqWorld: The World of Mathematical Equations.

302 http://eqworld.ipmnet.ru/en/solutions/lpde/lpde108.pdf 303 http://eqworld.ipmnet.ru/en/solutions/npde/npde1403.pdf

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The Schrödinger Equation in One Dimension 304 as well as the directory of the book 305. 306

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Source: http://en.wikipedia.org/wiki/Schr%C3%B6dinger_equation

Principal Authors: CYD, MathKnight, Passw0rd, Michael Hardy, Fresheneesz, Alex valavanis, Nommonomanac, Camembert, Voyajer, Linas

Schrödinger picture

In quantum mechanics, a state function is a linear combination (a superposition) of eigenstates. In the Schrödinger picture, all of these eigenstates are constantly rotating through time.

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This rotation is not in any ordinary spatial sense. Each eigenvector has an amplitude which is a complex number. This amplitude is a coefficient which multiplies one of the basis vectors. The complex coefficient has a magnitude and a direction. Therefore a state function is a linear combination of basis vectors, each one multiplied by a complex coefficient which has a magnitude and a direction in the complex plane. These coefficients can be thought of as phasors. In the Schrödinger picture, these phasor coefficients are constantly rotating in a circle through time. The rotation operator which causes their rotation is called the propagator. The time evolution of a Schrödinger wave function can be effected mathematically by multiplying the wave function with the propagator. The propagator effects a simultaneous rotation of all the phasor coefficients of all the (infinite) basis vectors which form the state function.

Let |ψe (0)i represent an energy eigenstate at time 0. Then the rotation of the phasor coefficient of this eigenstate through time can be described by: |ψe (t)i = e−iHt/~ |ψe (0)i.

where e−iHt/~ is a rotation in the complex plane, and H is the scalar Hamiltonian. Taking the time derivative of |ψe (t)i yields

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=

−i ~ H|ψe (t)i

which is the Schrödinger equation for time evolution.

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Thus, in the Schrödinger formulation of quantum mechanics, all unperturbed state functions are time-harmonic. State functions in the Schrödinger picture are never entirely static, they are always undulating. This is why state functions in the Schrödinger formulation are called wavefunctions. It reveals the undulatory nature of matter: the wave-particle duality. (Actually, wavefunctions also are also undulatory in space, independently of time.) The alternative to the Schrödinger picture is to switch to a rotating reference frame, which is itself being rotated by the propagator. Since the undulatory rotation is now being assumed by the reference frame itself, an undisturbed state function appears to be truly static. This is the →Heisenberg picture. See also interaction picture.

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Further reading •

Principles of Quantum Mechanics by R. Shankar, Plenum Press.

Source: http://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture Principal Authors: AugPi, Pen of bushido, Cyp, NawlinWiki, Lethe

Schrödinger’s cat

Schrödinger’s cat is a seemingly paradoxical thought experiment devised by Erwin Schrödinger that attempts to illustrate the incompleteness of an early interpretation of quantum mechanics when going from subatomic to macroscopic systems. The experiment proposes: A cat is placed in a sealed box. Attached to the box is an apparatus containing a radioactive atomic nucleus and a canister of poison gas. This apparatus is separated from the cat in such a way that the cat can in no way interfere with it. The experiment is set up so that there is exactly a 50% chance of the nucleus decaying in one hour. If the nucleus decays, it will emit a particle that triggers the apparatus, which opens the canister and kills the cat. If the nucleus does not decay, then the cat remains alive. According to quantum mechanics, the unobserved nucleus is described as a superposition (meaning it exists partly as each simultaneously) of "decayed Schrödinger’s cat

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Figure 43 Schrödinger’s Cat: If the nucleus in the bottom left decays, the geiger counter on its right will sense it and trigger the release of the gas. In one hour, there is a 50% chance that the nucleus will decay, and therefore that the gas will be released and kill the cat.

nucleus" and "undecayed nucleus". However, when the box is opened the experimenter sees only a "decayed nucleus/dead cat" or an "undecayed nucleus/living cat."

The question is: when does the system stop existing as a mixture of states and become one or the other? (See basis function.) The purpose of the experiment is to illustrate a paradox; as Schrödinger wrote, "The (wavefunction) for the entire system (has) the living and the dead cat (pardon the expression) mixed 307 or smeared out in equal parts". Because we cannot get along without making classical approximations, quantum mechanics is incomplete without some rules to relate the classical and quantum descriptions. One way of looking at this connection is to say that the wavefunction collapses and the cat becomes dead or remains alive instead of a mixture of both. The point of view that this thought experiment most clearly refutes is that the laws of physics are different for experiments than for other interactions. In the case of the cat dying, a necropsy would show a time of death that would be before the opening of the box. The geiger counter, in moving to one outcome

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The original article appeared in the German magazine Naturwissenschaften ("Natural Sciences") in 1935: E. Schrödinger: "Die gegenwärtige Situation in der Quantenmechanik" ("The present situation in quantum mechanics"), Naturwissenschaften, 48, 807, 49, 823, 50, 844 (November 1935). It was intended as a discussion of the EPR article published by Einstein, Podolsky and Rosen in the same year. Apart from introducing the cat, Schrödinger also coined the term "entanglement" (German: Verschränkung) in his article. Albert Einstein was impressed; in a letter to Schrödinger dated 1950 he wrote:

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You are the only contemporary physicist, besides Laue, who sees that one cannot get around the assumption of reality - if only one is honest. Most of them simply do not see what sort of risky game they are playing with reality - reality as something independent of what is experimentally established. Their interpretation is, however, refuted most elegantly by your system of radioactive atom + amplifier + charge of gun powder + cat in a box, in which the psi-function of the system contains both the cat alive and blown to bits. Nobody really doubts that the presence or absence of the cat is something independent of the act of observation.

Nowadays, the mainstream interpretation is that the triggering of the device is the actual observation that collapses the wave function.

Copenhagen interpretation

In the Copenhagen interpretation, a system stops being a superposition of states and becomes either one or the other when an observation takes place. This experiment makes apparent the fact that the nature of measurement, or observation, is not well defined in this interpretation. Some interpret the experiment to mean that while the box is closed, the system simultaneously exists in a superposition of the states "decayed nucleus/dead cat" and "undecayed nucleus/living cat", and that only when the box is opened and an observation performed does the wave function collapse into one of the two states. More intuitively, some feel that the "observation" is taken when a particle from the nucleus hits the detector. Recent developments in quantum physics show that measurements of quantum phenomena taken by non-conscious "observers" (such as a wiretap) most definitely alter the quantum state of the phenomena from the point of view of conscious observers reading the wiretap, lending support to this idea.

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A precise rule is that probability enters at the point where the classical approximation is first used to describe the system - almost by tautology, as the classical approximation is just a simplification of the quantum mathematics, and so must introduce imprecision in the measurement, which can be viewed as probability. Note, however, that this only applies to descriptions of the system, not the system itself.

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Under Copenhagen, the amount of uncertainty for a complex quantum system is predicted by quantum decoherence. Particles which exchange photons (and possibly other atomic or subatomic particles) become entangled with each other from the point of view of an observer, meaning that these particles can only be described accurately with reference to each other, which decreases the total uncertainty of those particles from the point of view of our observer. By the time one has reached "macroscopic" levels - such as a cat, which is made up of a number of atomic particles almost too large to express with words - so many particles have become entangled with each other so as to decrease the uncertainty to almost zero. (Quantum effects in huge collections of particles are only seen in very rare, and often man-made, situations, such as a Bose-Einstein condensate). Thus, at least from the point of view of the observer, any improbability regarding the cat as a system of quantum particles has disappeared due to the massive amount of entanglement between all of the particles that make it up, meaning that the cat does not truly exist as both alive and dead at the same time, at least from the point of view of any observer viewing the cat.

It is interesting to note that even before observation was noted to be fundamentally distinct from consciousness through experimentation, the experiment always contained at least two "observers" - the physicist and the cat. Even had the physicist been unaware of the cat’s state in the hypothetical experiment, one would have had to posit that the cat, at least, would have been quite sure of its status (at least, as long as the gas had not yet ended its ability to "observe"). However, since "observation" has been shown by experiment to have nothing to do with consciousness - or at the very least, any traditional definition of consciousness - most conjecture along these lines probably falls under the "interesting but physically irrelevant" category. Steven Weinberg in "Einstein’s Mistakes", Physics Today, November 2005, page 31, said: All this familiar story is true, but it leaves out an irony. Bohr’s version of quantum mechanics was deeply flawed, but not for the reason Einstein thought. The Copenhagen interpretation describes what happens when an observer makes a measurement, but the observer and the act of measurement are themselves treated classically. This is surely wrong: Physicists and their apparatus must be governed by the same quantum mechanical rules Schrödinger’s cat

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that govern everything else in the universe. But these rules are expressed in terms of a wavefunction (or, more precisely, a state vector) that evolves in a perfectly deterministic way. So where do the probabilistic rules of the Copenhagen interpretation come from? Considerable progress has been made in recent years toward the resolution of the problem, which I cannot go into here. It is enough to say that neither Bohr nor Einstein had focused on the real problem with quantum mechanics. The Copenhagen rules clearly work, so they have to be accepted. But this leaves the task of explaining them by applying the deterministic equation for the evolution of the wavefunction, the Schrödinger equation, to observers and their apparatus.

Everett many-worlds interpretation & consistent histories

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In the many-worlds interpretation of quantum mechanics, which does not single out observation as a special process, both states persist, but decoherent from each other. When an observer opens the box, he becomes entangled with the cat, so observer-states corresponding to the cat being alive and dead are formed, and each can have no interaction with the other. The same mechanism of quantum decoherence is also important for the interpretation in terms of Consistent Histories. Only the "dead cat" or "alive cat" can be a part of a consistent history in this interpretation.

In other words, when the box is opened, the universe (or at least the part of the universe containing the observer and cat) is split into two separate universes, one containing an observer looking at a box with a dead cat, one containing an observer looking at a box with a live cat.

Practical applications

The experiment is a purely theoretical one, and the machine proposed does not exist. This has some practical use in quantum computing and quantum cryptography. It is possible to send light that is in a superposition of states down a fiber optic cable. Placing a wiretap in the middle of the cable which intercepts and retransmits the transmission will collapse the wavefunction (in the Copenhagen interpretation, "perform an observation") and cause the light to fall into one state or another. By performing statistical tests on the light received at the other end of the cable, one can tell whether it remains in the superposition of Schrödinger’s cat

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states or has already been observed and retransmitted. In principle, this allows the development of communication systems that cannot be tapped without the tap being noticed at the other end. This experiment (which can be performed, although a workable quantum cryptographic communications system which can transmit large quantities of data has not yet been constructed) also illustrates that "observation" in the Copenhagen interpretation has nothing to do with consciousness, in that a perfectly unconscious wiretap will cause the statistics at the end of the wire to be different. In quantum computing, the phrase "cat state" often refers to the special entanglement of qubits where the qubits are in an equal superposition of all being 0 and all being 1, i.e. |00...0i + |11...1i.

A variant of the Schrödinger’s Cat experiment known as the quantum suicide machine has been proposed by cosmologist Max Tegmark. It examines the Schrödinger’s Cat experiment from the point of view of the cat, and argues that this may be able to distinguish between the Copenhagen interpretation and many worlds. Another variant on the experiment is Wigner’s friend.

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Physicist Stephen Hawking once exclaimed, "When I hear of Schrödinger’s cat, I reach for my gun," paraphrasing German playwright and Nazi "Poet Laureate", Hanns Johst’s famous phrase "Wenn ich ’Kultur’ höre, entsichere ich meinen Browning! " ("When I hear the word ’culture’, I release the safety on my Browning!") In fact, Hawking and many other physicists are of the opinion that the "Copenhagen School" interpretation of quantum mechanics unduly stresses the role of the observer. A final consensus on this point among physicists seems still to be out of reach.

Related humor

Figure 44 Another joke about Schrödinger’s cat and the tag in HTML.

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Curiosity may have killed the cat, Schrödinger only killed half of it.

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Many pet cats have been named "Schrödinger" as an allusion to Schrödinger’s thought experiment. Schrödinger’s cat

538 In 1982 Cecil Adams, in his column The Straight Dope, wrote a concise and humourous description of the thought experiment, and Einstein’s refutation of same, in the form of an epic poem. "The story of Schroedinger’s cat (an epic poem)" 308

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"Schrödinger’s Cat" is a novel by Robert Anton Wilson in which the ManyWorlds Interpretation is explored. Wilson refers to this model as the "Everett-Wheeler-Graham Model (after its postulators)". In his DVD biography entitled "Maybe Logic", Wilson suggests that our general misconceptions and confusion about quantum mechanics may stem from the use of logical constructs that are inadequate for describing the universe. See General Semantics.

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The term "Schrödinger’s Terrorist" has been used to semi-humorously label terrorists whose status as living or dead is unknown and/or subject to contradictory rumors such as Osama bin Laden.

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A "Schrödinger’s Date" is a meeting between two people that may or may not be a date, and whose status cannot be determined. Only the odds of it being one or the other can be estimated.

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Novelist Douglas Adams posits an amusing anecdote in his novel Dirk Gently’s Holistic Detective Agency, wherein Schrödinger’s cat is neither alive nor dead when the box is opened, and is, in fact, not there. It is later revealed that the cat simply became bored with the experiment and wandered off. They should have used a rat. A rat indeed obediently stays put until the cat shows up.

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The manga version of Hellsing features a catboy character named Schrödinger, who has the ability to be "everywhere and nowhere" at once.

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There’s also the mock headline "Schrödinger’s cat found half-alive: quantum theory a mistake!"

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From the syndicated comic strip, "Brewster Rockit": "Schrödinger’s cat-litter box. If you don’t observe it, you won’t have to change it."

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In Stargate SG-1, Carter gives a friend from a race of highly advanced physicists a tabby cat called Schrödinger as a gesture of good will.

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In in the pilot episode of Sliders, Quinn has a pet cat named Schrödinger.

308 http://www.straightdope.com/classics/a1_122.html

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539 In the NetHack computer game, one occasionally finds a quantum mechanic with a box with a cat named "Schrödinger’s Cat" inside. It is dead in half of the instances.

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Schrödinger’s Fridge 309, from the comic Bob the Angry Flower.

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In The Last Hero by Terry Pratchett, Albert tries to explain the Schrödinger’s Cat experiment to Death, who doesn’t quite grasp the concept of not knowing if something is alive or dead. Death finally gives up the experiment, remarking, "I don’t hold with cruelty to cats." Also, in Lords and Ladies, Pratchett describes the results of the experiment in three possible states: "Alive", "Dead", and "Bloody Furious."

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Tears for Fears Track on their Saturnine Martial & Lunatic Album is called Schrödinger’s cat.

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The online retailer ThinkGeek sells a t-shirt that has the words "Schrödinger’s cat is dead" on the front and "Schrödinger’s Cat is not dead" on the back.

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Cartoon: You see a picture of a refrigerator with a post-it-note saying "I’m out of town for a few days. Please be so nice and feed my cat. Thanks. Schrödinger".

See also

• Schroedinbug • →Schrödinger’s cat in fiction • →Double-slit experiment • Quantum suicide

External links •

Erwin Schrödinger, The Present Situation in Quantum Mechanics (Translation) 310 • A Lazy Layman’s Guide to Quantum Physics 311 • Quantum Mechanics and Schrodinger’s Cat 312 • The many worlds of quantum mechanics 313

309 http://www.angryflower.com/schrod.gif 310 http://www.tu-harburg.de/rzt/rzt/it/QM/cat.html

311 http://www.higgo.com/quantum/laymans.htm 312 http://www.harrymaugans.com/2006/05/03/in-search-of-schrodingers-cat/ 313 http://www.sankey.ws/qm.html

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Source: http://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat Principal Authors: David R. Ingham, DV8 2XL, Commonbrick, Sam Spade, David R. Ingharn, Florian-

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Schrödinger’s cat in fiction

→Schrödinger’s cat is a seemingly paradoxical thought experiment devised by Erwin Schrödinger that attempts to illustrate the incompleteness of the theory of quantum mechanics when going from subatomic to macroscopic systems. In 1935, Schrödinger published an essay describing the conceptual problems in quantum mechanics. A brief paragraph in this essay described the cat paradox:

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One can even set up quite ridiculous cases. A cat is penned up in a steel chamber, along with the following diabolical device (which must be secured against direct interference by the cat): in a Geiger counter there is a tiny bit of radioactive substance, so small that perhaps in the course of one hour one of the atoms decays, but also, with equal probability, perhaps none; if it happens, the counter tube discharges and through a relay releases a hammer which shatters a small flask of hydrocyanic acid. If one has left this entire system to itself for an hour, one would say that the cat still lives if meanwhile no atom has decayed. The first atomic decay would have poisoned it. The Psi function for the entire system would express this by having in it the living and the dead cat (pardon the expression) mixed or smeared out in equal parts.

Introduction

It was not long before science-fiction writers picked up this evocative concept, often using it in a humorous vein. Several have taken the thought experiment a step further, pointing out or extra complications which might arise should the experiment actually be performed. For example, in his novel American Gods, Neil Gaiman has a character observe, "if they don’t ever open the box to feed it it’ll eventually just be two different kinds of dead." Likewise, Terry Pratchett’s Lords and Ladies adds the issue of a third possible state, in the case of Greebo, "Bloody Furious". Douglas Adams describes an attempt to enact the experiment in Dirk Gently’s Holistic Detective Agency. By using clairvoyance to see inside the box, it was found that the cat was neither alive nor dead, but missing, and Dirk’s services were employed in order to recover it. Schrödinger’s cat in fiction

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On a somewhat more serious level, Ian Stewart’s novel Flatterland (a sequel to Flatland ) attempts to explain many concepts in modern mathematics and physics through the device of having a young female Flatlander explore other parts of the "Mathiverse". Schrödinger’s Cat is just one of the many strange Mathiverse denizens she and her guide meet; the cat is still uncertain whether it is alive or dead, long after it left the box. Her guide, the Space Hopper, reassures the Cat with a modern view of quantum decoherence. Ursula K. Le Guin wrote a story entitled "Schrödinger’s Cat" in 1974 (reprinted in The Compass Rose, published in 1982), which also deals with decoherence. Greg Egan’s novel Quarantine, billed as "a story of quantum catastrophe", features an alternate solution to the paradox: in Egan’s version of quantum mechanics, the wave function does not collapse naturally. Only certain living things — human beings among them — collapse the wave function of things they observe. Humans are therefore highly dangerous to other lifeforms which require the full diversity of uncollapsed wavefunctions in order to survive.

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As Egan notes, Schrödinger’s hypothetical cat is one of the most familiar illustrations of quantum-mechanical oddities. In Quarantine, a physicist asks the narrator, an ex-cop and private investigator, if he has ever heard of "the quantum measurement problem". The narrator is naturally confused, but when asked if he’s heard of Schrödinger’s Cat, he replies, "Of course."

Fiction writers have confined other animals besides cats in such contraptions. Dan Simmons’ novel Endymion begins with hero Raul Endymion trapped in a Schrödinger-style box. In the fortieth-anniversary Doctor Who audio drama "Zagreus" (2003), the Doctor is locked in a lead-lined box also containing cyanide in an effort to explain his situation of being neither dead nor alive. Afterwards, the Doctor does mention that he had met Schrödinger’s Cat. In addition, the name "Schrödinger" has itself become an inside joke, often employed to elicit a chuckle from those familiar with physics. In an episode of the 1940s radio drama "Maddox’s Bedtime Stories for Kid Geniuses" entitled "Little Eve and Gogo", the main character is a teenage female scientist named Kat Schrödinger. Cats named Schrödinger appear in the television series Sliders and Stargate SG-1, for example, as well as in Carol Hill’s feminist science fiction classic "The Eleven Million Mile High Dancer". The animated series Futurama, several of whose production team had advanced degrees in science and math, includes many jokes of this sort; in the episode "A Clone of My Own", a brief shot reveals a nightclub called "Schrödinger’s Kit-Kat Club" (also an allusion to Cabaret). Robert A. Heinlein’s novel The Cat Who Walks Through Walls features a cat named Pixel, affectionately termed "Schrödinger’s Cat" due to his ability to be wherever his favorite person is. Pixel’s ability to walk through walls is due to the fact that he does not know that it is impossible. Schrödinger’s cat in fiction

542

Other assorted examples

•

•

•

The Schrödinger’s Cat trilogy is the name commonly given to a trilogy of science fiction/conspiracy theory novels written by Robert Anton Wilson. It consists of The Universe Next Door, The Trick Top Hat and The Homing Pigeons. In The Last Hero, another of Terry Pratchett’s Discworld novels, Death’s surly assistant Albert tries to explain to Death the concept of "Quantum" with an object lesson similar to "Schrödinger’s Cat." Death waves it off, saying, I DON’T HOLD WITH CRUELTY TO CATS (since, of course, Death speaks entirely in capital letters). In The Coming of the Quantum Cats, by Frederik Pohl, a man meets his many corresponding selves as he travels into the multiverse. In the manga Hellsing, there is a catboy named Schrödinger. He looks human, save for the furry ears perched on his head. This being claims to be everywhere and nowhere at once, and as such, has the ability to teleport and to survive contact-range gunshots through the skull. Steve Martin’s 1998 book Pure Drivel includes a piece entitled "Schrödinger’s Cat", which presents a summary of the theory, followed by several fictitious, nonsensical theories, including "Wittgenstein’s Banana", "Apollo’s Non-Apple Non-Strudel", and "Chef Boyardee’s Bungee Cord" (which begins, "A bungee cord is hooked at one end to a neutrino, while the other end is hooked to a vibraphone...") Saturday by Ian McEwan refers to Schrödinger’s Cat. Schrödinger’s Cat is discussed in "The Summer of Love" by Lisa Mason. A two-headed cat, one head being conscious the other sleeping or presumably dead, appears in one of Christopher Stasheff’s series of "Her Majesties Wizard" novels, along with other thought experiment beings, the most notable being Maxwell’s Demon. George Alec Effinger’s novelette, "Schrödinger’s Kitten" (1988), received both the Hugo and the Nebula Award. Schrodinger’s Kitten was published in book form in 1992. In " Hopdinger’s Cat 314", a short story by the Roaming Janitors, Colin demonstrates that Schrödinger’s cat itself is the problem with the experiment, since it, of course, knows whether it is alive or dead. (Contrast the cat in Stewart’s Flatterland, which lacks even this information.)

DR A

•

FT

Books and stories

•

• • •

•

•

314 http://www.fictionpress.com/read.php?storyid=1822969

Schrödinger’s cat in fiction

543

•

•

Schrödinger’s cat also appears in Nano by John Robert Marlow, which includes the theoretical experiment as well as a cat that is quite real. The manga Ah! My Goddess features a "Schrödinger’s Whale". This whale has the ability to travel through space, though it cannot stay at the same space for more than three days, as that would make its existence certain. This rule makes it difficult for such whales to find companions and mates. In the DC Comics series Animal Man, there was a storyline in which Animal Man found himself displaced in time. To explain his situation to his friend, Nowhere Man, he proposed "Schrodinger’s Pizza" — one with peppers, and one without.

Audiovisual media •

FT

•

DR A

In the episode of NUMB3RS entitled "Identity Crisis", Charlie re-phrases Schrödinger’s Cat to illustrate the point that the tests used to disprove/prove a murder suspect’s identity are built on a false premise. • In the anime Master of Mosquiton, there is a discussion between Schrödinger and the vampire Saint-Germain, wherein Germain claims he can control fate. To prove this, Germain predicts the cast of a die. Just as the die is about to land as predicted, Schrödinger’s pet cat bats the die so that it lands on a different number. • Schrödinger’s cat is referenced in the movie John Carpenter’s Prince Of Darkness. • A Dutch short movie from 1990 is called Schrödingers kat. A teacher explains the theory to his class. At the same time a man called Schrödinger (played by Bruno Ganz) dies of a heart attack in his apartment, while neighbours assume he is still alive and send him Christmas cards. 315 • A track on the Tears for Fears album Saturnine Martial & Lunatic (1996) is called "Schrödinger’s Cat". • Schrödinger’s cat is referenced in the TV series Stargate SG1

Games •

In the computer game NetHack, monsters known as quantum mechanics may carry a chest containing Schrödinger’s Cat. When opened, there is a 50% chance of finding it dead and a 50% chance of it jumping out alive. • In both Shin Megami Tensei: Digital Devil Saga games, Schrödinger is the name of a cat that always seems to be at the right place at the right time. • In Wild ARMs 3, the Schrödinger family owns a talking flying cat named Shady. A scene shows Shady retrieving a gem only to trigger an invisible

315 http://www.imdb.com/title/tt0210966/

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544

FT

barrier around him. While Maya Schrödinger explains about Shady’s claustrophobia, Shady screams "Whaaaaaaaaaaaah! No poison gas, please!" • In the MMORPG Kingdom of Loathing, the PvP betting game known as "the Money Making Game" has one message that uses Schrödinger’s cat as a contest. The betting player guesses whether the cat is dead or alive when the box is opened. • In the roleplaying game supplement GURPS Infinite Worlds, Schrödinger’s Cat is used to illustrate the Many-worlds interpretation. Infinite Worlds, as the name suggests, is a science-fiction setting where player-controlled characters can physically travel to the alternate realities proposed by manyworlds theory through the science of Parachronics.

Webcomics •

DR A

A cat named Schrödinger appears as a semi-recurring character in Checkerboard Nightmare, a webcomic by Kristofer Straub. Schrödinger can see all possible states of existence at once, and as a result is very much insane, as detailed in his first appearance 316. • The webcomic Two Lumps has an episode 317 about Schrödinger’s cat. "They’re talking about schreddin’ cats and puttin’ ’em in a box!!!" • Stephen Notley’s Bob the Angry Flower cartoon strip "Schrödinger’s Fridge" 318 stated that it could not be determined if beer was in the fridge or not. Although Bob whipped open the fridge door and was observing his "frickin’ head off" the beers would not actualize. Bob quickly cursed Quantum Physics. • A "Schrödinger Box" (which no one is allowed to open) is used in Agnostica celebrations in the comic strip Nukees.

Reference •

↑ E. Schrödinger, Die gegenwartige Situation in der Quantenmechanik, Naturwissenschaftern. 23: pp. 807-812; 823-823, 844-849. (1935). English translation: John D. Trimmer, Proceedings of the American Philosophical Society 124, pp. 323-38 (1980), reprinted in Quantum Theory and Measurement, p. 152 (1983).

316 http://checkerboardnightmare.com/d/20010319.html 317 http://www.twolumps.net/d/20040714.html 318 http://www.angryflower.com/schrod.gif

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External links Poem by Cecil Adams 319 Stephen Notley’s Bob the Angry Flower: Schrödinger’s Fridge 320 "The Adventures of Schrodinger the Cat" comic strip 321

FT

• • •

Source: http://en.wikipedia.org/wiki/Schr%C3%B6dinger%27s_cat_in_fiction

Schwinger’s variational principle

In Schwinger’s variational approach to quantum field theory, introduced by Julian Schwinger, the quantum action is an operator. This is unlike the functional integral (path integral) approach where the action is a classical functional.

DR A

Suppose we have a complete set of commuting operators (or anticommuting ˆ Let |A> be the eigenstate of Aˆ with eigenfor fermions) Aˆ and another set B. value A and similarly for |B>. There is some ambiguity in the phase, but that ˆ can be taken care of in the quantum action S AB associated with Aˆ and B. Suppose also we have not just one model of quantum mechanics or quantum field theory but a whole family of them, varying smoothly. So, |A> and |B> are "different" for each model in the family. S AB also varies smoothly. Schwinger’s variational principle tells us δ < A|B >= i < A|δSAB |B >.

Source: http://en.wikipedia.org/wiki/Schwinger%27s_variational_principle Principal Authors: Phys, Joyous!, Conscious, Matt McIrvin, Aranel

319 http://www.straightdope.com/classics/a1_122.html 320 http://www.angryflower.com/schrod.gif 321 http://glp.customer.netspace.net.au/schrod/

Schwinger’s variational principle

546

Selection rule

FT

In physics, especially in the context of quantum mechanics, a selection rule is a condition constraining the physical properties of the initial system and the final system that is necessary for a process to occur with a nonzero probability. See also: angular momentum coupling

In many cases, a transition involves the emission of radiation, that is, a photon is emitted. In general, electric (charge) radiation or magnetic (current, magnetic moment radiation) can be classified into multipoles E (electric) or M (magnetic) of order 2, e.g. E1, E2, E3 for electric dipole, quadrupole or octupole. The radiation field will be a sum of the multipole contributions; however, usually one or two multipoles dominate.

DR A

The emitted particle carries away an angular momentum , which for the photon must be at least 1, since it is a vector particle (i.e., it has J P = 1 -). Thus there is no E0 (electric monopoles) or M0 (magnetic monopoles) radiation (the latter is forbidden because magnetic monopoles do not seem to exist). Since the total angular momentum has to be conserved during the transition, we have that

Ji = Jf + λ

p

where kλk = λ(λ + 1) ~ , and its z-projection is given by λz = µ ~. The corresponding quantum numbers , µ must satisfy |Ji − Jf | ≤ λ ≤ Ji + Jf

and

µ = Mi − Mf .

Parity is also preserved. For electric multipole transitions π(Eλ) = πi πf = (−1)λ

while for magnetic multipoles π(Mλ) = πi πf = (−1)λ+1 .

Thus, parity does not change for E-even or M-odd multipoles, while it changes for E-odd or M-even multipoles.

Selection rule

547

FT

These considerations generate different sets of transitions rules depending on the multipole order and type. The expression forbidden transitions is often used; this does not mean that these transitions cannot occur, only that they are electric-dipole forbidden. These transitions are perfectly possible, they merely occur at a lower rate. If the rate for an E1 transition is non-zero, the transition is said to be permitted; if it is zero, then M1, E2, etc. transitions can still produce radiation, albeit with much lower transitions rates. These are the so-called forbidden transitions. The transition rate decreases by a factor of approximately 10 -3 from one multipole to the next one, so the lowest multipole transitions are most likely to occur. Semi-forbidden transitions (resulting in so called intercombination lines) are electric dipole (E1) transitions for which the selection rule that the spin does not change is violated. This is a result of the failure of LS coupling.

Summary table

Magnetic Electric dipole (M1) quadrupole (E2)

Magnetic quadrupole (M2)

DR A

Electric dipole (E1) Rigorous rules

(1)

∆J = 0, ±1 (J = 0 6↔ 0)

∆MJ = 0, ±1, ±2

If ∆S =

∆L = 0,0∆L ±1 = 0 0 (L = 0 6↔ 0)

Intermediate coupling

(6) If ∆S = ±1∆L = 0, ±1, ±2

πf = πi

None or one One electron One electron One electron electron jump∆l = jump∆l = jump∆l = jump∆l = ±1 ±1, ±3 0, ±2 0, ±2 If ∆S =

If ∆S =

∆L = 0, ±1, ±2 0 (L = 0 6↔ 0, 1)

0 (L

If ∆S =

If ∆S =

If ∆S =

∆L = 0, ±1, ±2, ±3 = 0 6↔ 0, 1, 2; 1 6↔ 1) If ∆S =

∆L = 0, ±1, ∆L = 0, ±1 ∆L = 0, ∆L ±1, = 0, ±1, ±2, ±3 ±2, ±3, ±4 ±2 ±1 (L = 0 6↔ 0) 6↔ (L 0, 1) = 0 6↔ 0) ±1 (L = 0 6↔ 0) ±1 (L = 0 ±1

External links • •

∆MJ = 0, ±1, ±2, ±3

πf = −πi

πf = πi

LS coupling (4) One electron No electron jump∆l = jump∆l = ±1 0, ∆n = 0 (5) If ∆S =

Magnetic octupole (M3)

∆J = 0, ±1, ±2 ∆J = 0, ±1, ±2, ±3 (J = 0 6↔ 0, 1; 12 6↔(021 )6↔ 0, 1, 2; 12 6↔ 12 , 32 ; 1 6↔

(2) ∆MJ = 0, ±1 (3) πf = −πi

Electric octupole (E3)

National Institue of Standards and Technology 322 Lecture notes from The University of Sheffield 323

322 http://physics.nist.gov/Pubs/AtSpec/node17.html 323 http://www.shef.ac.uk/physics/teaching/phy332/atomic_physics3.pdf

Selection rule

548

Source: http://en.wikipedia.org/wiki/Selection_rule

Semiclassical

FT

Principal Authors: Joriki, Gaius Cornelius, Laurascudder, Jag123, Passw0rd

In physics, the adjective semiclassical has different precise meanings depending on the context. All these meanings usually refer to some approximation, limit or situation that combines quantum and classical aspects in a given problem. The plurality of meanings comes from the fact that the passage from quantum to classical mechanics is generally a very difficult task. Some of the possible significations are the following:

DR A

First, semiclassical approximation may refer to quantum-mechanical calculations that are obtained by considering a small perturbation of a classical calculation, for example the WKB approximation in non-relativistic quantum mechanics or the loop expansion or the instanton methods in quantum field theory. In quantum field theory, a semiclassical correction arises from one-loop Feynman diagrams. The semiclassical effective action is h i Γ[φ] = S[φ] + 21 T r ln S (2) [φ] + ...

Second, in the context of open quantum systems and measurement theory, where one considers the dynamics of a given quantum system in interaction with an environment, the semiclassical regime may refer to the situation in which the wavefunction of the system is approximately peaked around the solution of the corresponding classical equations of motion. Corrections to the classical trajectory and the dispersion of the solution around the mean value are usually considered.

Third, semiclassical gravity is the approximation to the yet unknown theory of quantum gravity in which one treats matter fields as being quantum and the gravitational field as being classical. The classical Einstein equations are computed with the expectation value of the quantum matter fields in the classical background. Semiclassical gravity has applications in black hole physics and physical cosmology. A semiclassical approximation is any high frequency approximation (or "high energy approximation"), less extreme than classical mechanics, that is used to approximate quantum mechanics.

Semiclassical

549 Source: http://en.wikipedia.org/wiki/Semiclassical

FT

Principal Authors: Daniel Arteaga, QFT, Lumidek, Scottbeck, Wahoofive

Separable states

In quantum mechanics, separable quantum states are states without quantum entanglement.

Separable pure states

For simplicity, the following assumes all relevant state spaces are finite dimensional. First, consider separability for pure states.

Let H1 and H2 be quantum mechanical state spaces, that is, finite dimensional →Hilbert spaces with basis states {|ai i}ni=1 and {|bj i}m j=1 , respectively. By a postulate of quantum mechanics, the state space of the composite system is given by the tensor product

DR A

H1 ⊗ H2

N with base states {|ai i |bj i}, or in more compact notation {|ai bj i}. From the very definition of the tensor product, any vector of norm 1, i.e. a pure state of the composite system, can be written as |ψi = Σi,j ci,j |ai i ⊗ bj i = Σi,j ci,j |ai bj i

If a pure states |ψi ∈ H1 ⊗ H2 that can be written in the form |ψi = |ψ1 i ⊗ |ψ2 i where |ψi i is a pure state of the i-th subsystem, it is said to be separable. Otherwise it is called entangled. Formally, the embedding of a product of states into the product space is given by the Segre embedding. That is, a quantummechanical pure state is separable if and only if it is in the image of the Segre embedding. A standard  example of an (un-normalized) entangled state is

1 0  |ψi =  0 ∈ H ⊗ H 1

where H is the Hilbert space of dimension 2. We see that when a system is in an entangled pure state, it is not possible to assign states to its subsystems. This will be true, in the appropriate sense, for the mixed state case as well. Separable states

550 The above discussion can be extended to the case of when the state space is infinite dimensional with virtually nothing changed.

FT

Separability for mixed states

Consider the mixed state case. A mixed state of the composite system is described by a density matrix ρ acting on H1 ⊗ H2 . ρ is separable if there exist pk ≥ 0, {ρk1 } and {ρk2 } which are mixed states of the respective subsystems such that P ρ = k pk ρk1 ⊗ ρk2 where P

k pk

= 1.

DR A

Otherwise ρ is called an entangled state. We can assume without loss of generality in the above expression that {ρk1 } and {ρk2 } are all rank-1 projections, that is, they represent pure ensembles of the appropriate subsystems. It is clear from the definition that the family of separable states is a convex set. Notice that, again from the definition of the tensor product, any density matrix, indeed any matrix acting on the composite state space, can be trivially written in the desired form, if we drop the requirement that {ρk1 } and {ρk2 } are P themselves states and k pk = 1.

In the language of quantum communication, a separable state can be created from any other state using local actions and classical communication while an entangled state cannot. When the state spaces are infinite dimensional, density matrices are replaced by positive trace class operators with trace 1, and a state is separable if it can be approximated, in trace norm, by states of the above form.

Extending to the multipartite case

The above discussion generalizes easily to the case of a quantum system consisting of more than two subsystems. Let a system have n subsystems and have state space H = H1 ⊗ · · · ⊗ Hn . A pure state |ψi ∈ H is separable if it takes the form |ψi = |ψ1 i ⊗ · · · ⊗ |ψn i.

Similarly, a mixed state ρ acting on H is separable if it is a convex sum P ρ = k pk ρk1 ⊗ · · · ρkn .

Separable states

551 Or, in the infinite dimensional case, ρ is separable if it can be approximated in the trace norm by states of the above form.

FT

Separability criterion

The problem of deciding whether a state is separable in general is sometimes called the separability problem in quantum information. It is considered to be a difficult problem. A feeling for this difficulty can be obtained if we attempt to solve the problem by employing the obvious naive approach, for a fixed dimension. We see that, using the naive techniqie, the problem quickly becomes intractable, even for low dimensions. Thus more sophisticated formulations are required. The separability problem is a subject of current research.

A separability criterion is a necessary condition a state must satisfy to be separable. In the low dimensional (2 X 2 and 2 X 3 ) cases, the →Peres-Horodecki criterion is actually a necessary and sufficient condition for separability. Other separability criterions include the →Range criterion and Reduction criterion.

DR A

Characterization via algebraic geometry

Quantum mechanics may be modelled on a projective Hilbert space, and the categorical product of two such spaces is the the Segre embedding. In the bipartite case, a quantum state is separable if and only if it lies in the image of the Segre embedding.

See also •

→Entanglement witness

Source: http://en.wikipedia.org/wiki/Separable_states

Principal Authors: Mct mht, Matthew Mattic, Vegalabs, Oleg Alexandrov, Keenan Pepper

Separable states

552

Shelter Island Conference

FT

The first Shelter Island Conference on the Foundations of Quantum Mechanics was held from June 2-4, 1947 at the Ram’s Head Inn in Shelter Island, New York. The most famous participant, J. Robert Oppenheimer, deemed it the most successful scientific meeting he had ever attended. A relatively young Richard Feynman would later observe, "There have been many conferences in the world since, but I’ve never felt any to be as important as this." The conference cost $850. Shelter Island was the first major opportunity since Pearl Harbor and the Manhattan Project for the leaders of the American physics community to escape the paranoia of war. As Julian Schwinger would later recall, "It was the first time that people who had all this physics pent up in them for five years could talk to each other without somebody peering over their shoulders and saying, ’Is this cleared?’ "

DR A

Organization

The conference was conceived by Duncan MacInnes, a scientist studying electrochemistry at the Rockefeller Institute for Medical Research. Once the president of the New York Academy of Sciences, MacInnes had already organized a number of small scientific conferences. However, he believed that the later conferences had suffered from a bloated attendance, and over this issue, he resigned from the Academy in January 1945. That fall, he approached the National Academy of Sciences (NAS) with the idea of a series of 2–3 day conferences limited to 20–25 people. Frank Jewett, the head of the NAS, liked the idea; he envisioned a "meeting at some quiet place where the men could live together intimately", possibly "at an inn somewhere", and suggested that MacInnes focus on a couple of pilot programs. MacInnes’ first choice was "The Nature of Biopotentials", a subject close to his own heart; the second would be "The Postulates of Quantum Mechanics", which later became "Foundations of Quantum Mechanics". K. K. Darrow, a theoetical physicist at Bell Labs and secretary of the American Physical Society, offered his help in organizing the quantum mechanics conference. The two decided to emulate the success of the early Solvay Congresses, and they consulted with Léon Brillouin, who had some experience in that area. In turn, Brillouin suggested consulting Wolfgang Pauli, the recent Nobel medalist at the Institute for Advanced Study at Princeton.

Shelter Island Conference

553

FT

In January 1946, MacInnes, Darrow, Brillouin, and Pauli met in New York and exchanged letters. Pauli was enthusiastic about the topic, but he was primarily interested in bringing together the international physics community after the ordeal of the war. He suggested a large conference, including many older, foreign physicists, much to MacInnes’ chagrin. With Jewett’s encouragement, MacInnes asked Pauli for suggestions of "younger men" such as John Archibald Wheeler, explaining that the Rockefeller Foundation would support only a small conference. Pauli and Wheeler replied that MacInnes’ conference might be merged with Niels Bohr’s conference on Wave Mechanics in Denmark in 1947; they pointed out that the Niels Bohr Institute had close ties with the Rockefeller Foundation anyway. Darrow wrote to Wheeler that Bohr’s conference was a poor replacement because it would draw few Americans. Finally, Shelter Island was explicitly an American conference. In the coming months, Wheeler...

Lamb shift

DR A

The muon

Participants • • • • • • • •

Hans Bethe David Bohm Gregory Breit Karl K. Darrow Herman Feshbach Richard Feynman Hendrik Kramers Willis Lamb

• • • • • • • •

Duncan MacInnes Robert Eugene Marshak John von Neumann Arnold Nordsieck J. Robert Oppenheimer Abraham Pais Linus Pauling Isidor Isaac Rabi

• • • • • • • •

Bruno Rossi Julian Schwinger Robert Serber Edward Teller George Uhlenbeck John Hasbrouck van Vleck Victor Frederick Weisskopf John Archibald Wheeler

See also •

Solvay Conference

References

Primary sources •

•

Bethe, Hans (August 15, 1947). "The Electromagnetic Shift of Energy Levels" 324. Phys. Rev. 72 (4): 339–341. Lamb and Retherford (August 1, 1947). "Fine Structure of the Hydrogen Atom by a Microwave Method" 325. Phys. Rev. 72 (3): 241–243.

324 http://link.aps.org/abstract/PR/v72/p339

Shelter Island Conference

554 •

FT

Marshak and Bethe (September 15, 1947). "On the Two-Meson Hypothesis" 326. Phys. Rev. 72 (6): 506–509. • Weisskopf, Victor (September 15, 1947). "On the Production Process of Mesons" 327. Phys. Rev. 72 (6): 510. Reviews

•

•

•

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•

Robert Crease and Charles Mann (1996). The second creation: Makers of the revolution in twentieth-century physics. Rutgers UP. ISBN 0-81352177-7. Schweber, Silvan (1985). R. Jackiw, N. Khuri, S. Weinberg, E. Witten "A Short History of Shelter Island I". Shelter Island II: Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics, 301–343, Cambridge, MA: MIT Press. ISBN 0-26210031-2. Schweber, Silvan (1986). "Feynman and the visualization of space-time processes" 328. Rev. Mod. Phys. 58 (2): 449–508. Schweber, Silvan (1994). QED and the men who made it: Dyson, Feynman, Schwinger, and Tomonaga. Princeton UP. ISBN 0-691-03685-3. Smith, Richard (1996). Notes on the 1947 Shelter Island Conference and Its Participants 329. Retrieved on January 17, 2006.

•

External links

The Shelter Island Conference 330 from the National Academy of Sciences Excerpt from 1983 issue 331 of Physics Today This Month in Physics History, June 2000 332 from The American Physical Society • Shelter island conference 333 from Issues in Science and Technology Summer 1997, provided by ProQuest

• • •

325 http://link.aps.org/abstract/PR/v72/p241 326 http://link.aps.org/abstract/PR/v72/p506

327 http://link.aps.org/abstract/PR/v72/i6/p510/s1 328 http://prola.aps.org/abstract/RMP/v58/i2/p449_1

329 http://www.sherryart.com/nano/shelter.html 330 http://www7.nationalacademies.org/archives/shelterisland.html

331 http://feynman.physics.lsa.umich.edu/~mduff/talks/1983%20-%20Shelter%20Island%20Conference

/1983%20-%20Shelter%20Island%20Conference.pdf

332 http://www.aps.org/apsnews/0600/060006.cfm 333 http://www.findarticles.com/p/articles/mi_qa3622/is_199707/ai_n8765069/print

Shelter Island Conference

555 Source: http://en.wikipedia.org/wiki/Shelter_Island_Conference

FT

Single particle reconstruction

DR A

In physics, in the area of microscopy, single particle reconstruction is a technique in which large numbers of images (10,000 - 1,000,000) of ostensibly identical individual molecules or macromolecular assemblies are combined to produce a 3 dimensional reconstruction. This is a complementary technique to crystallography of biological molecules. As molecules/assembies become larger, it becomes more difficult to prepare high resolution crystals. For single particle reconstruction, the opposite is true. Larger objects actually improve the resolution of the final structure. In single particle reconstruction, the molecules/assemblies in solution are prepared in a thin layer of vitreous (glassy) ice, then imaged on an electron cryomicroscope (see Transmission electron microscopy). Images of individual molecules/assemblies are then selected from the micrograph and then a complex series of algorithms is applied to produce a full volumetric reconstruction of the molecule/assembly. In the 1990’s this technique was limited to roughly 2 nm resolution, providing only gross features of the objects being studied. However, recent improvements in both microscope technology as well as available computational capabilities now make 0.5 nm resolution possible.

References •

The National Center for Macromolecular Imaging, Houston Texas USA 334

Source: http://en.wikipedia.org/wiki/Single_particle_reconstruction

334 http://ncmi.bcm.tmc.edu

Single particle reconstruction

556

Slater determinant

FT

A Slater determinant (named after the American physicist John C. Slater) is an expression in quantum mechanics for the wavefunction of a many-fermion system, which by construction satisfies the Pauli principle. The Slater determinant arises from the consideration of a wavefunction for a collection of electrons. The wavefunction for each individual electron is known as a spin-orbital, χ(x), where x indicates the position and spin of the electron.

Two-particle case

The simplest way to approximate the wavefunction of a many-particle system is to take the product of properly chosen one-electron wavefunctions of the individual particles. For the two-particle case, we have Ψ(x1 , x2 ) = χ1 (x1 )χ2 (x2 )

DR A

This expression occurs in Hartree theory and is known as a Hartree product. However, it is not satisfactory for fermions, such as electrons, because the wavefunction is not antisymmetric. An antisymmetric wavefunction can be mathematically described as follows: Ψ(x1 , x2 ) = −Ψ(x2 , x1 )

Therefore the Hartree product does not satisfy the Pauli principle. This problem can be overcome by taking a linear combination of both Hartree products Ψ(x1 , x2 ) =

√1 {χ1 (x1 )χ2 (x2 ) − χ1 (x2 )χ2 (x1 )} 2

where the coefficient is a normalization factor. This wavefunction is antisymmetric and no longer distinguishes between electrons. Moreover, it also goes to zero if any two wavefunctions or two electrons are the same. This is equivalent to satisfying the Pauli exclusion principle.

Generalization to the Slater determinant The expression can be generalised to any number of fermions by writing it as a determinant. For an N-electron system, the Slater determinant is defined as

χ1 (x1 ) χ1 (x2 ) · · · χ2 (x1 ) χ2 (x2 ) · · · Ψ(x1 , x2 , . . . , xN ) = √1N ! . .. .. . χN (x1 ) χN (x2 ) · · · Slater determinant

χ1 (xN ) χ2 (xN ) .. . χN (xN )

557

FT

The linear combination of Hartree products for the two-particle case can clearly be seen as identical with the Slater determinant for N =2. It can be seen that the use of (Slater) determinants assures an antisymmetrized function on the outset, symmetric functions are automatically rejected. In the same way, the use of Slater determinants assures the obeying of the Pauli principle; the determinant will vanish if any of the two spin-orbitals are identical, for this leads to two identical columns. A single Slater determinant is used as an approximation to the electronic wavefunction in Hartree-Fock theory. In more accurate theories (such as configuration interaction and MCSCF), a linear combination of Slater determinants is needed.

Source: http://en.wikipedia.org/wiki/Slater_determinant

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Principal Authors: Karada, Karol Langner, Charles Matthews, Syntax, CiaPan

Spin-1/2

In quantum mechanics, spin is an intrinsic property of all elementary particles. Fermions, the particles that make up ordinary matter, have half-integer spin. An important special case of this are the spin-1/2 particles. All known elementary particles that are fermions have spin 1/2.

Overview

Particles which are spin-1/2 include the electron, the proton, the neutron, the neutrino, and the quarks. Spin-1/2 objects have peculiar dynamics which are not accurately described using classical physics; they are among the simplest systems which require quantum mechanics to describe them. As such, the study of the behavior of spin-1/2 systems forms a central part of undergraduate and graduate-level instruction in quantum mechanics.

General properties

Spin-1/2 objects are all Fermions, a fact explained by the spin statistics theorem, and satisfy the Pauli exclusion principle. Spin-1/2 particles can have a permanent magnetic moment along the direction of their spin, and this magnetic moment gives rise to electromagnetic interactions that depend on the spin. One such effect that was important in the discovery of spin is the →Zeeman effect. Spin-1/2

558

FT

Unlike in more complicated quantum mechanical systems, the spin of a spin1/2 particle can be expressed as a linear combination of just two eigenstates, or eigenspinors. These are traditionally labeled spin up and spin down. Because of this the quantum mechanical spin operators can be represented as simple 2x2 matrices, as opposed to the infinite dimensional matrices commonly needed to represent operators like energy or position. These matrices are called the Pauli matrices. Raising and lowering operators can be constructed for spin 1/2 objects; these obey the same commutation relations as other angular momentum operators.

Connection to the uncertainty principle

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One consequence of the generalized uncertainty principle is that the spin projection operators (which measure the spin along a given direction like x, y, or z), cannot be measured simultaneously. Physically, this means that it is ill defined what axis a particle is spinning about. A measurement of the z component of spin destroys any information about the x and y components that might previously have been obtained.

Stern-Gerlach experiment

When a spin-1/2 particle with non-zero magnetic moment like an electron is placed in an inhomogenous magnetic field, it experiences a force. This acts to separate out particles in the spin up state from particles in the spin down state. This is the idea behind the →Stern-Gerlach experiment.

Mathematical Description

The quantum state of the spin of a spin-1/2 particle can be described by a complex-valued vector with two components called a two-component spinor. When spinors are used to describe the quantum states, quantum mechanical operators are represented by 2 x 2, complex-valued Hermitian matrices. For example, the spin projection operator Sz effects a measurement of the spin in the z direction.  

Sz = ~2 σz =

~ 2

1 0

0 −1

Sz operator has two eigenvalues, of ± ~2 , which correspond to the eigenvectors

   1 = sz = + 21 = |↑i 0

Spin-1/2

559

   0 = sz = − 21 = |↓i 1

See also • • • •

Spin Spinor Fermions Pauli matrices

References

FT

These vectors form a complete basis for the →Hilbert space describing the spin-1/2 particle. Thus, linear combinations of these two states can represent all possible states of the spin.

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Griffiths, David J. (2005) Introduction to Quantum Mechanics(2nd ed.). Upper Saddle River, NJ: Pearson Prentice Hall. ISBN 0-13-111892-7.

Source: http://en.wikipedia.org/wiki/Spin-1/2

Spin-orbital

In quantum mechanics, a spinorbital is a one-particle wavefunction taking both the position and spin angular momentum of a particle as its parameters. The spinorbital of a single electron, for example, is a complex-valued function of four real variables: the three scalars used to define its position, and a fourth scalar, m s, which can be either +1/2 or -1/2: χ(x, y, z, ms )

We can also write it more compactly as a function of a position vector ~r = (x, y, z) and the quantum number m s: χ(~r, ms ).

For a general particle with spin s, m s can take values between -s to s in integer steps. The electron has s=1/2. A spinorbital is usually normalized, such that the probability of finding the particle anywhere in space with any spin is equal to 1: Spin-orbital

560 Ps

R

ms =−s ∞ d

3~ r

|χ(~r, ms )|2 = 1.

FT

From a normalized spinorbital, one can calculate the probability that the particle is in an arbitrary volume of space V and has an arbitrary spin ms : R P (V, ms ) = V d3~r |χ(~r, ms )|2 .

Source: http://en.wikipedia.org/wiki/Spin-orbital

Squashed entanglement

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Squashed entanglement, also called CMI entanglement (CMI can be pronounced "see me"), is an information theoretic measure of quantum entanglement for a bipartite quantum system. If %A,B is the density matrix of a system (A, B) composed of two subsystems A and B, then the CMI entanglement ECM I of system (A, B) is defined by Eq.(1)

ECM I (%A,B ) =

1 S(A 2 % min A,B,Λ ∈K

: B|Λ),

where K is the set of all density matrices %A,B,Λ for a tripartite system (A, B, Λ) such that %A,B = trΛ (%A,B,Λ ). Thus, CMI entanglement is defined as an extremum of a functional S(A : B|Λ) of %A,B,Λ . We define S(A : B|Λ), the quantum Conditional Mutual Information (CMI), below. A more general version of Eq.(1) replaces the “min" (minimum) in Eq.(1) by an “inf" (infimum).

Motivation for definition of CMI entanglement CMI entanglement has its roots in classical (non-quantum) information theory, as we explain next. Given any two random variables A, B, classical information theory defines the mutual information, a measure of correlations, as Eq.(2)

H(A : B) = H(A) + H(B) − H(A, B) .

For three random variables A, B, C, it defines the CMI as H(A : B|Λ) = H(A|Λ) + H(B|Λ) − H(A, B|Λ) Eq.(3) = H(Λ) − H(A, Λ) − H(B, Λ) + H(A, B, Λ) .

Squashed entanglement

561 It can be shown that H(A : B|Λ) ≥ 0.

Eq.(4)

FT

Now suppose %A,B,Λ is the density matrix for a tripartite system (A, B, Λ). We will represent the partial trace of %A,B,Λ with respect to one or two of its subsystems by %A,B,Λ with the symbol for the traced system erased. For example, %A,B = traceΛ (%A,B,Λ ). One can define a quantum analogue of Eq.(2) by S(A : B) = S(%A ) + S(%B ) − S(%A,B ) ,

and a quantum analogue of Eq.(3) by Eq.(5)

S(A : B|Λ) = S(%Λ ) − S(%A,Λ ) − S(%B,Λ ) + S(%A,B,Λ ) .

It can be shown that S(A : B|Λ) ≥ 0. This inequality is often called the strongsubadditivity property of quantum entropy.

Consider three random variables A, B, Λ with probability distribution PA,B,Λ (a, b, λ), which we will abbreviate as P (a, b, λ). For those special P (a, b, λ) of the form P (a, b, λ) = P (a|λ)P (b|λ)P (λ) ,

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Eq.(6)

it can be shown that H(A : B|Λ) = 0. Probability distributions of the form Eq.(6) are in fact described by the Bayesian network shown in Fig.1.

One can define a classical CMI entanglement by Eq.(7)

ECM I (PA,B ) =

min

PA,B,Λ ∈K

H(A : B|Λ),

where K is the set of all probability distributions PA,B,Λ in three random variP ables A, B, Λ, such that λ PA,B,Λ (a, b, λ) = PA,B (a, b) for all a, b. Because, given a probability distribution PA,B , one can always extend it to a probability distribution PA,B,Λ that satisfies Eq.(6), it follows that the classical CMI entanglement, ECM I (PA,B ), is zero for all PA,B . The fact that ECM I (PA,B ) always

Squashed entanglement

562 vanishes is an important motivation for the definition of ECM I (%A,B ). We want a measure of quantum entanglement that vanishes in the classical regime.

FT

Suppose wλ for λ = 1, 2, ..., dim(Λ) is a set of non-negative numbers that add up to one, and |λ > for λ = 1, 2, ..., dim(Λ) is an orthonormal basis for the Hilbert space associated with a quantum system Λ. Suppose %λA and %λB , for λ = 1, 2, ..., dim(Λ) are density matrices for the systems A and B, respectively. It can be shown that the following density matrix P Eq.(8) %A,B,Λ = λ %λA %λB wλ |λ >< λ|

satisfies S(A : B|Λ) = 0. Eq.(8) is the quantum counterpart of Eq.(6). TracP ing the density matrix of Eq.(8) over Λ, we get %A,B = λ %λA %λB wλ , which is a separable state. Therefore, ECM I (%A,B ) given by Eq.(1) vanishes for all separable states.

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λ Next suppose |ψA,B > for λ = 1, 2, ..., dim(Λ) are some states in the Hilbert space associated with a quantum system (A, B). Let K be the set of density matrices defined previously for Eq.(1). Define Ko to be the set of all density matrices %A,B,Λ that are elements of K and have the special form %A,B,Λ = P λ λ λ |ψA,B >< ψA,B |wλ |λ >< λ| . It can be shown that if we replace in Eq.(1) the set K by its proper subset Ko , then Eq.(1) reduces to the definition of entanglement of formation for mixed states, as given in Ben96. K and Ko represent different degrees of knowledge as to how %A,B,Λ was created. K represents total ignorance.

History

Classical CMI, given by Eq.(3), first entered information theory lore, shortly after Shannon’s seminal 1948 paper and at least as early as 1954 in McG54. The quantum CMI, given by Eq.(5), was first defined by Cerf and Adami in Cer96. However, it appears that Cerf and Adami did not realize the relation of CMI to entanglement or the possibility of obtaining a measure of quantum entanglement based on CMI; this can be inferred, for example, from a later paper, Cer97, where they try to use S(A|B) instead of CMI to understand entanglement. The first paper to explicitly point out a connection between CMI and quantum entanglement appears to be Tuc99. The final definition Eq.(1) of CMI entanglement was first given by Tucci in a series of 6 papers. (See, for example, Eq.(8) of Tuc02 and Eq.(42) of Tuc01a). In Tuc00b, he pointed out the classical probability motivation of Eq.(1), and its connection to entanglement of formation for mixed states. In Tuc01a, he presented an algorithm and computer program, based on the Arimoto-Blahut method of information theory, for calculating CMI entanglement numerically. Squashed entanglement

563 In Tuc01b, he calculated CMI entanglement analytically, for a mixed state of two qubits.

FT

In Hay03, Hayden, Jozsa, Petz and Winter explored the connection between quantum CMI and separability.

Since CMI entanglement reduces to entanglement of formation if one minimizes over Ko instead of K, one expects that CMI entanglement inherits many desirable properties from entanglement of formation. As first shown in Ben96, entanglement of formation does not increase under LOCC (Local Operations and Classical Communication). In Chr03, Christandl and Winter showed that CMI entanglement also does not increase under LOCC, by adapting Ben96 arguments about entanglement of formation. In Chr03, they also proved many other interesting inequalities concerning CMI entanglement, and explored its connection to other measures of entanglement. The name squashed entanglement first appeared in Chr03. In Chr05, Christandl and Winter calculated analytically the CMI entanglement of some interesting states. In Ali03, Alicki and Fannes proved the continuity of CMI entanglement.

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References •

•

•

•

• • •

Ali03 R. Alicki, M. Fannes, “Continuity of quantum mutual information", quant-ph/0312081 335 Ben96 C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, “Mixed State Entanglement and Quantum Error Correction", quant-ph/9604024 336 Cer96 N. J. Cerf, C. Adami, “Quantum Mechanics of Measurement", quantph/9605002 337 Cer97 N.J. Cerf, C. Adami, R.M. Gingrich, “Quantum conditional operator and a criterion for separability", quant-ph/9710001 338 Chr03 M. Christandl, A. Winter, “Squashed Entanglement - An Additive Entanglement Measure", quant-ph/0308088 339 Chr05 M. Christandl, A. Winter, “Uncertainty, Monogamy, and Locking of Quantum Correlations", quant-ph/0501090 340 Chr06 M. Christandl, Ph.D. Thesis, quant-ph/0604183 341

335 http://arxiv.org/abs/quant-ph/0312081 336 http://arxiv.org/abs/quant-ph/quant-ph/9604024 337 http://arxiv.org/abs/quant-ph/9605002 338 http://arxiv.org/abs/quant-ph/9710001 339 http://arxiv.org/abs/quant-ph/0308088 340 http://arxiv.org/abs/quant-ph/0501090 341 http://arxiv.org/abs/quant-ph/0604183

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• • • • • •

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•

Hay03 P. Hayden, R. Jozsa, D. Petz, A. Winter, “Structure of states which satisfy strong subadditivity of quantum entropy with equality" quantph/0304007 342 McG54 W.J. McGill, “Multivariate Information Transmission", IRE Trans. Info. Theory 4(1954) 93-111. Tuc99 R.R. Tucci, “Quantum Entanglement and Conditional Information Transmission", quant-ph/9909041 343 Tuc00a R.R. Tucci,“Separability of Density Matrices and Conditional Information Transmission", quant-ph/0005119 344 Tuc00b R.R. Tucci, “Entanglement of Formation and Conditional Information Transmission", quant-ph/0010041 345 Tuc01a R.R. Tucci, “Relaxation Method For Calculating Quantum Entanglement", quant-ph/0101123 346 Tuc01b R.R. Tucci, “Entanglement of Bell Mixtures of Two Qubits", quantph/0103040 347 Tuc02 R.R. Tucci, “Entanglement of Distillation and Conditional Mutual Information", quant-ph/0202144 348

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•

Source: http://en.wikipedia.org/wiki/Squashed_entanglement

Squeezed coherent state

In physics, a squeezed coherent state is every state in the →Hilbert space of quantum mechanics that saturates the uncertainty principle that is the product of the corresponding two operators takes on its minimum value: ∆x∆p =

~ 2

The simplest such state is the ground state |0i of the quantum harmonic oscillator. The next simple class of states that satisfies this identity are the family of coherent states |αi.

342 http://arxiv.org/abs/quant-ph/0304007 343 http://arxiv.org/abs/quant-ph/9909041 344 http://arxiv.org/abs/quant-ph/0005119 345 http://arxiv.org/abs/quant-ph/0010041 346 http://arxiv.org/abs/quant-ph/0101123 347 http://arxiv.org/abs/quant-ph/0103040 348 http://arxiv.org/abs/quant-ph/0202144

Squeezed coherent state

565

Mathematical definition

FT

The most general wave function that satisfies the identity above is the squeezed coherent state (we work in units with ~ = 1)   (x−x )2 ψ(x) = C exp − 2w20 + ip0 x 0

where C, x0 , w0 , p0 are constants (a normalization constant, the center of the wavepacket, its width, and its average momentum). The new feature relative to a coherent state is the free value of the width w0 , which is the reason why the state is called "squeezed". The squeezed state above is an eigenstate of a linear operator x ˆ + iˆ pw02

and the corresponding eigenvalue equals x0 + ip0 w02 . In this sense, it is a generalization of the ground state as well as the coherent state.

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Examples of squeezed coherent states

Depending on at which phase the state’s quantum noise is reduced one can distinguish amplitude-squeezed and phase-squeezed states or general quadrature squeezed states. If no coherent excitation exists the state is called a squeezed vacuum. The figures below give a nice visual demonstration of the close connection between squeezed states and Heisenbergs uncertainty relation: Diminishing the quantum noise at a specific quadrature (phase) of the wave has as a direct consequence an enhancement of the noise of the complementary quadrature, that is the field at the phase shifted by π/2. From the top: • • • • •

Vacuum state Squeezed vacuum state Phase-squeezed state arbitrary squeezed state Amplitude-squeezed state

As can be seen at once in contrast to the coherent state the quantum noise is not independent of the phase of the light wave anymore. A characteristic broadening and narrowing of the noise during one oscillation period can be observed. The wave packet of a squeezed state is defined by the square of the wave function introduced in the last paragraph. They correspond to the probability distribution of the electric field strength of the light wave. The moving wave packets display an oscillatory motion combined with the widening and Squeezed coherent state

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FT

566

Figure 45 Figure 1: Measured quantum noise of the electric field of different squeezed states in dependence of the phase of the light field. For the first two states a 3π-interval is shown, for the last three states, belonging to a different set of measurements it is a 4πinterval. (source: link 1 and ref. 3)

narrowing of their distribution: The "breathing" of the wave packet. For an amplitude-squeezed state, the most narrow distribution of the wave packet is reached at the field maximum, resulting in an amplitude that is defined more precisely than the one of a coherent state. For a phase-squeezed state the most narrow distribution is reached at field zero, resulting in an average phase value that is better defined than the one of a coherent state. In phase space quantum mechanical uncertainties can be depicted by Wigner distributions. The intensity of the light wave, its coherent excitation is given by the displacement of the Wigner distribution from the origin. A change in the phase of the squeezed quadrature results in a rotation of the distribution.

Squeezed coherent state

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FT

567

Figure 46 Figure 2: Oscillating wave packets of the five states.

Photon number distributions and phase distributions of squeezed states The squeezing angle, that is the phase with minimum quantum noise, has a large influence on the photon number distribution of the light wave and its phase distribution as well.

For amplitude squeezed light the photon number distribution is usually narrower than the one of a coherent state of the same amplitude resulting in subpoissonian light, whereas its phase distribution ist wider. The opposite ist true for the phase-squeezed light, which displays a large intensity (photon number) noise but a narrow phase distribution. For the squeezed vacuum state the photon number distribution displays oddeven-oscillations. This can be explained by the mathematical form of the squeezing operator, that resembles the operator for two-photon generation and annihilation processes. Photons in a squeezed vacuum state are more likely to appear in pairs.

Squeezed coherent state

FT

568

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Figure 47 Figure 3: Wigner functions of the five states. The ripples are due to experimental inaccuracies.

Experimental realizations of squeezed coherent states There has been a whole variety of successful demonstrations of squeezed states. The most prominent ones were experiments with light fields using lasers and non-linear optics. But squeezed states have also been realized via motional states of an ion in a trap, phonon states in crystal lattices or atom ensembles. Even macroscopic oscillators were driven into classical motional states that were very similar to squeezed coherent states.

Applications

Squeezed states of the light field can be used to enhance precision measurements. For example phase-squeezed light can improve the phase read out of interferometric measurements (see for example gravitational waves). Amplitudesqueezed light can improve the read out of very weak spectroscopic signals. Various squeezed coherent states, generalized to the case of many degrees of freedom, are used in various calculations in quantum field theory, for example Unruh effect and Hawking radiation (generally: particle production in curved backgrounds).

Squeezed coherent state

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FT

569

Figure 48 Figure 1: Measured quantum noise of the electric field of different squeezed states in dependence of the phase of the light field. For the first two states a 3π-interval is shown, for the last three states, belonging to a different set of measurements it is a 4π-interval. (source: link 1 and ref. 3)

See also •

Quantum optics

External links •

An introduction to quantum optics of the light field 349

349 http://gerdbreitenbach.de/gallery

Squeezed coherent state

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FT

570

Figure 49 Figure 2: Oscillating wave packets of the five states.

References •

Loudon, Rodney, The Quantum Theory of Light (Oxford University Press, 2000), [ISBN 0198501773] • D.F. Walls and G.J. Milburn, Quantum Optics, Springer Berlin 1994 • G. Breitenbach, S. Schiller, and J. Mlynek, "Measurement of the quantum states of squeezed light", Nature, 387, 471 (1997) 350

Source: http://en.wikipedia.org/wiki/Squeezed_coherent_state

Principal Authors: Gerd Breitenbach, TimBentley, Gaius Cornelius, Alai, Matt McIrvin

350 http://www.exphy.uni-duesseldorf.de/Publikationen/1997/N387/471z.htm

Squeezed coherent state

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571

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Figure 50 Figure 3: Wigner functions of the five states. The ripples are due to experimental inaccuracies.

SQUID

For other uses, see Squid (disambiguation).

SQUIDs, or Superconducting Quantum Interference Devices, are used to measure extremely small magnetic fields; they are currently the most sensitive 1 such devices (magnetometers) known, with noise levels as low as 3 fT·Hz - /2. While a typical fridge magnet is ten thousand microteslas, some processes in animals produce very small magnetic fields; typically sized between a nanotesla (a million femto-tesla or fT) and a microtesla (1000 nanotesla), and SQUIDs are well suited to studying these.

History and Design

The DC SQUID was invented in 1964 by Robert Jaklevic, John Lambe, Arnold Silver, and James Mercereau of Ford Research Labs after B. D. Josephson postulated the Josephson junction in 1962 and the first Josephson Junction was made by John Rowell and Philip Anderson at Bell Labs in 1963. The RF SQUID was invented in 1965 by James Edward Zimmerman and Arnold Silver at Ford.

SQUID

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FT

572

Figure 51 Figure 1: Measured quantum noise of the electric field of different squeezed states in dependence of the phase of the light field. For the first two states a 3π-interval is shown, for the last three states, belonging to a different set of measurements it is a 4π-interval. (source: link 1 and ref. 3)

There are two main types of SQUID, DC and RF (or AC). RF SQUIDs have only one Josephson junction whereas DC SQUIDs have two or more junctions. This makes DC SQUIDs more difficult and expensive to produce, but DC SQUIDs are much more sensitive. Most SQUIDs are fabricated from lead or pure niobium. The lead is usually in the form of an alloy with 10% gold or indium, as pure lead is unstable when its temperature is repeatedly changed. The base electrode of the SQUID is made of a very thin niobium layer, formed by deposition, and the tunnel barrier is oxidised onto this niobium surface. The top electrode is a layer of lead alloy deposited on top of the other two, forming a sandwich arrangement.

SQUID

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573

Figure 52 Figure 2: Oscillating wave packets of the five states.

More recently developed "High Temperature" SQUIDS are made of a substance called YBCO (chemical formula YBa 2Cu 3O 7-x), and are cooled by liquid nitrogen which is cheaper and more easily handled than liquid helium. They are less sensitive than conventional "Low Temperature" SQUIDS but many applications do not require the extreme sensitivity of the LT SQUID. The basic principle of operation is closely linked to flux quantisation. This is the phenomenon that the favoured states for a loop of superconductor are those where the flux inside is a multiple of the flux quantum.

Uses for SQUIDs

The extreme sensitivity of SQUIDs make them ideal for studies in biology. Magnetoencephalography (MEG), for example, uses measurements from an array of SQUIDs to make inferences about neural activity inside brains. Because SQUIDs can operate at acquisition rates much higher than the highest temporal frequency of interest in the signals emitted by the brain (kHz), MEG achieves good temporal resolution. Another application is the scanning SQUID microscope, which uses a SQUID immersed in liquid helium as the probe. The use of SQUIDs in oil prospecting, earthquake prediction and geothermal energy surveying is becoming more widespread as superconductor technology develops; SQUID

FT

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Figure 53 Figure 3: Wigner functions of the five states. The ripples are due to experimental inaccuracies.

they are also used as precision movement sensors in a variety of scientific applications, such as the detection of gravity waves. Four SQUIDs are currently employed on Gravity Probe B in order to test the limits of the theory of general relativity.

SQUIDs in Fiction

The science fiction writer William Gibson made reference to SQUIDs in his story Johnny Mnemonic, where a genetically engineered ex-military dolphin uses a SQUID implant to read a memory device in the title character’s brain. SQUIDs are also referenced in the film Strange Days, where they are used to record and play back human memories, which are exchanged on the black market.

See also

Spallation Neutron Source Superconducting RF Cavities

External links •

Dr. John Bland’s explanation 351.

351 http://www.cmp.liv.ac.uk/frink/thesis/thesis/node47.html

SQUID

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575

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Figure 54 Figure 4: Measured photon number distributions for an amplitude-squeezed state, a coherent state, and a phase squeezed state. Bars refer to theory, dots to experimental values. (source: link 1 and ref. 2)

Figure 55 Figure 5: Pegg-Barnett phase distribution of the three states.

Source: http://en.wikipedia.org/wiki/SQUID Principal Authors:

Gene Nygaard, Lostart, TwoOneTwo, Cspalletta, Tzartzam, Glenn, Slicky,

Frencheigh, McE, Maximus Rex

SQUID

FT

576

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Figure 56 Figure 4: Measured photon number distributions for an amplitudesqueezed state, a coherent state, and a phase squeezed state. Bars refer to theory, dots to experimental values. (source: link 1 and ref. 2)

Figure 57 Figure 5: Pegg-Barnett phase distribution of the three states.

Stark effect

In atomic physics, the Stark effect is the splitting and shift of a spectral line into several components in the presence of an electric field. The amount of splitting itself is called the Stark shift. It is analogous to the →Zeeman effect where a spectral line is split into several components in the presence of a magnetic field. Stark effect

FT

577

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Figure 58 Figure 4: Measured photon number distributions for an amplitudesqueezed state, a coherent state, and a phase squeezed state. Bars refer to theory, dots to experimental values. (source: link 1 and ref. 2)

Figure 59 Figure 5: Pegg-Barnett phase distribution of the three states.

The Stark effect is responsible for the pressure broadening (Stark broadening) of spectral lines by charged particles.

History

The effect is named after Johannes Stark, who discovered it in 1913. It was independently discovered in the same year by the Italian physicist Antonino Lo Surdo, and is thus sometimes called the Stark-Lo Surdo effect. Earlier, Stark effect

FT

578

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Figure 60 Figure 4: Measured photon number distributions for a squeezed-vacuum state. (source: link 1 and ref. 3)

unsuccessful, attempts to compute the magnitude of the effect, and to discover the perturbation, had been made by Voigt in 1899. In 1916, Epstein and Schwarzschild were able to perform computations using the Bohr model of the atom to exactly fit the magnitude of the Stark effect in hydrogen. In 1920, Hendrik Kramers was able to perform calculations within the Bohr model to estimate the relative intensities of the lines in the line pattern. While first-order perturbation effects for the Stark effect in hydrogen are in agreement for the Bohr model and the quantum-mechanical theory of the atom, higher order effects are not. Measurements of the Stark effect under high field strengths confirmed the correctness of the quantum theory over the Bohr model.

Mechanism

The effect arises because of the interaction between the electric dipole moment of an electron with an external electric field. If the electric field is uniform over the length scale of the atom, then the perturbing Hamiltonian is of the form ~ = eEz zˆ. H 1 = p~ · E

The first order energy shift of the state |ψm i due to the perturbation is given by ∆Em = eEz hψm | zˆ |ψm i (see Perturbation theory). Since the unperturbed Stark effect

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579

Figure 61 (SQUID).

A prototype of a Semiconductor Superconducting Quantum Interference Device

Figure 62 The inner workings of a early Superconducting Quantum Interference Device (SQUID).

states may be degenerate, we normally need to use the eigenvectors of H 1 Stark effect

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580

Figure 63 Second order Stark shifts in hydrogen. Magnetic quantum number: m = 1, parabolic quantum number: (n1 - n2) = -n+4,-n+2,...,n-2,n-4. States with higher principle quantum number n experience a greater energy shift for a given electric field strength.actual size

when calculating the energy shifts. The effect of H 1 is therefore to lift this degeneracy, which is observed experimentally as a splitting of spectral lines.

Quantum-Confined Stark Effect

In a semiconductor heterostructure, where a small bandgap material is sandwiched between two layers of a larger bandgap material, the Stark effect can be dramatically enhanced by bound excitons. This is due to the fact that the electron and hole which form the exciton are pulled in opposite directions by the applied electric field, but they remain confined in the smaller bandgap material, so the exciton is not merely pulled apart by the field. The quantum-confined Stark effect is widely used for semiconductor-based optical modulators, particularly for optical fiber communications.

References

• • •

Voigt, Annalen der Physik, 69, 297 (1899), and 4, 197 (1901). Epstein, Annalen der Physik, 50, 489 (1916). Schwarzschild, Sitzber. Berliner Akad., (1916) p. 548.

Stark effect

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Kramers, Danske Vidensk. Selsk. Skrifter (8), III, 3, 287. (1920), and Zeitschrift fur Physik, 3. 169 (1920). • E. U. Condon and G. H. Shortley (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN 521-09209-4. (Chapter 17 provides a comprehensive treatment, as of 1935.)

Source: http://en.wikipedia.org/wiki/Stark_effect

Principal Authors: Linas, Laurascudder, Lionelbrits, Piil, That Guy, From That Show!

Stationary state

In quantum mechanics, a stationary state is an eigenstate of a Hamiltonian. It is called stationary because, as an eigenstate, it is not subject to change or decay (to a lower energy state) over time.

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In practice, stationary states are never truly "stationary" for all time. Rather, they refer to the eigenstate of a Hamiltonian where small perturbative effects have been ignored. The language allows one to discuss the eigenstates of the unperturbed Hamiltonian, whereas the perturbation will eventually cause the stationary state to decay. The only true stationary state is the ground state.

Ground state

The ground state of a quantum mechanical system is its lowest-energy state. An excited state is any state with energy greater than the ground state. The ground state of a quantum field theory is usually called the vacuum state or the vacuum. If more than one ground state exists, they are said to be degenerate. Many systems have degenerate ground states, for example, the hydrogen atom. It turns out that degeneracy occurs whenever a nontrivial unitary operator commutes with the Hamiltonian of the system. According to the third law of thermodynamics, a system at absolute zero temperature exists in its ground state; thus, its entropy is determined by the degeneracy of the ground state. Many systems, such as a perfect crystal lattice, have a unique ground state and therefore have zero entropy at absolute zero (because ln(1) = 0). The condition of an atom, ion, or molecule, when all of its electrons are in their lowest possible energy levels, is called, not excited. When an atom is in its ground state, its electrons fill the lowest energy orbitals completely before they Stationary state

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See also • • •

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begin to occupy higher energy orbitals, and they fill subshells in accordance with Hund’s rule (usually!).

Quantum number Quantum mechanic vacuum or vacuum state Virtual particle

Source: http://en.wikipedia.org/wiki/Stationary_state

Stern-Gerlach experiment

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In quantum mechanics, the Stern-Gerlach experiment, named after Otto Stern and Walther Gerlach, is a celebrated experiment in 1920 on deflection of particles, often used to illustrate basic principles of quantum mechanics. It can be used to demonstrate that electrons and atoms have intrinsically quantum properties, and how measurement in quantum mechanics affects the system being measured.

Basic theory and description

Figure 64 Classical model of spinning particle.

Otto Stern and Walther Gerlach devised an experiment that would determine whether particles had any intrinsic angular momentum. If we imagine a classical system, such as the earth orbiting the sun, then the earth has angular momentum from both its orbit around the sun and the orbit around its axis (its spin). We are seeking to determine whether individual particles like electrons have any "spin" angular momentum. To do this, we consider the electron to be like a classical dipole with two halves of charge spinning quickly. If this particle is in a magnetic field, it will begin to precess because of the torque that the magnetic field is exerting on the dipole (see Torque-induced precession). Stern-Gerlach experiment

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If the particle is travelling in a homogeneous magnetic field, then the forces exerted on opposite ends of the dipole cancel each other out and the motion of the particle is unaffected. If we are conducting the experiment using electrons, then we must compensate for the tendency of any charged particle to curl in its path through a magnetic field (see cyclotron motion). This can be done easily using an electric field of appropriate magnitude and oriented transverse to the charged particle’s path. We can then ignore the fact that electrons are charged. The Stern-Gerlach experiment can be conducted using electrically neutral particles and the same conclusion is reached. We are concerning ourselves with angular momentum only, not any electrostatic phenomena.

Figure 65

Basic elements of the Stern-Gerlach experiment.

If our particle is travelling through an inhomogeneous magnetic field, then the force on one end of the dipole will be slightly greater than the opposing force on the other end of the dipole. This leads to the particle being deflected in the inhomogeneous magnetic field. The direction in which the particles are deflected is typically called the "z" direction. If our particles are classical, "spinning" particles, then we would expect the distribution of their spin angular momentum vectors to be truly random and thus each particle would be deflected up or down by a different amount. Thus, we would expect an even distribution on the screen of our detector. Instead, we find that the particles passing through the device are deflected either up or down by a specific amount! This can only mean that spin angular momentum is quantized, i.e. it can only take on discrete values. There is not a continuous distribution of possible angular momenta!

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Figure 66

Spin values for fermions.

Electrons are spin-1/2 particles. These have only two possible spin values, called spin-up and spin-down. The exact value of their spin is +h/2 or -h/2. What does this quantity imply about the nature of these particles? If this value arises as a result of the particles rotating the way a planet rotates, then the individual particles would have to be spinning impossibly fast! The speed of rotation would be in excess of the speed of light and thus impossible. We conclude that the spin angular momentum has nothing to do with rotation and is a purely quantum mechanical phenomenon. That is why it is sometimes known as the "intrinsic angular momentum." For electrons, two possible values for spin exist, as well as for the proton and the neutron, which are composite particles made up of three quarks each, which are themselves spin 1/2 particles. Other particles may have a different number of possible values. Delta baryons (∆ ++, ∆ +, ∆ 0, ∆ -), for example, are spin-3/2 particles and have four possible values for spin angular momentum. Vector mesons, as well as photons, W and Z bosons and gluons are spin-1 particles and have three possible values for spin angular momentum. To describe the experiment with spin-1/2 particles mathematically, it is easiest to use Dirac’s bra-ket notation. As the particles pass through the Stern-Gerlach device, they are "being observed." The act of observation in quantum mechanics is equivalent to measuring them. Our observation device is the detector and Stern-Gerlach experiment

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2

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in this case we can observe one of two possible values, either spin up or spin down. These are described by the angular momentum quantum number j, which can take on one of the two possible allowed values, either +h/2 or h/2. The act of observing (measuring) corresponds to the operator J z. In mathematical terms, E E |ψi = c1 ψj=+ ~ + c2 ψj=− ~ 2

The constants c 1 and c 2 are complex numbers. The square of their absolute values determines the probability of the state |ψ> being found with one of the two possible values for j. The constants must also be normalized so the probability of finding the wavefunction in one of either state is unity. Here we know that the probability of finding the particle in each state is 0.5. Therefore we also the values of the constants c 1 and c 2. These are √1 2

c2 =

√1 2

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c1 =

Sequential experiments

If we combine some Stern-Gerlach apparati we can clearly see that they do not act as simple selectors, but alter the states observed (as in light polarization), according to quantum mechanics laws:

History

The Stern-Gerlach experiment was performed in Frankfurt, Germany in 1920 by Otto Stern and Walther Gerlach. At the time, Stern was an assistant to Max Born at the University of Frankfurt’s Institute for Theoretical Physics, and

Stern-Gerlach experiment

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Figure 67

A plaque at the Frankfurt institute commemorating the experiment

Gerlach was an assistant at the same university’s Institute for Experimental Physics. At the time of the experiment, the most prevalent model for describing the atom was the →Bohr model, which described electrons as going around the positively-charged nucleus only in certain discrete atomic orbitals or energy levels. Since the electron was quantized to be only in certain positions in space, the separation into distinct orbits was referred to as space quantization.

Impact

The Stern-Gerlach experiment, had one of the biggest impacts on modern physics: •

In the decade that followed, scientists showed using similar techniques, that the nucleus of some atoms also have quantized angular momentum. It is the interaction with the spin of the electron that is responsible for the hyperfine structure of the spectroscopic lines.

•

In the thirties, using an extended version of the S-G apparatus, Isidor Rabi and colleagues showed that by using a varying magnetic field, one can force the magnetic momentum to go from one state to the other. The series of experiments culminated in 1937 when they discovered that state transitions Stern-Gerlach experiment

587 could be induced using time varying fields or RF fields. The so called Rabi oscillation is the working mechanism for the Magnetic Resonance Imaging equipment found in hospitals. Later Norman F. Ramsey, modified the Rabi apparatus to increase the interaction time with the field. The extreme sensitivity due to frequency of the radiation makes this very useful for keeping accurate time, and is still used today in atomic clocks.

•

In the early sixties, Ramsey and Daniel Kleppner used a S-G system to produce a beam of polarized hydrogen as the source of energy for the Hydrogen Maser, which is still one of the most popular atomic clocks.

•

The direct observation of the spin is the most direct proof of quantization and in quantum mechanics.

External links

Stern-Gerlach Experiment Java Applet Animation 352 Detailed explanation of the Stern-Gerlach Experiment 353

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• •

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•

References •

Friedrich, Bretislav and Herschbach, Dudley. "Stern and Gerlach: How a Bad Cigar Helped Reorient Atomic Physics" 354 Physics Today, December 2003.

Source: http://en.wikipedia.org/wiki/Stern-Gerlach_experiment

Principal Authors: BeardedPhysicist, Michael Hardy, Creidieki, Sonett72, Theresa knott

352 http://www.if.ufrgs.br/~betz/quantum/SGPeng.htm 353 http://galileo.phys.virginia.edu/classes/252/Angular_Momentum/Angular_Momentum.html 354 http://www.physicstoday.org/vol-56/iss-12/p53.html

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Subatomic particle

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A subatomic particle is a particle smaller than an atom: it may be elementary or composite. Particle physics and nuclear physics concern themselves with the study of these particles, their interactions, and matter made up of them which do not aggregate into atoms.

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These particles include atomic constituents such as electrons, protons, and neutrons (protons and neutrons are actually composite particles, made up of quarks), as well as other particles such as photons and neutrinos which are produced copiously in the sun. However, most of the particles that have been discovered and studied are not encountered under normal earth conditions; they are produced in cosmic rays and during scattering processes in particle accelerators.

Figure 68 Helium atom (schematic) Showing two protons (red), two neutrons (green) and two electrons (yellow).

Dividing an atom

The study of electrochemistry led G. Johnstone Stoney to postulate the existence of the electron (denoted e -) in 1874 as a constituent of the atom. It was observed in 1897 by J. J. Thomson. Subsequent speculation about the structure of atoms was severely constrained by the 1907 experiment of Ernest Rutherford which showed that the atom was mostly empty space, and almost all its mass was concentrated into the (relatively) tiny atomic nucleus. The development of the quantum theory led to the understanding of chemistry in Subatomic particle

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terms of the arrangement of electrons in the mostly empty volume of atoms. Protons (p +) were known to be the nucleus of the hydrogen atom. Neutrons (n) were postulated by Rutherford and discovered by James Chadwick in 1932. The word nucleon denotes both the neutron and the proton. Electrons, which are negatively charged, have a mass of 1/1836 of a hydrogen atom, the remainder of the atom’s mass coming from the positively charged proton. The atomic number of an element counts the number of protons. Neutrons are neutral particles with a mass almost equal to that of the proton. Different isotopes of the same nucleus contain the same number of protons but differing numbers of neutrons. The mass number of a nucleus counts the total number of nucleons.

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Chemistry concerns itself with the arrangement of electrons in atoms and molecules, and nuclear physics with the arrangement of protons and neutrons in a nucleus. The study of subatomic particles, atoms and molecules, their structure and interactions, involves quantum mechanics and quantum field theory (when dealing with processes that change the number of particles). The study of subatomic particles per se is called particle physics. Since many particles need to be created in high energy particle accelerators or cosmic rays, sometimes particle physics is also called high energy physics.

Classification of subatomic particles

Symmetries play a very important role in the physics of subatomic particles by providing intrinsic quantum numbers which are used to classify particles. Poincare symmetry, which is the full symmetry of special relativity, is enjoyed by any Hamiltonian which describes these particles. Hence all particles have the following quantum numbers —

• •

•

the mass (m) of the particle, its spin (J): all particles with integer values of spin are called bosons, those with half-integer spins are called fermions. its intrinsic parity (P), which is a multiplicative quantum number.

In addition, some particles may have a definite C-parity (C). Particles may also carry other quantum numbers related to internal symmetries, such as charges and flavour quantum numbers. Corresponding to every particle there exists an antiparticle. Every additive quantum number of a particle is reversed in sign for the antiparticle. Equality of the masses and lifetimes of particle and antiparticle follows in local quantum field theories through CPT symmetry, and hence tests of these equalities constitute important tests of this symmetry. Subatomic particle

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Elementary particles

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A full classification of subatomic particles involves understanding the fundamental forces that they are subject to: the electromagnetic, weak and strong forces. In the modern unified quantum field theory of these three forces, called the standard model, the elementary particles are

spin J = 1 particles called gauge bosons. These include • photons, which are carriers of the electromagnetic force, • W bosons and Z bosons which mediate the weak forces, and • gluons, which carry the strong force. • spin J = 1/2 fermions which constitute all matter in the universe and come in two varieties— • leptons such as the electron, muon, tau lepton, the three corresponding neutrinos (these are called six flavours of leptons), and their antiparticles. These are affected essentially only by the weak and electromagnetic forces. The former allow flavour changes (for example, from a muon to an electron) • quarks which come in six other flavours, and are affected by all three forces unified into the standard model. The weak interactions cause flavour changes. • spin J = 0 (and P = +1) Higgs boson which is responsible for the masses of the quarks, leptons, W and Z bosons. This remains to be actually seen in experiments; a major purpose of the Large Hadron Collider (LHC) is to search for this particle.

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•

Conjectures and predictions

Further structures beyond the standard model are often invoked. In particular, there is a search for a theory that unifies the standard model with gravity. There is strong evidence that when such a theory is found it will include gravitons (constrained to have spin J = 2), to mediate this fourth fundamental interaction. A further structure called supersymmetry is often invoked, although direct experimental evidence for it is lacking. Supersymmetric extensions of the standard model would contain a bosonic partner for each of the fermions described above (called selectrons, smuons, staus, sneutrinos, squarks), and a fermionic partner for each boson (called gauginos and Higgsinos). Supersymmetric extensions which include a theory of gravity (called supergravity) also involve a partner of the graviton, called the gravitino, which has spin J = 3/2. In many versions of these theories there are extra bosons called axions with J = 0 and P = -1. Relic particles are postulated to be remnants of the early cosmological expansion of the Big Bang. Subatomic particle

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Composite particles

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There were attempts to build theories which posited that the elementary particles in the standard model are actually composites built out of really elementary particles variously called preons, rishons or quinks. However, these theories are so strongly constrained by experimental data now that they are almost ruled out. Extended supersymmetric theories have also been postulated; these allow particles such as leptoquarks, which transmute leptons into quarks.

All observed subatomic composite particles are called hadrons. All bosonic hadrons are called mesons and all fermionic hadrons are baryons. The most well-known baryons are the constituents of atomic nuclei called protons and neutrons, and collectively named nucleons. The quark model of hadrons posits that mesons are built out of a quark and an antiquark, whereas a baryon is made up of three quarks. As of 2005, searches for exotic hadrons are currently under way.

History

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J. J. Thomson discovered electrons in 1897. In 1905 Albert Einstein demonstrated the physical reality of the photons which were postulated by Max Planck in order to solve the problem of black body radiation in thermodynamics. Ernest Rutherford discovered in 1907 in the gold foil experiment that the atom is mainly empty space, and that it contains a heavy but small atomic nucleus. The early successes of the quantum theory involved explaining properties of atoms in terms of their electronic structure. The proton was soon identified as the nucleus of hydrogen. The neutron was postulated by Rutherford following his discovery of the nucleus, but was discovered by James Chadwick much later, in 1932. Neutrinos were postulated in 1931 by Wolfgang Pauli (and named by Enrico Fermi) to be produced in beta decays (the weak interaction) of neutrons, but were not discovered till 1956. Pions were postulated by Hideki Yukawa as mediators of the strong force which binds the nucleus together. The muon was discovered in 1936 by Carl D. Anderson, and initially mistaken for the pion. In the 1950s the first kaons were discovered in cosmic rays. The development of new particle accelerators and particle detectors in the 1950s led to the discovery of a huge variety of hadrons, prompting Wolfgang Pauli’s remark: "Had I foreseen this, I would have gone into botany". The classification of hadrons through the quark model in 1961 was the beginning of the golden age of modern particle physics, which culminated in the completion of the unified theory called the standard model in the 1970s. The discovery of the gauge bosons through the 1980s, and the verification of their properties through the 1990s is considered to be an age of consolidation in particle physics. Among the standard model particles the existence of the Higgs boson remains to be Subatomic particle

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See also •

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verified—this is seen as the primary physics goal of the accelerator called the Large Hadron Collider in CERN. All particles found till now fit into the standard model.

Poincare symmetry, CPT invariance, spin statistics theorem, bosons and fermions. • Particle physics, list of particles, the quark model and the standard model.

External links

particleadventure.org: The Standard Model 355 particleadventure.org: Particle chart 356 University of California: Particle Data Group 357 Annotated Physics Encyclopædia: Quantum Field Theory 358 Jose Galvez: Chapter 1 Electrodynamics (pdf) 359

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• • • • •

Source: http://en.wikipedia.org/wiki/Subatomic_particle

Principal Authors: Bambaiah, Xerxes314, Ahoerstemeier, Glenn, Bevo, Doug Bell, LittleDan, GraemeL

Superdense coding

Superdense coding is a technique used in quantum information theory to send two bits of classical information using only one qubit, with the aid of entanglement.

The idea

Suppose Alice would like to send classical information to Bob using qubits. Alice would encode the classical information in a qubit and send it to Bob. After receiving the qubit, Bob recovers the classical information via measurement. The question is: how much classical information can be transmitted per qubit? Since non-orthogonal quantum states can not be distinguished reliably,

355 http://particleadventure.org/particleadventure/frameless/standard_model.html 356 http://particleadventure.org/particleadventure/frameless/chart.html 357 http://pdg.lbl.gov/ 358 http://web.mit.edu/redingtn/www/netadv/qft.html

359 http://jgalvez.home.cern.ch/jgalvez/School/pdf/LM-WeakIteractions.pdf

Superdense coding

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one would guess that Alice can do no better than one classical bit per qubit. Indeed this bound on efficiency has been proven formally. Thus there is no advantage gained in using qubits instead of classical bits. However, with the additional assumption that Alice and Bob share an entangled state, two classical bits per qubit can be achieved. The term superdense refers to this doubling of efficiency. We next describe this prodedure.

Crucial to the procedure is the shared entangled state between Alice and Bob, and the property of entangled state that a (maximally) entangled states can be transformed into another such state via local manipulation. Suppose parts of a Bell state, say |Ψ+ i =

√1 (|0iA 2

⊗ |1iB + |1iA ⊗ |0iB )

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is distributed to Alice and Bob. The first subsystem, denoted by subscript A, belongs to Alice and the second, B, system to Bob. By only manipulating her particle locally, Alice can transform the composite system into any one of the Bell states (this is not so surprising, since entanglement can not be broken using local operations): •

Obviously, if Alice does nothing, the system remains in the state |Ψ+ i.

•

If Alice sends her particle through the unitary gate



0 1 σ1 = 1 0



(notice this is one of the Pauli matrices), the total two-particle system now is in state (σ1 ⊗ I)|Ψ+ i = |Φ+ i.

•

If σ1 is replaced by σ3 , the initial state |Ψ+ i is transformed into |Ψ− i.

•

Similarly, if Alice applies iσ2 ⊗ I to the system, the resulting state is |Φ− i

So, depending on the message she would like to send, Alice performs one of the four local operations given above and sends her qubit to Bob. By performing a projective measurement in the Bell basis on the two particle system, Bob decodes the desired message. Superdense coding

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References •

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Notice that if some mischievous person, Carol, intercepts Alice’s qubit en route to Bob, all that is obtained by Carol is part of an entangled state. No useful information whatsoever is gained by Carol, unless she can interact with Bob’s qubit.

C. Bennett and S.J. Wiesner. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys. Rev. Lett., 69:2881, 1992 360

Source: http://en.wikipedia.org/wiki/Superdense_coding

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Superselection sector

A superselection sector is a concept used in quantum mechanics. One of the insights of quantum mechanics is that not all self-adjoint operators are observables. Suppose we are given an operator unital *-algebra A and an observable unital *-subalgebra O (The observables would then correspond to the self-adjoint elements of O). A reducible unitary representation of O is decomposable into the direct sum of inequivalent irreducible unitary representations of O (I’ll explain why they have to be inequivalent in a moment). Each irrep is called a superselection sector. →Observables map a state in each irrep into another state in the same irrep. The relative phase of a superposition of nonzero states from different irreps is not observable (the expectation values of the observables can’t distinguish between them). In the density state formulation where states are positive linear functionals of O where the unit of O is mapped to 1 (the unit, 1 O is "intuitively" an observable, a trivial one, no doubt), this would correspond to a mixed state; and in fact, all possible values for the relative phase would give rise to the same state. (In the density state formulation, the direct sum of two or more equivalent irreps would give rise to exactly the same states as a single rep alone, so by Occam’s razor, we can cut off all but one irrep)

360 http://prola.aps.org/abstract/PRL/v69/i20/p2881_1

Superselection sector

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Symmetries often give rise to superselections sectors; but this is not the only reason for the occurrence of superselection sectors. In the study of decoherence, for example, if we restrict ourselves to observations in a local region, we can have approximate superselection sectors. Say a group G acts upon A, and H is a unitary rep of A, and also a unitary rep of G such that for all g in G, a in A and |ψi in H, g [a|ψi] = g [a] g [|ψi]

(i.e. the representation of H, as a unitary rep of A is a G -intertwiner).

O is an invariant subalgebra of A under G (all observables are invariant under G, but not every self-adjoint operator invariant under G is necessarily an observable). H decomposes into superselection sectors, each of which is the direct product of an irrep of G and a rep of O. The irrep of G acts trivially under O. Using the same Occam’s razor argument as above, we can reduce it to a rep of O alone. However, we can still keep the irrep of G as a label for the superselection sector, if we wish.

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Actually, insisting that H is a rep of G is unnecessarily restrictive. To take an example, let G be the Lorentz group and A be an operator algebra rep of G. H could be a Hilbert space of A which contains, among other things states with an odd number of fermions. This would mean H is not a rep of the Lorentz group, although it IS a rep of its double cover. Or let’s say G is a group with an extension K. A is again an algebra rep of G. (Any rep of G can be turned into a rep of K) Then, it’s possible to have a unitary rep of A, H which is a unitary rep of K but not G. Actually, we can be more general than that. Replacing G with a Lie algebra, Lie superalgebra or a Hopf algebra would still work. See algebra representation of a Lie superalgebra, unitary representation of a star Lie superalgebra, algebra representation of a Hopf algebra and representation of a Hopf algebra.

Examples

A simple example would be a quantum mechanical particle confined to a closed loop (i.e. a periodic line of period L ). The superselection sectors are labeled by an angle θ between 0 and 2π. All the wave functions within a single superselection sector satisfy ψ (x + L) = eiθ ψ (x)

This would be the →Aharonov-Bohm effect, if we introduced a locally flat connection A. Source: http://en.wikipedia.org/wiki/Superselection_sector

Superselection sector

596 Principal Authors: Phys, Kebes, Charles Matthews, Sam Hocevar, Woohookitty

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Supersymmetric quantum mechanics

In theoretical physics, supersymmetric quantum mechanics is an area of research where mathematical concepts from high-energy physics are applied to the seemingly more prosaic field of quantum mechanics.

Introduction

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Understanding the consequences of supersymmetry has proven mathematically daunting, and it has likewise been difficult to develop theories that could account for symmetry breaking, i.e., the lack of observed partner particles of equal mass. To make progress on these problems, physicists developed supersymmetric quantum mechanics, an application of the SUSY superalgebra to quantum mechanics as opposed to quantum field theory. It was hoped that studying SUSY’s consequences in this simpler setting would lead to new understanding; remarkably, the effort created new areas of research in quantum mechanics itself. For example, as of 2004 students are typically taught to "solve" the hydrogen atom by a laborious process which begins by inserting the Coulomb potential into the →Schrödinger equation. After a considerable amount of work using many differential equations, the analysis produces a recursion relation for the Laguerre polynomials. The final outcome is the spectrum of hydrogen-atom energy states (labeled by quantum numbers n and l ). Using ideas drawn from SUSY, the final result can be derived with significantly greater ease, in much the same way that operator methods are used to solve the harmonic oscillator. Oddly enough, this approach is analogous to the way Erwin Schrödinger first solved the hydrogen atom. Of course, he did not call his solution supersymmetric, as SUSY was thirty years in the future—but it is still remarkable that the SUSY approach, both older and more elegant, is taught in so few universities. The SUSY solution of the hydrogen atom is only one example of the very general class of solutions which SUSY provides to shape-invariant potentials, a category which includes most potentials taught in introductory quantum mechanics courses. SUSY quantum mechanics involves pairs of Hamiltonians which share a particular mathematical relationship, which are called partner Hamiltonians. (The potential energy terms which occur in the Hamiltonians are then called partner potentials.) An introductory theorem shows that for every eigenstate of Supersymmetric quantum mechanics

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one Hamiltonian, its partner Hamiltonian has a corresponding eigenstate with the same energy (except possibly for zero energy eigenstates). This fact can be exploited to deduce many properties of the eigenstate spectrum. It is analogous to the original description of SUSY, which referred to bosons and fermions. We can imagine a "bosonic Hamiltonian", whose eigenstates are the various bosons of our theory. The SUSY partner of this Hamiltonian would be "fermionic", and its eigenstates would be the theory’s fermions. Each boson would have a fermionic partner of equal energy—but, in the relativistic world, energy and mass are interchangeable, so we can just as easily say that the partner particles have equal mass. SUSY concepts have provided useful extensions to the WKB approximation. In addition, SUSY has been applied to non-quantum statistical mechanics through the Fokker-Planck equation, showing that even if the original inspiration in high-energy particle physics turns out to be a blind alley, its investigation has brought about many useful benefits.

The SUSY QM superalgebra

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In fundamental quantum mechanics, we learn that an algebra of operators is defined by commutation relations among those operators. For example, the canonical operators of position and momentum have the commutator [x,p]=i. (Here, we use "natural units" where Planck’s constant is set equal to 1.) A more intricate case is the algebra of angular momentum operators; these quantities are closely connected to the rotational symmetries of three-dimensional space. To generalize this concept, we define an anticommutator, which relates operators the same way as an ordinary commutator, but with the opposite sign: {A, B} = AB + BA.

If operators are related by anticommutators as well as commutators, we say they are part of a Lie superalgebra. Let’s say we have a quantum system described by a Hamiltonian H and a set of N self-adjoint operators Q i. We shall call this system supersymmetric if the following anticommutation relation is valid for all i, j = 1, . . . , N :

{Qi , Qj } = Hδij .

If this is the case, then we call Q i the system’s supercharges.

Supersymmetric quantum mechanics

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Example

Q2 =

i 2

FT

Let’s look at the example of a one-dimensional nonrelativistic particle with a 2D (i.e., two state) internal degree of freedom called "spin" (it’s not really spin because "real" spin is a property of 3D particles). Let b be an operator which transforms a "spin up" particle into a "spin down" particle. Its adjoint b † then transforms a spin down particle into a spin up particle; the operators are normalized such that the anticommutator {b,b †}=1. And of course, b 2=0. Let p be the momentum of the particle and x be its position with [x,p]=i. Let W (the "superpotential") be an arbitrary complex analytic function of x and define the supersymmetric operators   Q1 = 21 (p − iW )b + (p + iW † )b†   (p − iW )b − (p + iW † )b†

Note that Q 1 and Q 2 are self-adjoint. Let the Hamiltonian (p+={W })2 2

+

<{W }2 2

+

<{W }0 † 2 (bb

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H = {Q1 , Q1 } = {Q2 , Q2 } =

− b† b)

where W’ is the derivative of W. Also note that {Q 1,Q 2}=0. This is nothing other than N = 2 supersymmetry. Note that ={W } acts like an electromagnetic vector potential. Let’s also call the spin down state "bosonic" and the spin up state "fermionic". This is only in analogy to quantum field theory and should not be taken literally. Then, Q 1 and Q 2 maps "bosonic" states into "fermionic" states and vice versa.

Let’s reformulate this a bit: Define

Q = (p − iW )b

and of course,

Q† = (p + iW † )b†

{Q, Q} = {Q† , Q† } = 0

and

{Q† , Q} = 2H

Supersymmetric quantum mechanics

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An operator is "bosonic" if it maps "bosonic" states to "bosonic" states and "fermionic" states to "fermionic" states. An operator is "fermionic" if it maps "bosonic" states to "fermionic" states and vice versa. Any operator can be expressed uniquely as the sum of a bosonic operator and a fermionic operator. Define the supercommutator [,} as follows: Between two bosonic operators or a bosonic and a fermionic operator, it is none other than the commutator but between two fermionic operators, it is an anticommutator. Then, x and p are bosonic operators and b, b† , Q and Q† are fermionic operators. Let’s work in the →Heisenberg picture where x, b and b† are functions of time. Then, [Q, x} = −ib [Q, b} = 0 dx dt

− i<{W }

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[Q.b† } =

[Q† , x} = ib† [Q† , b} =

dx dt

+ i<{W }

[Q† , b† } = 0

This is nonlinear in general: i.e., x(t), b(t) and b† (t) do not form a linear SUSY representation because <{W } isn’t necessarily linear in x. To avoid this problem, define the self-adjoint operator F = <{W }. Then, [Q, x} = −ib [Q, b} = 0 [Q.b† } =

dx dt

− iF

[Q, F } = − db dt

[Q† , x} = ib†

Supersymmetric quantum mechanics

600 [Q† , b} =

dx dt

+ iF

[Q† , F } =

FT

[Q† , b† } = 0 db† dt

and we see that we have a linear SUSY representation. Now let’s introduce two "formal" quantities, θ; and θ¯ with the latter being the adjoint of the former such that ¯ θ} ¯ = {θ, ¯ θ} = 0 {θ, θ} = {θ,

and both of them commute with bosonic operators but anticommute with fermionic ones.

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Next, we define a construct called a superfield: ¯ θ) = x(t) − iθb(t) − iθb ¯ † (t) + θθF ¯ (t) f (t, θ, f is self-adjoint, of course. Then, [Q, f } = ∂ f − iθ¯ ∂ f, ∂θ

[Q† , f } =

∂ f ∂ θ¯

∂t

∂ − iθ ∂t f.

Refererences

Supersymmetric quantum mechanics on arxiv.org 361

Source: http://en.wikipedia.org/wiki/Supersymmetric_quantum_mechanics Principal Authors: Phys, Anville, Agentsoo, Phil Boswell, Charles Matthews

361 http://xstructure.inr.ac.ru/x-bin/theme2.py?arxiv=hep-th&level=2&index1=19

Supersymmetric quantum mechanics

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Thermal de Broglie wavelength

FT

In physics, the Thermal de Broglie wavelength is defined for a free ideal gas of massive particles in equilibrium as: q h2 Λ = 2πmkT where • • • •

h is Planck’s constant m is the mass of a gas particle k is Boltzmann’s constant T is the Temperature of the gas

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The thermal de Broglie wavelength is roughly the average de Broglie wavelength of the gas particles in an ideal gas at the specified temperature. We can take the average interparticle spacing in the gas to be approximately (V/N) 1/3 where V is the volume and N is the number of particles. When the thermal de Broglie wavelength is much smaller than the interparticle distance, the gas can be considered to be a classical or Maxwell-Boltzmann gas. On the other hand, when the thermal de Broglie wavelength is on the order of, or larger than the interparticle distance, quantum effects will dominate and the gas must be treated as a →Fermi gas or a →Bose gas, depending on the nature of the gas particles. The critical temperature is the transition point between these two regimes, and at this critical temperature, the thermal wavelength will be approximately equal to the interparticle distance. That is, the quantum nature of the gas will be evident for V N Λ3

≤1

and in this case the gas will obey →Bose-Einstein statistics or →Fermi-Dirac statistics, whichever is appropriate. On the other hand, for V N Λ3

>> 1

the gas will obey →Maxwell-Boltzmann statistics.

Massless particles

For a massless particle, the thermal wavelength may be defined as: Λ=

ch 2kT π 1/3

Thermal de Broglie wavelength

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FT

where c is the speed of light. As with the massive thermal wavelength, this is of the order of the average wavelength of the particles in the gas. This is derived from the more general definition of the thermal wavelength due to Yan (Yan 2000) described below.

General definition of the thermal wavelength

A general definition of the thermal wavelength for an ideal quantum gas in any number of dimensions and for a generalized relationship between energy and momentum (dispersion relationship) has been given by Yan (Yan 2000). It is of practical importance, since there are many experimental situations with different dimensionality and dispersion relationships. If n is the number of dimensions, and the relationship between energy (E) and momentum (p) is given by: E = aps

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where a and s are constants, then the thermal wavelength is defined as: i1/n
h a 1/s Γ(n/2+1) Λ = √hπ kT Γ(n/s+1)

where Γ is the Gamma function. For example, in the usual case of massive particles in a 3-D gas we have n=3 , and E=p 2/2m which gives the above results for massive particles. For massless particles in a 3-D gas, we have n=3 , and E=pc which gives the above results for massless particles.

References •

Zijun Yan, "General thermal wavelength and its applications", Eur. J. Phys. 21 (2000) 625-631. http://www.iop.org/EJ/article/0143-0807/21/6/314 /ej0614.pdf

Source: http://en.wikipedia.org/wiki/Thermal_de_Broglie_wavelength Principal Authors: PAR, Feezo, Charles Matthews, VivaEmilyDavies

Thermal de Broglie wavelength

603

Topological order

FT

In physics, topological order is a new kind of order (a new kind of organization of particles) in a quantum state that is beyond the Landau symmetrybreaking description. It cannot be described by local order parameters and long range correlations. However, topological orders can be described by a new set of quantum numbers, such as ground state degeneracy, quasiparticle fractional statistics, edge states, topological entropy, etc. Roughly speaking, topological order is a non-local quantum entanglement in quantum states. States with different topological orders can change into each other only through a phase transition.

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A large class of topological orders is realized through a mechanism called string-net condensation. This class of topological orders is described and classified by a beautiful mathematical theory — tensor category theory. One finds that string-net condensation can generate infinitely many different types of topological orders, which may indicate that there are many different new types of materials remaining to be discovered.

Background

Although all matter is formed by atoms, matter can have very different properties and appear in very different forms, such as solid, liquid, superfluid, magnet, etc. According to condensed matter physics and the principle of emergence, the different properties of materials originate from the different ways in which the atoms are organized in the materials. Those different organizations of the atoms (or other particles) are formally called the orders in the materials. Atoms can organize in many ways which lead to many different orders and many different types of materials. With so many different orders, we need a general understanding of the orders. Landau symmetry-breaking theory provides such a general understanding. It points out that different orders really correspond to different symmetries in the organizations of the constituent atoms. As a material changes from one order to another order (i.e., as the material undergoes a phase transition), what happens is that the symmetry of the organization of the atoms changes.

For example, atoms have a random distribution in a liquid, so a liquid remains the same as we displace it by an arbitrary distance. We say that a liquid has a continuous translation symmetry. After a phase transition, a liquid can turn into a crystal. In a crystal, atoms organize into a regular array (a lattice). A lattice remains unchanged only when we displace it by a particular distance, so a crystal has only discrete translation symmetry. The phase transition between Topological order

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FT

a liquid and a crystal is a transition that changes the continuous translation symmetry of the liquid to the discrete symmetry of the crystal. Such change in symmetry is called symmetry breaking. The essence of the difference between liquids and crystals is therefore that the atoms have different symmetries in the two phases. Landau symmetry-breaking theory is a very successful theory. For a long time, physicists believed that Landau symmetry-breaking theory describes all possible orders in materials, and all possible (continuous) phase transitions.

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However, in last twenty years, it has become more and more apparent that Landau symmetry-breaking theory may not describe all possible orders. In 1989, physicists introduced chiral spin state in an attempt to explain high temperature superconductivity. At first people still wanted to use Landau symmetrybreaking theory to describe the chiral spin state. They identified the chiral spin state as a state that breaks the time reversal and parity symmetries, but not the spin rotation symmetry. However, it was quickly realized that there are many different chiral spin states that have exactly the same symmetry, so symmetry alone was not enough to characterize different chiral spin states. This means that the chiral spin states contain a new kind of order that is beyond symmetry description. This new kind of order was named topological order. A new quantum number, ground state degeneracy, was introduced to characterize the different topological orders in chiral spin states. (The name "topological order" is motivated by the low energy effective theory of the chiral spin states, which is a topological quantum field theory.) But experiments soon indicated that chiral spin states do not describe hightemperature superconductors, and the theory of topological order became a theory with no experimental realization. However, the similarity between chiral spin states and quantum Hall states allows one to use the theory of topological order to describe different quantum Hall states. Just like chiral spin states, different quantum Hall states all have the same symmetry and are beyond the Landau symmetry-breaking description. One finds that the different orders in different quantum Hall states can indeed be described by topological orders, so the topological order does have experimental realizations.

Applications

The materials described by Landau symmetry-breaking theory have had a substantial impact on technology. For example, Ferromagnetic materials that break spin rotation symmetry can be used as the media of digital information storage. A hard drive made by ferromagnetic materials can store gigabytes of information on it. Liquid crystals that break the rotational symmetry of molecules find wide application in display technology. Nowadays one can hardly find a Topological order

605

FT

household without a liquid crystal display somewhere in it. Crystals that break translation symmetry lead to well defined electronic bands which in turn allow us to make semiconducting devices such as transistors. Topologically ordered states are a new class of materials that are even richer than symmetry breaking states. This may suggest an exciting potential for applications.

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One theorized application would be to use topologically ordered states as media for quantum computing. A topologically ordered state is a state with complicated non-local quantum entanglement. The non-locality means that the quantum entanglement in a topologically ordered state is distributed among many different particles. As a result, the pattern of quantum entanglements cannot be destroyed by local perturbations. This significantly reduces the effect of decoherence. This suggests that if we use different quantum entanglements in a topologically ordered state to encode quantum information, the information may last much longer. The quantum information encoded by the topological quantum entanglements can also be manipulated by dragging the topological defects around each other. This process may provide a physical appartus for performing quantum computations. Therefore, topologically ordered states may provide natural media for both quantum memory and quantum computation. Such realizations of quantum memory and quantum computation may potentially be made fault tolerant.

Potential impact

Why is topological order important? Landau symmetry-breaking theory is a cornerstone of condensed matter physics. It used to define the territory of condensed matter research. The existence of topological order appears to indicate that nature is much richer than Landau symmetry-breaking theory has so far indicated. Some suggest a potential for topological order (or more precisely, string-net condensation) to provide a unified origin for photons, electrons and other elementary particles in our universe.

References

Fractional quantum Hall states: •

Two-Dimensional Magnetotransport in the Extreme Quantum Limit, D. C. Tsui and H. L. Stormer and A. C. Gossard, Phys. Rev. Lett., 48, 1559 (1982) • Anomalous Quantum Hall Effect: An Incompressible Quantum Fluid with Fractionally Charged Excitations, R. B. Laughlin, Phys. Rev. Lett., 50, 1395 (1983)

Chiral spin states:

Topological order

606 •

Topological order: • • • •

•

Quantum field theory and the Jones polynomial, E. Witten, Comm. Math. Phys., 121, 351 (1989) Vacuum Degeneracy of Chiral Spin State in Compactified Spaces, Xiao-Gang Wen, Phys. Rev. B, 40, 7387 (1989) Topological Orders in Rigid States, Xiao-Gang Wen, Int. J. Mod. Phys., B4, 239 (1990) Off-diagonal long-range order, oblique confinement, and the fractional quantum Hall effect, S. M. Girvin and A. H. MacDonald, Phys. Rev. Lett., 58, 1252 (1987) Effective-Field-Theory Model for the Fractional Quantum Hall Effect, S. C. Zhang and T. H. Hansson and S. Kivelson, Phys. Rev. Lett., 62, 82 (1989) Fractional Statistics and the Quantum Hall Effect, D. Arovas and J. R. Schrieffer and F. Wilczek, Phys. Rev. Lett., 53, 722 (1984) Gapless Boundary Excitations in the FQH States and in the Chiral Spin States, Xiao-Gang Wen, Phys. Rev. B, 43, 11025 (1991)

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•

FT

Equivalence of the resonating-valence-bond and fractional quantum Hall states, V. Kalmeyer and R. B. Laughlin, Phys. Rev. Lett., 59, 2095 (1987) • Chiral Spin States and Superconductivity, Xiao-Gang Wen, F. Wilczek and A. Zee, Phys. Rev., B39, 11413 (1989)

•

String-net condensation: •

Photons and electrons as emergent phenomena, Michael A. Levin, XiaoGang Wen, Rev. Mod. Phys., 77, 871 (2005) • String-net condensation: A physical mechanism for topological phases, Michael Levin, Xiao-Gang Wen, Phys. Rev. B, 71, 045110 (2005) Quantum computing: •

Fault-tolerant quantum computation by anyons, A. Yu. Kitaev Ann. Phys. (N.Y.), 303, 2 (2003) • Topological quantum computation, Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang, Bull. Amer. Math. Soc., 40, 31 (2003) • Topological quantum memory, Eric Dennis, Alexei Kitaev, Andrew Landahl, and John Preskill, J. Math. Phys., 43, 4452 (2002) • Proposed Experiments to probe the Non-Abelian nu=5/2 Quantum Hall State, Ady Stern and Bertrand I. Halperin, Phys. Rev. Lett., 96, 016802 (2006) Topological order

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FT

Source: http://en.wikipedia.org/wiki/Topological_order

Topological quantum number

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In physics, a topological quantum number is any quantity, in a physical theory, that takes on only one of a discrete set of values, due to topological considerations. Most commonly, topological quantum numbers are topological invariants associated with topological defects or soliton-type solutions of some set of differential equations modeling a physical system, as the solitons themselves owe their stability to topological considerations. The specific "topological considerations" are usually due to the appearance of the fundamental group or a higher-dimensional homotopy group in the description of the problem, quite often because the boundary, one which the boundary conditions are specified, has a non-trivial homotopy group that is preserved by the differential equations. The topological quantum number of a solution is sometimes called the winding number of the solution. Recent ideas about the nature of phase transitions indicates that topological quantum numbers, and their associated solitons, can be created or destroyed during a phase transition.

Particle physics

In particle physics, an example is given by the Skyrmion, for which the baryon number is a topological quantum number. The origin comes from the fact that the isospin is modelled by SU(2), which is isomorphic to the 3-sphere S3 . By taking real three-dimensional space, and closing it with a point at infinity, one also gets a 3-sphere. Solutions to Skyrme’s equations in real three dimensional space map a point in "real" (physical; Euclidean) space to a point on the 3manifold SU(2). Topologically distinct solutions "wrap" the one sphere around the other, such that one solution, no matter how it is deformed, cannot be "unwrapped" without creating a discontinuity in the solution. In physics, such discontinuities are associated with infinite energy, and are thus not allowed. In the above example, the topological statement is that the 3rd homotopy group of the three sphere is

π3 (S3 ) = Z

and so the baryon number can only take on integer values.

Topological quantum number

608 A generalization of these ideas is found in the Wess-Zumino-Witten model.

Exactly solvable models

π1 (S1 ) = Z

FT

Additional examples can be found in the domain of exactly solvable models, such as the sine-Gordon equationand the Korteweg-de Vries equation. The onedimensional sine-Gordon equation makes for a particularly simple example, as the fundamental group at play there is

and so is literally a winding number: a circle can be wrapped around a circle an integer number of times.

Solid state physics

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In solid state physics, certain types of crystalline dislocations, such as screw dislocations, can be described by topological solitons. An example includes screw-type dislocations associated with Germanium whiskers.

See also •

Inverse scattering transform

Source: http://en.wikipedia.org/wiki/Topological_quantum_number

Principal Authors: Linas, MarSch, Conscious, Charles Matthews, Kymara

Transformation theory (quantum mechanics) The term transformation theory refers to a procedure used by P. A. M. Dirac in his early formulation of quantum theory, from around 1927. The term is related to the famous wave-particle duality, according to which a particle (a "small" physical object) may display either particle or wave aspects, depending on the observational situation. Or, indeed, a variety of intermediate aspects, as the situation demands. This "transformation" idea also refers to the changes a physical object may undergo in the course of time, whereby it may "move" between "positions" in its Hilbert "space". Transformation theory (quantum mechanics)

609

FT

Remaining in full use today, it would be regarded as a topic in the mathematics of →Hilbert space, although technically speaking it is somewhat more general in scope. While the terminology is reminiscent of motion in ordinary space, the Hilbert space of a quantum object is more general, and holds its entire quantum state.

Source: http://en.wikipedia.org/wiki/Transformation_theory_%28quantum_mechanics%29 Principal Authors: Charles Matthews

T-symmetry

T-symmetry is the symmetry of physical laws under a time-reversal transformation—

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T : t 7→ −t.

The universe is not symmetric under time reversal, although in restricted contexts one may find this symmetry. Physicists distinguish time asymmetries that are intrinsic to the dynamic laws of nature, and those that are due to the initial conditions of our universe. The T-asymmetry of the weak nuclear force is of the first kind, while the T-asymmetry of the second law of thermodynamics is of the second kind. Physicists also discuss the time-reversal invariance of local and/or macroscopic descriptions of physical systems, independent of the invariance of the underlying microscopic physical laws. For example, Maxwell’s equations with material absorption or Newtonian mechanics with friction are not time-reversal invariant at the macroscopic level where they are normally applied, even if they are invariant at the microscopic level when one includes the atomic motions into which the "lost" energy is translated.

T-symmetry

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FT

610

Macroscopic phenomena: the second law of thermodynamics Our daily experience shows that T-symmetry does not hold for the behavior of bulk materials. Of these macroscopic laws, most notable is the second law of thermodynamics. Many other phenomena, such as the relative motion of bodies with friction, or viscous motion of fluids, reduce to this, because the underlying mechanism is the dissipation of usable energy (for example, kinetic energy) into heat. Is this time-asymmetric dissipation really inevitable? This question has been considered by many physicists, often in the context of Maxwell’s demon. The name comes from a thought experiment described by James Clerk Maxwell in which a microscopic demon guards a gate between two halves of a room. It only lets slow molecules into one half, only fast ones into the other. By eventually making one side of the room cooler than before and the other, hotter, it seems to reduce the entropy of the room, and reverse the arrow of time. Many analyses have been made of this; all show that when the entropy of room and demon are taken together, this total entropy does increase. Modern analyses of this problem have taken into account Claude E. Shannon’s relation between T-symmetry

611

FT

entropy and information. Many interesting results in modern computing are closely related to this problem— reversible computing, quantum computing and physical limits to computing, are examples. These seemingly metaphysical questions are today, in these ways, slowly being converted to the stuff of the physical sciences.

The consensus nowadays hinges upon the Boltzmann-Shannon identification of the logarithm of phase space volume with negative of Shannon information, and hence to entropy. In this notion, a fixed initial state of a macroscopic system corresponds to relatively low entropy because the coordinates of the molecules of the body are constrained. As the system evolves in the presence of dissipation, the molecular coordinates can move into larger volumes of phase space, becoming more uncertain, and thus leading to increase in entropy.

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However, one can equally well imagine a state of the universe in which the motions of all of the particles at one instant were the reverse (strictly, the CPT reverse). Such a state would then evolve in reverse, so presumably entropy would decrease (Loschmidt’s paradox). Why is ’our’ state preferred over the other?

One position is to say that the constant increase of entropy we observe happens only because of the initial state of our universe. Other possible states of the universe (for example, a universe at heat death equilibrium) would actually result in no increase of entropy. In this view, the apparent T-asymmetry of our universe is a problem in cosmology: why did the universe start with a low entropy? This view, if it remains viable in the light of future cosmological observation, would connect this problem to one of the big open questions beyond the reach of today’s physics— the question of initial conditions of the universe.

Microscopic phenomena: time reversal invariance

Since most systems are asymmetric under time reversal, it is interesting to ask whether there are any phenomena which do have this symmetry. In classical mechanics, a velocity v reverses under the operation of T, but an acceleration does not. Therefore, one models dissipative phenomena through terms which are odd in v. However, delicate experiments in which known sources of dissipation are removed reveal that the laws of mechanics are time reversal invariant. However, the motion of a charged body in a magnetic field, B involves the velocity through the Lorentz force term v×B, and might seem at first to be asymmetric under T. A closer look assures us that B also changes sign under time reversal. This happens because a magnetic field is produced by an electric current, J, which reverses sign under T. Thus, the motion of classical charged particles in electromagnetic fields is also time reversal invariant. (Despite this, T-symmetry

612

FT

it is still useful to consider the time-reversal non-invariance in a local sense when the external field is held fixed, as when the magneto-optic effect is analyzed. This allows one to analyze the conditions under which optical phenomena that locally break time-reversal, such as Faraday isolators, can occur.) The laws of gravity also seem to be time reversal invariant in classical mechanics. In physics one separates the laws of motion, ie, kinematics, from the laws of force, called dynamics. Following the classical kinematics of Newton’s laws of motion, the kinematics of quantum mechanics is built in such a way that it presupposes nothing about the time reversal symmetry of the dynamics. In other words, if the dynamics is invariant, then the kinematics will allow it to remain invariant; if the dynamics is not, then the kinematics will also show this. The structure of the quantum laws of motion are richer, and we examine these next.

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Time reversal in quantum mechanics

This section contains a discussion of the three most important properties of time reversal in quantum mechanics; namely,

T-symmetry

613 • •

FT

•

that it must be represented as an anti-unitary operator, that it protects non-degenerate quantum states from having an electric dipole moment, that it has two-dimensional representations with the property T 2 = -1.

The strangeness of this result is clear if one compares it with parity. If parity transforms a pair of quantum states into each other, then the sum and difference of these two basis states are states of good parity. Time reversal does not behave like this. It seems to violate the theorem that all Abelian groups be represented by one dimensional irreducible representations. The reason it does this, is that it is represented by an anti-unitary operator. It thus opens the way to spinors in quantum mechanics.

Anti-unitary representation of time reversal

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Eugene Wigner showed that a symmetry operation S of a Hamiltonian is represented, in quantum mechanics either by an unitary operator, S = U, or an antiunitary one, S = UK where U is unitary, and K denotes complex conjugation. For parity (physics) one has PxP = -x and PpP = -p, where x and p are the position and momentum operators. In canonical quantization, one has the commutator [x, p] = ih /2π, where h is the Planck’s constant. This commutator is invariant if P is chosen to be unitary, ie, PiP = i. Such an argument can be attempted for time reversal, T. one has TxT = x and TpT = -p, and the commutator is invariant only if T is chosen to be anti-unitary, ie, TiT = -i. For a particle with spin, one can use the representation T = e−iπSy /~ K,

where S y is the y-component of the spin, to find that TJT = -J.

Electric dipole moments

This has an interesting consequence on the electric dipole moment (EDM) of any particle. The EDM is defined through the shift in the energy of a state when it is put in an external electric field: ∆e = d·E + E ·δ·E, where d is called the EDM and δ, the induced dipole moment. One important property of an EDM is that the energy shift due to it changes sign under a parity transformation. However, since d is a vector, its expectation value in a state |ψ> it must be proportional to <ψ|J |ψ>. Thus, under time reversal, an invariant state must have vanishing EDM. In other words, a non-vanishing EDM signals both P and T symmetry-breaking.

T-symmetry

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It is interesting to examine this argument further, since one feels that some molecules, such as water, must have EDM irrespective of whether T is a symmetry. This is correct: if a quantum system has degenerate ground states which transform into each other under parity, then time reversal need not be broken to give EDM.

Experimentally observed bounds on the electric dipole moment of the nucleon currently set stringent limits on the violation of time reversal symmetry in the strong interactions, and their modern theory: quantum chromodynamics. Then, using the CPT invariance of a relativistic quantum field theory, this puts strong bounds on strong CP violation.

Kramer’s theorem

For T, which is an anti-unitary Z 2 symmetry generator T 2 = UKUK = U U * = U (U T ) -1 = Φ,

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where Φ is a diagonal matrix of phases. As a result, U = ΦU T and U T = U Φ, showing that U = Φ U Φ.

This means that the entries in Φ are ±1, as a result of which one may have either T 2 = ±1. This is specific to the anti-unitarity of T. For an unitary operator, such as the parity, any phase is allowed.

Next, take a Hamiltonian invariant under T. Let |a> and T |a> be two quantum states of the same energy. Now, if T 2 = -1, then one finds that the states are orthogonal: a result which goes by the name of Kramer’s theorem. This implies that if T 2 = -1, then there is a two-fold degeneracy in the state. This result in non-relativistic quantum mechanics presages the spin statistics theorem of quantum field theory. →Quantum states which give unitary representations of time reversal, ie, have T 2=1, are characterized by a multiplicative quantum number, sometimes called the T-parity. Time reversal transformation for fermions in quantum field theories can be represented by an 8-component spinor 362 in which the above mentioned T-parity can be a complex number with unit radius. The CPT invariance is not a theorem but a better to have propert in these class of theories.

362 http://arxiv.org/abs/hep-th/0010074

T-symmetry

615

Time reversal of the known dynamical laws Unsolved problems in physics: Why are some of the fundamental forces symmetric under time reversal whereas others are not?

See also

The second law of thermodynamics and Maxwell’s demon (also Loschmidt’s paradox). Applications to reversible computing and quantum computing, including limits to computing. The standard model of particle physics, CP violation, the CKM matrix and the strong CP problem Neutrino masses, CPT invariance and tests of CPT violation.

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•

FT

The study of particle physics has culminated in a codification of the basic laws of dynamics into the standard model. This is formulated as a quantum field theory which has CPT symmetry, ie, the laws are invariant under simultaneous operation of time reversal, parity and charge conjugation. However, time reversal itself is seen not to be a symmetry (this is usually called CP violation). There are two possible origins of this asymmetry, one through the mixing of different flavours of quarks in their weak decays, the second through a direct CP violation in strong interactions. The first is seen in experiments, the second is strongly constrained by the non-observation of the EDM of a neutron.

• • •

References and external links •

•

•

•

•

Maxwell’s demon: entropy, information, computing, edited by H.S.Leff and A.F. Rex (IOP publishing, 1990) [ISBN 0750300574] Maxwell’s demon, 2: entropy, classical and quantum information, edited by H.S.Leff and A.F. Rex (IOP publishing, 2003) [ISBN 0750307595] The emperor’s new mind: concerning computers, minds, and the laws of physics, by Roger Penrose (Oxford university press, 2002) [ISBN 0192861980] CP violation, by I.I. Bigi and A.I. Sanda (Cambridge University Press, 2000) [ISBN 0521443490] Particle Data Group on CP violation 363 • the Babar 364 experiment in SLAC • the BELLE 365 experiment in KEK

363 http://pdg.lbl.gov/2004/reviews/cpviolrpp.pdf 364 http://www-public.slac.stanford.edu/babar/ 365 http://belle.kek.jp

T-symmetry

616 the KTeV 366 experiment in Fermilab the CPLEAR 367 experiment in CERN

FT

• •

Source: http://en.wikipedia.org/wiki/T-symmetry

Principal Authors: Bambaiah, Roadrunner, Michael Hardy, Pcarbonn, Jheald

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Two interfering electron wave-packets

Figure 69

The green plane is the x-y-plane, where two (non-interacting) electron wavepackets meet. The vertical direction shows the real part of ψ (x, y). The semitransparent white plane in the top shows the density of detection probability, i.e, |ψ (x, y) |2 , as blue spots. The blue in the middle is the same again. Before interfering, both electrons have circular detection probabilities. During the interference, you can see lines, where there are strong wave movements and others, with no motion in between. The lines without wave motion are

366 http://kpasa.fnal.gov:8080/public/ktev.html 367 http://cplear.web.cern.ch/cplear/Welcome.html

Two interfering electron wave-packets

617 called knot-lines. As you see, the detection probability is zero on knots. After the interference, both electrons move as if they never had seen the other one.

See also • • •

→Double-slit experiment →Wave-particle duality Light

FT

The smearing out of electrons is their usual behavior due to dispersion. It is independent of interference.

Source: http://en.wikipedia.org/wiki/Two_interfering_electron_wave-packets Principal Authors: Linas, Uyanga, SimonP, Conscious, Nikki chan

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Ultraviolet catastrophe

The ultraviolet catastrophe, also called the Rayleigh-Jeans catastrophe, was a prediction of early 20th century classical physics that an ideal black body at thermal equilibrium will emit radiation with infinite power. As experimental observation showed this to be clearly false, it was one of the first clear indications of problems with classical physics. The solution to this problem led to the development of an early form of quantum mechanics. The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, although the concept goes back to 1905; the word "ultraviolet" refers to the fact that the problem appears in the short wavelength region of the electromagnetic spectrum. Since the first appearance of the term, it has also been used for other predictions of a similar nature, e.g. in quantum electrodynamics (also used in those cases: ultraviolet divergence). The ultraviolet catastrophe results from the equipartition theorem of classical statistical mechanics which states that all modes (degrees of freedom) of a system at equilibrium have an average energy of kT /2. According to classical electromagnetism, the number of electromagnetic modes in a 3-dimensional cavity, per unit frequency, is proportional to the square of the frequency. This therefore implies that the radiated power per unit frequency should follow the Rayleigh-Jeans law, and be proportional to frequency squared. Thus, both the power at a given frequency and the total radiated power go to infinity as higher and higher frequencies are considered: this is clearly an impossibility, a Ultraviolet catastrophe

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point that was made independently by Einstein and by Lord Rayleigh and Sir James Jeans in the year 1905. Einstein pointed out that the difficulty could be avoided by making use of a hypothesis put forward five years earlier by Max Planck. Planck had postulated that electromagnetic energy did not follow the classical description, but could only oscillate or be emitted in discrete packets of energy proportional to the frequency (as given by Planck’s law). This has the effect of reducing the number of possible modes with a given energy at high frequencies in the cavity described above, and thus the average energy at those frequencies by application of the equipartition theorem. The radiated power eventually goes to zero at infinite frequencies, and the total predicted power is finite. The formula for the radiated power for the idealized system (black body) was in line with known experiments, and came to be called Planck’s law of black body radiation. Based on past experiments, Planck was also able to determine the value of its parameter, now called Planck’s constant. The packets of energy later came to be called photons, and played a key role in the quantum description of electromagnetism.

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Many popular histories of physics, as well as a number of textbooks, present an incorrect version of the history of the Ultraviolet Catastrophe. In this version, the "catastrophe" was first noticed by Planck, who developed his formula in response. In fact Planck never concerned himself with this aspect of the problem, because he did not believe that the equipartition theorem was fundamental his motivation for introducing "quanta" was entirely different. It was several years later that physicists realized that Planck’s law resolved a fundamental crisis of classical physics.

References •

Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 0716710889. (See Chapter 4) • Cohen-Tannoudji, Claude; Diu, Bernard; Laloë, Franck (1977). Quantum Mechanics: Volume One. Hermann, Paris. 624–626 • Kragh, Helge. "Max Planck: The reluctant revolutionary" 368. Physics World. December 2000

External Links •

Ultraviolet Catastrophe and Schrodinger’s Cat 369

368 http://www.physicsweb.org/articles/world/13/12/8/1 369 http://www.harrymaugans.com/2006/05/03/in-search-of-schrodingers-cat/

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619 Source: http://en.wikipedia.org/wiki/Ultraviolet_catastrophe

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Principal Authors: Stevenj, Roadrunner, Jonathan F, Pcarbonn, Rparson, RTC

Uncertainty principle

In quantum physics, the Heisenberg uncertainty principle or just Uncertainty principle (sometimes also the Heisenberg indeterminacy principle - a name given to it by Niels Bohr) states that one cannot measure values (with arbitrary precision) of certain conjugate quantities, which are pairs of observables of a single elementary particle. These pairs include the position and momentum. Mathematics provides a positive lower bound for the product of the uncertainties of measurements of the conjugate quantities. The uncertainty principle is one of the cornerstones of quantum mechanics and was discovered by Werner Heisenberg in 1927.

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The Uncertainty principle follows from the mathematical definition of operators in quantum mechanics; it is represented by a set of theorems of functional analysis. It is often confused with the observer effect.

Overview

The concept of probability distributions pervades the science of measurement. Until the beginning of the discovery of quantum physics, it was thought that the only uncertainty in measurement was caused by the limitations of a measuring tool’s precision. But it is now understood that no treatment of any scientific subject, experiment, or measurement is said to be accurate without disclosing the nature of the probability distribution (sometimes called the error) of the measurement. Uncertainty is the characterization of the relative narrowness or broadness of the distribution function applied to a physical observation. Illustrative of this is an experiment in which a particle is prepared in a definite state and two successive measurements are performed on the particle. The first one measures the particle’s position and the second immediately after measures its momentum. Each time the experiment is performed, some value x is obtained for position and some value p is obtained for momentum. Depending upon the precision of the instrument taking the measurements, each successive measurement of the positions and momenta respectively should be nearly identical, but in practice they will exhibit some deviation due to constraints of measurement using a real world instrument that is not infinitely

Uncertainty principle

620 precise. However, Heisenberg showed that, even in theory with a hypothetical infinitely precise instrument, no measurement could be made to arbitrary accuracy of both the position and the momentum of a physical object.

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The Heisenberg uncertainty principle (developed in an essay published in 1927) provides a quantitative relationship between the uncertainties of the hypothetical infinitely precise measurements of p and x as measured by the sizes of their distributions in the following way: If the particle state is such that the first measurement yields a dispersion of values ∆x, then the second measurement will have a distribution of values whose dispersion ∆p is at least inversely proportional to ∆x. For the limiting case, the constant of proportionality is derivable using commutator arithmetic. It is equal to Planck’s constant divided by 4π.

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This stipulates that the product of the uncertainties in position and momentum is equal to or greater than about 10−35 joule-seconds. Therefore, the product of the uncertainties only becomes significant for regimes where the uncertainty in position or momentum measurements is small. Thus, the uncertainty principle governs the observable nature of atoms and subatomic particles while its effect on measurements in the macroscopic world is negligible and can be usually ignored. The Heisenberg uncertainty relations are a theoretical bound over all measurements. They hold for so-called ideal measurements, sometimes called von Neumann measurements. They hold even more so for non-ideal or Landau measurements.

Wave-particle duality and the relationship to the uncertainty principle A fundamental consequence of the Heisenberg Uncertainty Principle is that no physical phenomena can be (to arbitrary accuracy) described as a "classic point particle" or as a wave but rather the microphysical situation is best described in terms of wave-particle duality. The uncertainty principle, as initially considered by Heisenberg, is concerned with cases in which neither the wave nor the point particle descriptions are fully and exclusively appropriate, such as a particle in a box with a particular energy value. Such systems are characterized neither by one unique "position" (one particular value of distance from a potential wall) nor by one unique value of momentum (including its direction). Any observation that determines either a position or a momentum of such a waveparticle to arbitrary accuracy - known as wavefunction collapse is subject to the condition that the width of the wavefunction collapse in position, multiplied by the width of the wavefunction collapse in momentum, is Uncertainty principle

621 constrained by the principle to be greater than or equal to Planck’s constant divided by 4π.

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Every measured particle in quantum mechanics exhibits wavelike behaviour so there is an exact, quantitative analogy between the Heisenberg uncertainty relations and properties of waves or signals. For example, in a time-varying signal such as a sound wave, it is meaningless to ask about the frequency spectrum at a single moment in time because the measure of frequency is the measure of a repetition recurring over a period of time. In order to determine the frequencies accurately, the signal needs to be sampled for a finite (non-zero) time. This necessarily implies that time precision is lost in favor of a more accurate measurement of the frequency spectrum of a signal. This is analogous to the relationship between momentum and position, and there is an equivalent formulation of the uncertainty principle which states that the uncertainty of energy of a wave (directly proportional to the frequency) is inversely proportional to the uncertainty in time with a constant of proportionality identical to that for position and momentum.

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Common incorrect explanation of the uncertainty principle

The uncertainty principle in quantum mechanics is sometimes erroneously explained by claiming that the measurement of position necessarily disturbs a particle’s momentum. Heisenberg himself may have initially offered explanations which suggested this view. That this disturbance does not describe the essence of the uncertainty principle in current theory has been demonstrated above. The fundamentally non-classical characteristics of the uncertainty measurements in quantum mechanics were clarified due to the EPR paradox which arose from Einstein attempting to show flaws in quantum measurements that used the uncertainty principle. Instead of Einstein succeeding in showing uncertainty was flawed, Einstein guided researchers to examine more closely what uncertainty measurements meant and led to a more refined understanding of uncertainty. Prior to the publication of the EPR paper in 1935, a measurement was often visualized as a physical disturbance inflicted directly on the measured system, being sometimes illustrated as a thought experiment called Heisenberg’s microscope. For instance, when measuring the position of an electron, one imagines shining a light on it, thus disturbing the electron and producing the quantum mechanical uncertainties in its position. Such explanations, which are still encountered in popular expositions of quantum mechanics, are debunked by the EPR paradox, which shows that a "measurement" can be performed on a particle without disturbing it directly, by performing a measurement on a distant entangled particle. Heisenberg’s original argument used the ’old’ quantum theory (namely, the Einstein-deBroglie relations) and provided a heuristic argument that the position and momentum observables Uncertainty principle

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were not simultaneously observable with infinite precision. The more modern uncertainty relations deal with independent measurements being done on an ensemble of systems.

Formulation and characteristics

Measurements of position and momentum taken in several identical copies of a system in a given state will vary according to known probability distributions. This is the fundamental postulate of quantum mechanics.

If we compute the uncertainty ∆x of the position measurements and the standard deviation ∆p of the momentum measurements, then ∆x∆p ≥ where

~ 2

~ is the reduced Planck’s constant (Planck’s constant divided by 2π).

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Heisenberg did not just use any arbitrary number to describe the minimum standard deviation between position and momentum of a particle. Heisenberg knew that particles behaved like waves and he knew that the energy of any wave is the frequency multiplied by Planck’s constant. In a wave, a cycle is defined by the return from a certain position to the same position such as from the top of one crest to the next crest. This actually is equivalent to a circle of 360 degrees, or 2π radians. Therefore, dividing h by 2π describes a constant that when multiplied by the frequency of a wave gives the energy of one radian. Heisenberg took 1/2 of ~ as his standard deviation. This can be written as ~ over 2 as above or it can be written as h/(4π). Normally one will see ~ over 2 as this is simpler.

Two years earlier in 1925 when Heisenberg had developed his matrix mechanics the difference in position and momentum were already showing up in the formula. In developing matrix mechanics Heisenberg was measuring amplitudes of position and momentum of particles such as the electron that have a period of 2π, like a cycle in a wave, which are called Fourier series variables. When amplitudes of position and momentum are measured and multiplied together, they give intensity. However, Heisenberg found that when the position and momentum were multiplied together in that respective order or in the reverse order, there was a difference between the two calculated intensities of h/(2π). In other words, the two quantities position and momentum did not commute. In 1927, to develop the standard deviation for the uncertainty principle, Heisenberg took the gaussian distribution or bell curve for the imprecision in the measurement of the position q of a moving electron to the Uncertainty principle

623 corresponding bell curve of the measured momentum p. That gave the minimum standard deviation to be 1/2 of h/(2π), or, ~/2.

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In some treatments, the "uncertainty" of a variable is taken to be the smallest width of a range which contains 50% of the values, which, in the case of normally distributed variables, leads to a larger lower bound of h/(2π) for the product of the uncertainties. Note that this inequality allows for several possibilities: the state could be such that x can be measured with high precision, but then p will only approximately be known, or conversely p could be sharply defined while x cannot be precisely determined. In yet other states, both x and p can be measured with "reasonable" (but not arbitrarily high) precision.

Common observables which obey the uncertainty principle

An uncertainty relation arises between any two observable quantities that can be defined by non-commuting operators. This means that the uncertainty principle arises in measuring the position and the velocity of an object, or in measuring the position and momentum of an object. The most common one is the uncertainty relation between position and momentum of a particle in space:

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•

∆xi ∆pi ≥

•

~ 2

The uncertainty relation between two orthogonal components of the total angular momentum operator of a particle is as follows: ∆Ji ∆Jj ≥

~ 2

|hJk i|

where i, j, k are distinct and J i denotes angular momentum along the x i axis.

One of the theorems

Theorem. For arbitrary symmetric operators A : H → H and B : H → H, and any element x of H such that A B x and B A x are both defined (so that in particular, A x and B x are also defined), then hBAx|xihx|BAxi = hABx|xihx|ABxi = |hBx|Axi|2 ≤ ||Ax||2 ||Bx||2

This is an immediate consequence of the Cauchy-Bunyakovski-Schwarz inequality.

Uncertainty principle

624 Consequently, the following general form of the uncertainty principle, first pointed out in 1930 by Howard Percy Robertson and (independently) by Erwin Schrödinger, holds: − BA)x|xi|2 ≤ ||Ax||2 ||Bx||2 .

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1 4 |h(AB

This inequality is called the Robertson-Schrödinger relation.

The operator A B - B A is called the commutator of A, B and is denoted [A, B ]. It is defined on those x for which A B x and B A x are both defined. From the Robertson-Schrödinger relation, the following Heisenberg uncertainty relation is immediate:

Suppose A and B are two observables which are identified to self-adjoint (and in particular symmetric) operators. If B A ψ and A B ψ are defined then ∆ψ A ∆ψ B ≥ 12 h[A, B]iψ where

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hXiψ = hψ|Xψi

is the operator mean of observable X in the system state ψ and q ∆ψ X = hX 2 iψ − hXi2ψ is the operator standard deviation of observable X in the system state ψ The above definitions of mean and standard deviation are defined formally in purely operator-theoretic terms. The statement becomes more meaningful however, once we note that these actually are the mean and standard deviation for the measured distribution of values. See quantum statistical mechanics. It may be evaluated not only for pairs of conjugate operators (e.g. those defining measurements of distance and of momentum, or of duration and of energy) but generally for any pair of Hermitian operators. There is also an uncertainty relation between the field strength and the number of particles which is responsible for the phenomenon of virtual particles.

Note that it is possible to have two non-commuting self-adjoint operators A and B which share an eigenvector ψ, in this case ψ represents a pure state in which it is predictable with probability one what the result of measuring A or B will be in spite of their not being simultaneously measurable.

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Energy, time and further generalizations

∆E∆t ≥ ~.

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General arguments, connected with the theory of relativity, point out that seemingly a relation like the following should exist: But its correct mathematical formulation was given 370 only in 1945 by L. I. Mandelshtam and I. E. Tamm.

History and interpretations

Main article: →Interpretation of quantum mechanics

The Uncertainty Principle was developed as an answer to the question: How does one measure the location of an electron around a nucleus?

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In the summer of 1922 Heisenberg met Niels Bohr, the founding father of quantum mechanics, and in September 1924 Heisenberg went to Copenhagen, where Bohr had invited him as a research associate and later as his assistant. In 1925 Werner Heisenberg laid down the basic principles of a complete quantum mechanics. In his new matrix theory he replaced classical commuting variables with non-commuting ones. Heisenberg’s paper marked a radical departure from previous attempts to solve atomic problems by making use of observable quantities only. He wrote in a 1925 letter, "My entire meagre efforts go toward killing off and suitably replacing the concept of the orbital paths that one cannot observe." Rather than struggle with the complexities of threedimensional orbits, Heisenberg dealt with the mechanics of a one-dimensional vibrating system, an anharmonic oscillator. The result was formulae in which quantum numbers were related to observable radiation frequencies and intensities. In March 1926, working in Bohr’s institute, Heisenberg formulated the principle of uncertainty thereby laying the foundation of what became known as the Copenhagen interpretation of quantum mechanics. Albert Einstein was not happy with the uncertainty principle, and he challenged Niels Bohr and Werner Heisenberg with a famous thought experiment (See the Bohr-Einstein debates for more details): we fill a box with a radioactive material which randomly emits radiation. The box has a shutter, which is opened and immediately thereafter shut by a clock at a precise time, thereby allowing some radiation to escape. So the time is already known with precision. We still want to measure the conjugate variable energy precisely. Einstein proposed doing this by weighing the box before and after. The equivalence between mass and energy from special relativity will allow you to determine precisely how much energy was left in the box. Bohr countered as follows: should energy leave,

370 http://daarb.narod.ru/mandtamm-eng.html

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then the now lighter box will rise slightly on the scale. That changes the position of the clock. Thus the clock deviates from our stationary reference frame, and again by special relativity, its measurement of time will be different from ours, leading to some unavoidable margin of error. In fact, a detailed analysis shows that the imprecision is correctly given by Heisenberg’s relation.

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The term Copenhagen interpretation of quantum mechanics was often used interchangeably with and as a synonym for Heisenberg’s Uncertainty Principle by detractors who believed in fate and determinism and saw the common features of the Bohr-Heisenberg theories as a threat. Within the widely but not universally accepted Copenhagen interpretation of quantum mechanics (i.e. it was not accepted by Einstein or other physicists such as Alfred Lande), the uncertainty principle is taken to mean that on an elementary level, the physical universe does not exist in a deterministic form—but rather as a collection of probabilities, or potentials. For example, the pattern (probability distribution) produced by millions of photons passing through a diffraction slit can be calculated using quantum mechanics, but the exact path of each photon cannot be predicted by any known method. The Copenhagen interpretation holds that it cannot be predicted by any method, not even with theoretically infinitely precise measurements. It is this interpretation that Einstein was questioning when he said "I cannot believe that God would choose to play dice with the universe." Bohr, who was one of the authors of the Copenhagen interpretation responded, "Einstein, don’t tell God what to do." Niels Bohr himself acknowledged that quantum mechanics and the uncertainty principle were counter-intuitive when he stated, "Anyone who is not shocked by quantum theory has not understood a single word."

The basic debate between Einstein and Bohr (including Heisenberg’s Uncertainty Principle) was that Einstein was in essence saying: "Of course, we can know where something is; we can know the position of a moving particle if we know every possible detail, and thereby by extension, we can predict where it will go." Bohr and Heisenberg were saying the opposite: "There is no way to know where a moving particle is ever even given every possible detail, and thereby by extension, we can never predict where it will go." Einstein was convinced that this interpretation was in error. His reasoning was that all previously known probability distributions arose from deterministic events. The distribution of a flipped coin or a rolled dice can be described with a probability distribution (50% heads, 50% tails). But this does not mean that their physical motions are unpredictable. Ordinary mechanics can be used to calculate exactly how each coin will land, if the forces acting on it are known. And the heads/tails distribution will still line up with the probability distribution (given random initial forces). Uncertainty principle

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Einstein assumed that there are similar hidden variables in quantum mechanics which underlie the observed probabilities and that these variables, if known, would show that there was what Einstein termed "local realism", a description opposite to the uncertainty principle, being that all objects must already have their properties before they are observed or measured. For the greater part of the twentieth century, there were many such hidden variable theories proposed, but in 1964 John Bell theorized the Bell inequality to counter them, which postulated that although the behavior of an individual particle is random, it is also correlated with the behavior of other particles. Therefore, if the uncertainty principle is the result of some deterministic process in which a particle has local realism, it must be the case that particles at great distances instantly transmit information to each other to ensure that the correlations in behavior between particles occur. The interpretation of Bell’s theorem explicitly prevents any local hidden variable theory from holding true because it shows the necessity of a system to describe correlations between objects. The implication is, if a hidden local variable is the cause of particle 1 being at a position, then a second hidden local variable would be responsible for particle 2 being in its own position - and there is no system to correlate the behavior between them. Experiments have demonstrated that there is correlation. In the years following, Bell’s theorem was tested and has held up experimentally time and time again, and these experiments are in a sense the clearest experimental confirmation of quantum mechanics. It is worth noting that Bell’s theorem only applies to local hidden variable theories; non-local hidden variable theories can still exist (which some, including Bell, think is what can bridge the conceptual gap between quantum mechanics and the observable world). Whether Einstein’s view or Heisenberg’s view is true or false is not a directly empirical matter. One criterion by which we may judge the success of a scientific theory is the explanatory power it gives us, and to date it seems that Heisenberg’s view has been the better at explaining physical subatomic phenomena.

The uncertainty principle in popular culture The uncertainty principle is stated in popular culture in many ways, for example by stating that it is impossible to know both where an electron is and where it is going at the same time. This is roughly correct, although it fails to mention an important part of the Heisenberg principle, which is the quantitative bounds on the uncertainties. The uncertainty principle is frequently, but incorrectly, confused with the "observer effect", wherein the observation of an event changes the event. The

Uncertainty principle

628 observer effect is an important effect in many fields, from electronics to psychology and social science.

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In some science fiction stories, a device to circumvent the uncertainty principle is called a Heisenberg compensator, most famously in Star Trek for use on the transporter; however, it is not clear what compensating means. In Stephen Donaldson’s Gap Cycle science fiction book series, one of the characters postulates a socio-political version of the uncertainty principle: namely, that by determining his precise "location" in the current political landscape, he is prevented from simultaneously calculating the likely direction of political events in the near future. In software programming, a Heisenbug is a software error that disappears or alters its characteristics when it is researched. The Heisenberg Principle was referenced in "Prophecy", an episode of Stargate SG-1.

Humor

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The unusual nature of Heisenberg’s uncertainty principle, and its distinctive name, has made it the source of several jokes.

It is said that a popular item of graffiti at the physics department of university campuses is the slogan "Heisenberg may have been here." In another uncertainty principle joke, a quantum physicist is stopped on the highway by a police officer who asks "Do you know how fast you were going, sir?", to which the physicist responds, "No, but I know exactly where I am!". The biggest flop since the Edsel... The Heisenbergmobile. The problem was that when you look at the speedometer you got lost. The character, Phillip Richbourg, of Night Court got into a heated debate with Judge Harry about this very principle. He argued the "ball theory", claiming that if you roll a ball down a hill, and you know exactly when it has been exactly 10sec, and you can measure distance exactly, I don’t see how you can’t know where the ball is. It was very comical and fed in with the show’s theme. A reference to the uncertainty principle is made in an animation on a popular online animations site, albinoblacksheep.com . Jokes with Einstein 5 refers to a failed experiment to undermine this principle.

See also • • •

→Quantum indeterminacy Basics of quantum mechanics Correspondence principle

Uncertainty principle

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References

•

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Journal articles W. Heisenberg, "Über den anschaulichen Inhalt der quantentheoretischen Kinematik und Mechanik", Zeitschrift für Physik, 43 1927, pp. 172-198. English translation: J. A. Wheeler and H. Zurek, Quantum Theory and Measurement Princeton Univ. Press, 1983, pp. 62-84. • L. I. Mandelshtam, I. E. Tamm " The uncertainty relation between energy and time in nonrelativistic quantum mechanics 371", Izv. Akad. Nauk SSSR (ser. fiz.) 9, 122-128 (1945). English translation: J. Phys. (USSR) 9, 249-254 (1945). • G. Folland, A. Sitaram, "The Uncertainty Principle: A Mathematical Survey", Journal of Fourier Analysis and Applications, 1997 pp 207-238.

External links

Stanford Encyclopedia of Philosophy entry 372 aip.org: Quantum mechanics 1925-1927 - The uncertainty principle 373 Eric Weisstein’s World of Physics - Uncertainty principle 374 Schrödinger equation from an exact uncertainty principle 375 John Baez on the time-energy uncertainty relation 376 Beating the uncertainty principle in finite-parameter systems 377

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• • • • • •

Source: http://en.wikipedia.org/wiki/Uncertainty_principle

Principal Authors: CSTAR, Voyajer, Light current, ScienceApologist, Fwappler, Stevenj, Linas, CarlHewitt, AxelBoldt

371 http://daarb.narod.ru/mandtamm-eng.html 372 http://plato.stanford.edu/entries/qt-uncertainty/

373 http://www.aip.org/history/heisenberg/p08.htm 374 http://scienceworld.wolfram.com/physics/UncertaintyPrinciple.html 375 http://arxiv.org/abs/quant-ph/0102069 376 http://math.ucr.edu/home/baez/uncertainty.html 377 http://www.seeingwithsound.com/freqtime.htm

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Unitarity

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In mathematics and physics, unitarity is the property of an operator (or a matrix) that is unitary.

In physics, the requirement of unitarity of the evolution operator or the Smatrix (the evolution operator from t = −∞ to t = +∞) is essential for the physical interpretation of any complete theory. These operators must preserve the squared length of the original vector in the →Hilbert space simply because the physical interpretation of this squared length, according to the basic principles of quantum mechanics, is the total probability of all possible alternatives of the evolution - and the total probability of all alternatives must always be equal to 100 percent. (In various approximate theories, non-unitary evolution operators can represent a nonzero probability of "other" alternatives that we are not interested in - for example, a reduced total probability of different states of a kaon represents its decay.)

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The evolution operator is automatically unitary if it can be calculated from a Hamiltonian that is Hermitian in a consistent theory.

Source: http://en.wikipedia.org/wiki/Unitarity

Principal Authors: Lumidek, MathMartin, Charles Matthews, Beland, Keenan Pepper

Unitarity bound

In theoretical physics, a unitarity bound is any inequality that follows from the unitarity of the evolution operator, i.e. from the statement that probabilities are numbers between 0 and 1 whose sum is conserved. Unitarity implies, among other things, the optical theorem. According to the optical theorem, the imaginary part of a probability amplitude Im(M) of the forward scattering is related to the total cross section, up to some numerical factors. Because |M |2 for the forward scattering process is one of the terms that contributes to the total cross section, it cannot exceed the total cross section i.e. Im(M). The inequality |M |2 ≤ Im(M )

Unitarity bound

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implies that the complex number M must belong to a certain disk in the complex plane. Similar unitarity bounds imply that the amplitudes and cross section can’t increase too much with energy or they must decrease as quickly as a certain formula dictates.

Source: http://en.wikipedia.org/wiki/Unitarity_bound

Variational method (quantum mechanics)

The variational method is, in quantum mechanics, one way of finding approximations to the lowest energy eigenstate or ground state. The basis for this method is the variational principle.

Introduction

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Suppose we are given a →Hilbert space and a Hermitian operator over it called the Hamiltonian H. Ignoring complications about continuous spectra, we look at the discrete spectrum of H and the corresponding eigenspaces of each eigenvalue (see spectral theorem for Hermitian operators for the mathematical background): P 1 = λ∈Spec(H) |ψλ ihψλ | with

hψα |ψβ i = δαβ

and

E ˆ λ = λ|ψλ i. H|ψ

Physical states are normalized, meaning that their norm is equal to 1. Once again ignoring complications involved with a continuous spectrum of H, suppose it is bounded from below and that its greatest lower bound is E 0. Suppose also that we know the corresponding state |ψ>. The expectation value of H is then





 P hψ|H|ψi = λ1 ,λ2 ∈Spec(H) ψ|ψλ1 ψλ1 |H|ψλ2 ψλ2 |ψ =

2 λ∈Spec(H) λ| hψλ |ψi |

P

≥

2 λ∈Spec(H) E0 | hψλ |ψi |

P

Variational method (quantum mechanics)

= E0

632

Ansatz

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Obviously, if we were to vary over all possible states with norm 1 trying to minimize the expectation value of H, the lowest value would be E 0 and the corresponding state would be an eigenstate of E 0. Varying over the entire Hilbert space is usually too complicated for physical calculations, and a subspace of the entire Hilbert space is chosen, parametrized by some (real) differentiable parameters α i (i =1,2..,N). The choice of the subspace is called the ansatz. Some choices of ansatzes lead to better approximations than others, therefore the choice of ansatz is important. Let’s assume there is some overlap between the ansatz and the ground state (otherwise, it’s a bad ansatz). We still wish to normalize the ansatz, so we have the constraints hψ(αi )|ψ(αi )i = 1 and we wish to minimize

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ε(αi ) hψ(αi )|H|ψ(αi )i.

This, in general, is not an easy task, since we are looking for a global minimum and finding the zeroes of the partial derivatives of  over α i is not sufficient. If ψ (α i) is expressed as a linear combination of other function (α i being the ceofficients), as in the Ritz method, there is only one minimum and the problem is straightforward. There are other, non-linear methods, however, such as the Hartree-Fock method, that are also not characterized by a multitude of minima and are therefore comfortable in calcualtions.

There is an additional complication in the calculations described. As  tends toward E 0 in minimization calculations, there is no guarantee that the corresponding trial wavefunctions will tend to the actual wavefunction. This has been demonstrated by calculations using a modified harmonic oscillator as a model system, in which an exactly solvable system is approached using the variational method. A wavefunction different from the exact one is obtained by use of the method described above.

See also • •

Hartree-Fock method →Ritz method

Source: http://en.wikipedia.org/wiki/Variational_method_%28quantum_mechanics%29

Variational method (quantum mechanics)

633 Principal Authors: Karol Langner, Vb, Gparker, MarSch, Agentsoo

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Variational perturbation theory

In mathematics, variational perturbation theory is a mathematical method to convert divergent power series in a small expansion parameter, say P n s= ∞ n=0 an g , into a convergent series in powers P ω n s= ∞ n=0 bn /(g ) ,

where ω is a critical exponent. This is possible with the help of variational parameters, which are determined by optimization order by order in g.

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Variational perturbation theory is an important mathematical tool in the theory of critical phenomena. It has led to the most accurate predictions of critical exponents.

See also •

→Variational method (quantum mechanics)

References •

Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 3. Auflage, World Scientific (Singapore, 2004) 378 (readable online here 379) (see Chapter 15) • Hagen Kleinert and Verena Schulte-Frohlinde, Critical Properties of φ 4Theories, World Scientific (Singapur, 2001) 380; Paperback ISBN 981-024658-76 (readable online here 381) (see Chapter 19)

Source: http://en.wikipedia.org/wiki/Variational_perturbation_theory Principal Authors: Nimur, GangofOne, Alektzin, Charles Matthews

378 http://www.worldscibooks.com/physics/5057.html 379 http://www.physik.fu-berlin.de/~kleinert/b5 380 http://www.worldscibooks.com/physics/4733.html 381 http://www.physik.fu-berlin.de/~kleinert/b8

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Wavefunction

Definition

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This article discusses the concept of a wavefunction as it relates to quantum mechanics. The term has a significantly different meaning when used in the context of classical mechanics or classical electromagnetism.

The modern usage of the term wavefunction refers to any vector or function which describes the state of a physical system. Typically, a wavefunction is either: •

a complex vector with finitely many components

  c1  ..  ~ ψ= .

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cn

,

•

a complex vector with infinitely many components

  c1  ..    ~= . ψ  cn    .. . ,

•

or a complex function of one or more real variables (a "continuously indexed" complex vector) ψ(x1 , . . . xn ).

In all cases, the wavefunction provides a complete description of the associated physical system.

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Interpretation

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The physical interpretation of the wavefunction is context dependent. Several examples are provided below, followed by a detailed discussion of the three cases described above.

One particle in one spatial dimension

The spatial wavefunction associated with a particle in one dimension is a complex function ψ(x) defined over the real line. The positive function |ψ|2 is interpreted as the probability density associated with the particle’s position. That is, the probability of a measurement of the particle’s position yielding a value in the interval [a, b] is given by Rb Pab = a |ψ(x)|2 dx. This leads to the normalization condition R∞ 2 −∞ |ψ(x)| dx = 1 .

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since, clearly, the probability of the particle being somewhere on the line must be 1.

One particle in three spatial dimensions

The three dimensional case is analogous to the one dimensional case; the wavefunction is a complex function ψ(x, y, z) defined over three dimensional space, and its complex square is interpreted as a three dimensional probability density function: R PR = R |ψ(x, y, z)|2 dV The normalization condition is likewise R |ψ(x, y, z)|2 dV = 1

where the preceding integral is taken over all space.

Two distinguishable particles in three spatial dimensions In this case the wavefunction is a complex function of six spatial variables, ψ(x1 , y1 , z1 , x2 , y2 , z2 ) , and |ψ|2 is the joint probability density associated with the positions of both particles. Thus the probability that a measurement of the positions of both particles indicates particle one is in region R and particle two is region S is

Wavefunction

636 PR,S =

R R R S

|ψ|2 dV2 dV1

where dV1 = dx1 dy1 dz1 , and similarly for dV2 .

FT

The normalization condition is then: R |ψ(x, y, z)|2 dV2 dV1 = 1

where the preceding integral is taken over the full range of all six variables.

Given a wave function of ψ of a systems consisting of two (or more) particles, it is in general not possible to assign a definite wavefuction to a single-particle subsystem. In other words, the particles in the system can be entangled.

One particle in one dimensional momentum space

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The wavefunction for a one dimensional particle in momentum space is a complex function ψ(p) defined over the real line. The quantity |ψ|2 is interpreted as a probability density function in momentum space: Rb Pab = a |ψ(p)|2 dp As in the position space case, this leads to the normalization condition: R∞ 2 −∞ |ψ(p)| dp = 1.

Spin 1/2

The wavefunction for a spin 1/2 particle (ignoring its spatial degrees of freedom) is acolumn vector

~ = c1 ψ c2 .

The meaning of the vector’s components depends on the basis, but typically c1 and c2 are respectively the coefficients of spin up and spin down in the z direction. In Dirac notation this is: |ψi = c1 | ↑z i + c2 | ↓z i

The values |c1 |2 and |c2 |2 are then respectively interpreted as the probability of obtaining spin up or spin down in the z direction when a measurement of the particle’s spin is performed. This leads to the normalization condition |c1 |2 + |c2 |2 = 1 .

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Interpretation

Finite vectors

FT

A wavefunction describes the state of a physical system by expanding it in terms of other states of the same system. We shall denote the state of the system under consideration as |ψi and the states into which it is being expanded as |φi i. Collectively the latter are referred to as a basis or representation. In what follows, all wavefunctions are assumed to be normalized.

~ with n components describes how to exA wavefunction which is a vector ψ press the state of the physical system |ψi as the linear combination of finitely many basis elements |φi i, where i runs from 1 to n. In particular the equation

  c1  ..  ~ ψ= . cn

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,

which is a relation between column vectors, is equivalent to P |ψi = ni=1 ci |φi i,

which is a relation between the states of a physical system. Note that to pass between these expressions one must know the basis in use, and hence, two column vectors with the same components can represent two different states of a system if their associated basis states are different. An example of a wavefunction which is a finite vector is furnished by the spin state of a spin-1/2 particle, as described above. ~ is given by the wavefunction The physical meaning of the components of ψ

collapse postulate:

If the states |φi i have distinct, definite values, λi , of some dynamical variable (e.g. momentum, position, etc) and a measurement of that variable is performed on a system in the state |ψi =

P

i ci |φi i

then the probability of measuring λi is |ci |2 , and if the measurement yields λi , the system is left in the state |φi i.

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638

Infinite vectors

c1  ..    ~= . ψ  cn    .. .

is equivalent to P |ψi = i ci |ψi i,

FT

The case of an infinite vector with a discrete index is treated in the same manner a finite vector, except the sum is extended over all the basis elements. Hence 

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~ where it is understood that the above sum includes all the components of ψ. The interpretation of the components is the same as the finite case (apply the collapse postulate).

Continuously indexed vectors (functions)

In the case of a continuous index, the sum is replaced by an integral; an example of this is the spatial wavefunction of a particle in one dimension, which expands the physical state of the particle, |ψi, in terms of states with definite position, |xi. Thus R∞ |ψi = −∞ ψ(x)|xi dx.

Note that |ψi is not the same as ψ(x) . The former is the actual state of the particle, whereas the latter is simply a wavefunction describing how to express the former as a superposition of states with definite position. In this case the base states themselves can be expressed as R∞ |x0 i = −∞ δ(x − x0 )|xi dx and hence the spatial wavefunction associated with |x0 i is δ(x − x0 ) .

Formalism

Given an isolated physical system, the allowed states of this system (i.e. the states the system could occupy without violating the laws of physics) are part of a →Hilbert space H. Some properties of such a space are 1. If |ψi and |φi are two allowed states, then

Wavefunction

639 a|ψi + b|φi

FT

is also an allowed state, provided |a|2 + |b|2 = 1. (This condition is due to normalisation.) 2. There is always an orthonormal basis of allowed states of the vector space H. The wavefunction associated with a particular state may be seen as an expansion of the state in a basis of H. For example, {| ↑z i, | ↓z i}

is a basis for the space associated with the spin of a spin-1/2 particle and consequently the spin state of any such particle can be written uniquely as

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a| ↑z i + b| ↓z i.

Sometimes it is useful to expand the state of a physical system in terms of states which are not allowed, and hence, not in H. An example of this is the spacial wavefunction associated with a particle in one dimension which expands the state of the particle in terms of states with definite position. Every Hilbert space H is equipped with an inner product. Physically, the nature of the inner product is contingent upon the kind of basis in use. When the basis is a countable set {|φi i} , and orthonormal, i.e. hφi |φj i = δij .

Then an arbitrary vector |ψi can be expressed as P |ψi = i ci |φi i where ci = hφi |ψi.

If one chooses a "continuous" basis as, for example, the position or coordinate basis consisting of all states of definite position {|xi}, the orthonormality condition holds similarly: hx|x0 i = δ(x − x0 ).

We have the analogous identity R R hx| ψ(x0 )|x0 i dx0 = ψ(x0 )δ(x − x0 ) dx0 = ψ(x).

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See also

• • • •

→Wave packet Boson - particles with symmetric wavefunction under permutation (i.e. switching positions) Fermion - particles with antisymmetric wavefunction under permutation →Quantum mechanics →Schrödinger equation →Normalisable wavefunction

References •

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• •

Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X.

Source: http://en.wikipedia.org/wiki/Wavefunction

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Principal Authors: CSTAR, Joshua Barr, CygnusPius, Oleg Alexandrov, Dhc529

Wave packet

The neutrality of this article is disputed. Please see the discussion on the talk page.

In physics, a wave packet is an envelope or packet containing an arbitrary number of wave forms. In quantum mechanics the wave packet is ascribed a special significance: it is interpreted to be a "probability wave" describing the probability that a particle or particles in a particular state will have a given position and momentum. By applying the →Schrödinger equation in quantum mechanics it is possible to deduce the time evolution of a system, similar to the process of the Hamiltonian formalism in classical mechanics. The wave packet is a mathematical solution to the Schrödinger equation. The square of the area under the wave packet solution is interpreted to be the probability density of finding the particle in a region. In the coordinate representation of the wave (such as the Cartesian coordinate system) the position of the wave is given by the position of the packet. Moreover, the narrower the wave packet, and therefore the better defined the position of the wave packet, the larger the uncertainty in the momentum of the wave. This tradeoff is known as the Heisenberg uncertainty principle. Wave packet

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Background

FT

In the early 1900s it became apparent that classical mechanics had some major failings. Isaac Newton originally proposed the idea that light came in discrete packets which he called "corpuscles", but the wave-like behavior of many light phenomena quickly led scientists to favor a wave description of electromagnetism. It wasn’t until the 1930s that the particle nature of light really began to be widely accepted in physics. The development of quantum mechanics — and its success at explaining confusing experimental results — was at the foundation of this acceptance. One of the most important concepts in the formulation of quantum mechanics is the idea that light comes in discrete bundles called photons. The energy of light is a discrete function of frequency: E = nhf

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The energy is an integer, n, multiple of Planck’s constant, h, and frequency, f. This resolved a significant problem in classical physics, called the ultraviolet catastrophe. The ideas of quantum mechanics continued to be developed throughout the 20th century. The picture that was developed was of a particulate world, with all phenomena and matter made of and interacting with discrete particles; however, these particles were described by a probability wave. The interactions, locations, and all of physics would be reduced to the calculations of these probability amplitude waves. The particle-like nature of the world was significantly confirmed by experiment, while the wave-like phenomena could be characterized as consequences of the wave packet nature of particles.

Mathematics of wave packets

As an example, consider wave solutions to the following wave equation: ∂2u ∂t2

= c2 ∇2 u

where c is the speed of the wave’s propagation in a given medium. The wave equation has plane-wave solutions u(x, t) = eik·x−iωt

where |k| =

ω c.

To simplify, consider only waves propagating in one dimension. Then the general solution is

Wave packet

642 u(x, t) = Aeikx−iωt + Be−ikx−iωt

2π

FT

A wave packet is a localized disturbance that results from the sum of many different wave forms. If the packet is strongly localized, more frequencies are needed to allow the constructive superposition in the region of localization and destructive superposition outside the region. From the basic solutions in the one dimension, a general form of a wave packet can be expressed as R∞ f (x, t) = √1 −∞ A(k)eikx−iω(k)t dk.

√ The factor 1/ 2π comes from Fourier transform conventions. The amplitude A(k) contains the coefficients of the linear superposition of the plane wave solutions. These coefficients can in turn be expressed as a function of f (x, t) evaluated at t = 0: R∞ A(k) = √1 −∞ f (x, 0)e−ikx dk. 2π

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Quantum mechanical waves

A quantum mechanical wave in its most salient and simple form is a solution to a differential equation. It is a bridge that provides mathematical insight into physical problems. The solutions of these mathematical models are postulated in quantum mechanics to provide the possible or observable outcomes for any experiment. Unfortunately, the wave packet combined with the probabilistic nature of measurement leads to some peculiarities. First, the solutions do not provide answers about single experiments. These solutions can only be confirmed by measurement, and measurement is probabilistic in nature. Experimental confirmation of a prediction of quantum mechanics is only given in the outcomes of repeated similar experiments. The first mistake that many people make in thinking about quantum mechanical predictions is in thinking about only single experiments. In reality, quantum mechanics has little to say about the observations made in single experiments unless the system is prepared in an energy eigenstate. Next, the quantum mechanical wave is a representation of a particle. The wave carries the information about particle position and momentum and also any other observable that can be derived from position and momentum. However, there is no reason to believe that a quantum wave actually is a particle. In the words of Dirac, the wave expresses information. A quantum mechanical wave can be nothing more than a mathematical model.

Wave packet

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Superposition

FT

Nevertheless, it is reasonable to think about experiments in terms of quantum mechanical intuition. It is this blurring of the line between mathematical modeling and the quantum picture of the world that so often leads to confusion. For the purposes of the uninitiated it would be much safer to consider only the model, and leave the intuition to those who are better acquainted with the mathematical intricacies of quantum mechanics.

One of the most common classes of problems discussed in a quantum mechanics are interference phenomena. These interference phenomena apparently arise from the self-interaction of particles and the wave-like nature of these interactions. Such self-interaction is enabled by the principle of superposition. A particle does not negotiate any single path through a diffraction grating; its probability wave actually coincidently traverses all possible paths. Ultimately it is the act of measurement that collapses the wave packet to the single observed outcome.

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The collapse of the wave packet

The superposition principle of quantum mechanics allows any solution to the Schrödinger equation to be composed of a linear combination of any number of possible states in a complete set of commuting observables. The act of measurement typically collapses these superimposed states into a single outcome. For instance, the state of an electron passing through a double slit is most correctly described as a combination of each of the possible individual paths it might take, resulting in the famous double slit diffraction pattern. If, however, a measurement is made at the slits, then the double slit diffraction pattern disappears because the electron now travels through only a single slit. Quantum mechanics places no constraints on how we interpret the collapse of the wave packet. We might say that during the act of observation the electron suddenly and probabilistically jumps into the state we measure. On the other hand we might suppose, as many early physicists, including Albert Einstein, erroneously did, that the particle was in the observed state all along. This is the classical interpretation and does not agree with experimental evidence. Because none of the modern interpretations can be falsified experimentally, quantum mechanics provides no way of knowing how or why the wave packet collapsed, and what if any significance the event holds. Pointedly, Dirac intimates that this question has less significance than many people assume. The importance of quantum mechanics lies in the predictions that it makes, and the abject success of those predictions, culminating in nearly overwhelming experimental confirmation of the theory. The fact that we don’t Wave packet

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Metaphysical claims

FT

really know how to interpret the collapse of the wave packet at this time is irrelevant, what is important for physics is that the theory works at all.

Many discussions about the collapse of the wave packet and quantum superposition occur in metaphysical and speculative fiction circles, primarily based on attempts to draw analogies between the language used in quantum physics with the world of the layman. Common themes are the many-worlds interpretation and the related multiverse, proof of the existence of God, teleportation, faster-than-light travel, and proof of human super-consciousness. Generally, like most fiction, these themes sacrifice scientific rigor in favor of evocative concepts, as the physics of quantum mechanics is not easily described relative to human experience.

References

Jackson, J.D. (1975). Classical Electrodynamics (2nd Ed.). New York: John Wiley & Sons, Inc. ISBN 0-471-43132-X

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•

Source: http://en.wikipedia.org/wiki/Wave_packet

Principal Authors: John187, Laurascudder, Art LaPella, CLW, Hansm

Wave-particle duality

In physics, wave-particle duality holds that light and matter exhibit properties of both waves and of particles. It is a central concept of quantum mechanics. The idea is rooted in a debate over the nature of light and matter dating back to the 1600s, when competing theories of light were proposed by Christiaan Huygens and Isaac Newton. The photon was the first entity that was seen to exhibit these dualistic properties. And so wave-particle duality is often stated like this: "A photon sometimes acts like a wave, and sometimes acts like a particle, but not at the same time." However, this is slightly misleading, because a photon always acts like both to varying degrees. For example, when shooting single photons through a slit, a detector can detect each photon when it hits a photosensitive screen (its position is recorded) - but over time, the detector will detect the same diffraction

Wave-particle duality

645 pattern as it would if the photons were given off all in one burst. This is because any given trajectory the photon could take has a certain probability that is dictated by the properties of an electromagnetic wave.

History

FT

Once it was realised that all particles exhibit wave-particle duality this lead rapidly into the development of "new" quantum mechanics, superseding the old Bohr atomic planetary model. The new quantum mechanics incorporated wave-particle duality into the core of the formalism, where it remains to this day. Through the work of Albert Einstein, Louis de Broglie, Arthur Compton and many others, it is now accepted that all objects have both wave and particle nature (though this phenomenon is only detectable on small scales, such as with atoms), and that quantum mechanics provides the over-arching theory resolving this paradox.

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At the close of the 19th century, the case for atomic theory, that matter was made of particulate objects or atoms, was well established. Electricity, first thought to be a fluid, was understood to consist of particles called electrons, as demonstrated by J.J. Thomson by his research into the work of Rutherford, who had investigated using cathode rays that an electrical charge would actually travel across a vacuum from cathode to anode. In brief, it was understood that much of nature was made of particles. At the same time, waves were well understood, together with wave phenomena such as diffraction and interference. Light was believed to be a wave, as Thomas Young’s double-slit experiment and effects such as Fraunhofer diffraction had clearly demonstrated the wave-like nature of light.

But as the 20th century turned, problems had emerged with this viewpoint. The photoelectric effect, as analyzed in 1905 by Albert Einstein, demonstrated that light also possessed particle-like properties, further confirmed with the discovery of the Compton effect in 1923. Later on, the diffraction of electrons would be predicted and experimentally confirmed, thus showing that electrons must have wave-like properties in addition to particle properties. This confusion over particle versus wave properties was eventually resolved with the advent and establishment of quantum mechanics in the first half of the 20th century, which ultimately explained wave-particle duality. It provided a single unified theoretical framework for understanding that all matter can behave in both a wave-like and a particle-like fashion in the appropriate circumstances. Quantum mechanics holds that every particle in nature, be it a photon, electron or atom, is described by a solution to a differential equation, most typically, the Schroedinger equation. The solutions to this equation are known as wave functions, as they are inherently wave-like in their form. Wave-particle duality

646

FT

They can diffract and interfere, leading to the wave-like phenomena that are observed. Yet also, the wave functions are interpreted as describing the probability of finding a particle at a given point in space. Thus, if one is looking for a particle, one will find one, with a probability density given by the square of the magnitude of the wave function.

One does not observe the wave-like quality of everyday objects because the associated wavelengths of people-sized objects are exceedingly small. The wavelength is given essentially as the inverse of the size of the object, with the factor given by Planck’s constant h, an extremely small number.

Huygens and Newton; earliest theories of light

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The earliest comprehensive theory of light was advanced by Christiaan Huygens, who proposed a wave theory of light, and in particular demonstrated how waves might interfere to form a wave-front, propagating in a straight line. However, the theory had difficulties in other matters, and was soon overshadowed by Isaac Newton’s corpuscular theory of light. That is, Newton proposed that light consisted of small particles, with which he could easily explain the phenomenon of reflection. With considerably more difficulty, he could also explain refraction through a lens, and the splitting of sunlight into a rainbow by a prism. Because of Newton’s immense intellectual stature, his theory went essentially unchallenged for over a century, with Huygens’ theories all but forgotten. With the discovery of diffraction in the early 19th century, the wave theory was revived, and so by the advent of the 20th century, a scientific debate over waves vs. particles had already been thriving for a very long time.

Fresnel, Maxwell, and Young

In the early 1800s, the double-slit experiments by Young and Fresnel provided evidence for Huygens’ theories: these experiments showed that when light is sent through a grid, a characteristic interference pattern is observed, very similar to the pattern resulting from the interference of water waves; the wavelength of light can be computed from such patterns. Maxwell, during the late1800s, explained light as the propagation of electromagnetic waves with the Maxwell equations. These equations were verified by experiment, and Huygens’ view became widely accepted.

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Einstein and photons

FT

In 1905, Albert Einstein provided a remarkable explanation of the photoelectric effect, a hitherto troubling experiment which the wave theory of light seemed incapable of explaining. He did so by postulating the existence of photons, quanta of light energy with particulate qualities.

In the photoelectric effect, it was observed that shining a light on certain metals would lead to an electric current in a circuit. Presumably, the light was knocking electrons out of the metal, causing them to flow. However, it was also observed that while a dim blue light was enough to cause a current, even the strongest, brightest red light caused no current at all. According to wave theory, the strength or amplitude of a light wave was in proportion to its brightness: a bright light should have been plenty strong enough to create a large current. Yet, oddly, this was not so. Einstein explained this conundrum by postulating that the electrons were knocked free of the metal by incident photons, with each photon carrying an amount of energy E that was related to the frequency, ν of the light by

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E = hν ,

where h is Planck’s constant (6.626 x 10 -34 J seconds). Only photons of a highenough frequency, (above a certain threshold value) could knock an electron free. For example blue light, but not red light, had sufficient energy to free an electron from the metal. More intense light above the threshold frequency could release more electrons, but no amount of light below the threshold frequency could release an electron. Einstein was awarded the Nobel Prize in Physics in 1921 for his theory of the photoelectric effect.

De Broglie

In 1924, Louis-Victor de Broglie formulated the de Broglie hypothesis, claiming that all matter has a wave-like nature; he related wavelength, (lambda), and momentum, p: λ=

h p

This is a generalization of Einstein’s equation above since the momentum of a photon is given by p = E / c where c is the speed of light in vacuum, and = c / ν. De Broglie’s formula was confirmed three years later for electrons (which have a rest-mass) with the observation of electron diffraction in two independent Wave-particle duality

648

FT

experiments. At the University of Aberdeen, George Paget Thomson passed a beam of electrons through a thin metal film and observed the predicted interference patterns. At Bell Labs Clinton Joseph Davisson and Lester Halbert Germer guided their beam through a crystalline grid. De Broglie was awarded the Nobel Prize for Physics in 1929 for his hypothesis. Thomson and Davisson shared the Nobel Prize for Physics in 1937 for their experimental work.

Wave nature of large objects

Similar experiments have since been conducted with neutrons and protons. Among the most famous experiments are those of Estermann and Otto Stern in 1929. Authors of similar recent experiments with atoms and molecules claim that these larger particles also act like waves.

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A dramatic series of experiments emphasizing the action of gravity in relation to wave-particle duality were conducted in the 1970’s using the neutron interferometer. Neutrons, subatomic particles in atomic nuclei, provide much of the mass of a nucleus and thus of ordinary matter. Neutrons are fermions, and thus obey the Pauli Exclusion Principle. In the neutron interferometer, they act as quantum-mechanical waves directly subject to the force of gravity. While the results were not surprising since gravity was known to act on everything - even deflecting light on large scales and acting on photons as well on smaller scales (the Pound-Rebka falling photon experiment), the self-interference of the quantum mechanical wave of a massive fermion in a gravitational field had never been experimentally confirmed before.

In 1999, the diffraction of C 60 fullerenes by researchers from the University of Vienna was reported 1. Fullerenes are rather large and massive objects, having an atomic mass of about 720. The de Broglie wavelength is 2.5 picometers, whereas the diameter of the molecule is about 1 nanometer, i.e. about 400 times larger. As of 2005, this is the largest object for which quantum-mechanical wave-like properties have been directly observed in farfield diffraction. The experimenters have assumed the arguments of waveparticle duality and have assumed the validity of de Broglie’s equation in their argument. In 2003 the Vienna group has meanwhile also demonstrated the wave-nature of tetraphenylporphyrin 4 - a flat biodye with an extension of about 2 nm and a mass of 614 amu. For this demonstration they employed a nearfield Talbot Lau interferometer 2,3. In the same interferometer they also found interference fringes for C60F48, a fluorinated buckyball with a mass of about 1600 amu, composed of 108 atoms 4. Large molecules are already so complex that they give experimental access to some aspects of the quantum-classical interface, i.e. to certain decoherence mechanisms 5,6. Wave-particle duality

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Whether objects heavier than the Planck mass (about the weight of a large bacterium) have a de Broglie wavelength is theoretically unclear and experimentally unreachable. The wavelength would be smaller than the Planck length, a scale at which current theories of physics may break down or need to be replaced by more general ones.

Theoretical sketch and remarks on philosophical inquiry The wave-particle paradox is resolved in the theoretical framework of quantum mechanics. This framework is deep and complex and therefore impossible to adequately summarize in brief.

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Every particle in nature can be described as a superposition of solutions to a differential equation. The most basic is the Schroedinger equation, but this does not include any relativistic effects and is not generally realistic. The KleinGordon Equation is a relativistic version of the Schroedinger equation that is applicable to spin-0 particles, and the Dirac Equation is the relativistic version of Schroedinger’s equation for spin-1/2 particles. The solutions to these equations contain oscillatory mathematical components and are hence inherently wave-like in nature. In practice these are referred to as wave functions and they can describe diffractions, interferance with one another or themselves, and otherwise accurately predict observed wave-like phenomena such as is described in the double-slit experiment.

Wave functions are often interpreted as describing the probability of finding their corresponding particle at a given point in space at a given time. For example, upon setting up an experiment involving a moving particle, one can ’look’ for that particle to arrive at some particular location using a detection apparatus set up at that location. While quantum behavior follows well-defined deterministic equations (such as the wave function), the solutions to these equations are probabilistic. The probability of the detector detecting the particle is calculated by taking the integral of the product of the wave function and its complex conjugate. While the wave function can be thought of as smeared out in space, in practice the detector will always either *see* or *not see* the entire particle in question; it will never see a fractional piece of the particle, like two-thirds of an electron. Hence the strange duality: The particle propagates in space in a distributed, probabilistic wavelike fashion but arrives at a detector as a localized, complete corpuscle. This paradoxical conceptual framework has some explanations in the forms of the Copenhagen interpretation, Path Integral Formulation, or the Many Worlds Interpretation. It is important to realize that all of these interpretations are equivalent and result in the same predictions even though they offer widely different philosophical interpretations. Wave-particle duality

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A more mathematically concise formulation of the wave function is to treat it as a ket. Essentially each quantum object can be described by making use of an infinite dimensional →Hilbert space. The wave functions, or kets reside in the Hilbert space, and are eigenfunctions of an eigenequation in which an operator acts on the ket to return the ket multiplied by a scalar eigenvalue. In this sense, the quantum system is neither a particle nor a wave. It is described by an abstract ket that will appear to behave as one or the other depending on what kind of observation you are making at the time.

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While quantum mechanics makes astoundingly accurate predictions about the outcomes of such experiments, its philosophical meaning is still sought after and debated. This debate has evolved as a broadening of the original struggles to comprehend wave-particle duality. What does it mean for a proton, to behave both as a particle and as a wave? How can an antimatter electron be mathematically equivalent to a regular electron moving backwards in time under certain circumstances, and what implications does this have for our experience of time as one-directional? How can a particle seemingly teleport through a barrier while soccer balls regularly fail to pass through cement walls? The implications of these facets of quantum mechanics continue to puzzle many who delve into the subject. The discussion currently can be investigated further under headings of local realism and quantum measurement. As to the assumption of (nonlocal) hidden variables in a hypothesized sub-quantum domain, see Bohm interpretation, or quantum cybernetics, respectively.

In John Cramer’s transactional interpretation of QM, the probabilistic or particle aspect of matter is de-emphasized, in favor of a more comprehensive use of waves to explain all the same phenomena. This approach is also used in Carver Mead’s Collective Electrodynamics approach to quantum electron phenomena. Whether these approaches lead to verifiably distinct physical predictions remains to be seen, but they at least provide an alternative philosophy to the Copenhagen interpretation, a way to understand QM and entanglement without Einstein’s so-called "spooky action at a distance". They substitute instead the non-causality of an advanced wave to allow a mutual wave transaction between events at zero interval in spacetime, as in Wheeler–Feynman electrodynamics. Some physicists intimately associated with the historical struggle to arrive at the rules of quantum mechanics have viewed these philosophical debates on wave-particle duality and related matters as attempts to impose human experience on the quantum (microscopic) world. Since by its nature this world is completely non-intuitive, quantum theory (they would assert) must be learned on its own terms independent of experience-based human intuition. The scientific merit of searching too deeply for a ’meaning’ to quantum mechanics Wave-particle duality

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Applications

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is thereby suspect; Bell’s theorem and experiments it inspires provide a good example of such testing of the foundations of quantum mechanics. From a physics viewpoint, the inability of a new quantum philosophy to satisfy the testability criterion or alternatively the inability to find a flaw in the predictive power of the existing theory reduces to a null proposition, perhaps even risking degeneration into pseudoscience.

Wave-particle duality is exploited in electron microscopy, where the small wavelengths associated with the electron can be used to view objects much smaller than what is visible using visible light.

See also •

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This entry is still rather rudimentary. It would be very desirable to add a information on the huge amount of literature on atomic beam interferometry, Bose Einstein Condensation, decoherence experiments as well as interesting papers on diffraction of cold molecules and several recent electron interference studies. • Afshar experiment • Arago spot • Hanbury-Brown and Twiss effect • →Scattering theory

References •

Note 1: Arndt, Markus, O. Nairz, J. Voss-Andreae, C. Keller, G. van der Zouw, A. Zeilinger (14 October 1999). "Wave-particle duality of C60" 382. Nature 401: 680-682. • Note 2: Clauser, John F., S. Li (1994). "Talbot von Lau interefometry with cold slow potassium atoms." 383. Phys. Rev. A 49: R2213-17. • Note 3: Brezger, Björn, Lucia Hackermüller, Stefan Uttenthaler, Julia Petschinka, Markus Arndt and Anton Zeilinger (2002). "Matter-wave interferometer for large molecules" 384. Phys. Rev. Lett. 88: 100404. • Note 4: Hackermüller, Lucia, Stefan Uttenthaler, Klaus Hornberger, Elisabeth Reiger, Björn Brezger, Anton Zeilinger and Markus Arndt (2003). "The

382 http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v401/n6754/full/401680a0_fs.

html&content_filetype=pdf

383 http://ojps.aip.org 384 http://ojps.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PRLTAO000088000010100404000001

&idtype=cvips

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•

•

•

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•

wave nature of biomolecules and fluorofullerenes" 385. Phys. Rev. Lett. 401: 680-682. Note 5: Hornberger, Klaus, Stefan Uttenthaler,Björn Brezger, Lucia Hackermüller, Markus Arndt and Anton Zeilinger (2003). "Observation of Collisional Decoherence in Interferometry" 386. Phys. Rev. Lett. 90: 160401. Note 6: Hackermüller, Lucia, Klaus Hornberger, Björn Brezger, Anton Zeilinger and Markus Arndt (2004). "Decoherence of matter waves by thermal emission of radiation" 387. Nature 427: 711-714. R. Nave. Wave-Particle Duality 388. (Web page) HyperPhysics. Georgia State University, Department of Physics and Astronomy. Retrieved on December 12, 2005. Markus Arndt (2006). Interferometry and decoherence experiments with large molecules 389. (Web page) University of Vienna. Retrieved on May 6, 2006.

Source: http://en.wikipedia.org/wiki/Wave-particle_duality

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Principal Authors: Linas, StuRat, Cdang, Pizza Puzzle, Srleffler

Wien’s displacement law

Wien’s displacement law is a law of physics that states that there is an inverse relationship between the wavelength of the peak of the emission of a black body and its temperature. λmax =

b T

where

λmax is the peak wavelength in meters,

T is the temperature of the blackbody in kelvins (K), and

385 http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v401/n6754/full/401680a0_fs.

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386 http://ojps.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=PRLTAO000090000016160401000001

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387 http://www.nature.com/cgi-taf/DynaPage.taf?file=/nature/journal/v427/n6976/full/nature02276_fs.

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388 http://hyperphysics.phy-astr.gsu.edu/hbase/mod1.html 389 http://www.quantum.at/typo/index.php?id=177&tx_jppageteaser_pi1=35

Wien’s displacement law

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FT

653

Figure 70 The wavelength corresponding to the peak emission in various black body spectra as a function of temperature

b is a constant of proportionality, called Wien’s displacement constant, in kelvin-meters.

The value of this constant (CODATA 2002 recommended) is b = 2.8977685 × 10−3 ± 5.1 × 10−9 m · K For optical wavelengths, it is often more convenient to use the nanometer in place of the meter as the unit of measure. In this case, the constant becomes 2.8977685 × 10 6 kelvin-nanometers.

Explanation

Fundamentally, Wien’s law states that the hotter an object is, the shorter the wavelength at which it will emit most of its radiation. For example, the surface temperature of the Sun is 5780 K. Using Wien’s law, this temperature corresponds to a peak emission at a wavelength of 500 nm. This wavelength is fairly in the middle of the visual spectrum (see for example the article color), because of the spread resulting in white light. Due to the Rayleigh scattering of Wien’s displacement law

654 blue light by the atmosphere this white light is separated somewhat, resulting in a blue sky and a yellow sun.

FT

A lightbulb has a glowing wire with a somewhat lower temperature, resulting in yellow light, and something that is "red hot" is again a little less hot. The law is named for Wilhelm Wien, who formulated the relationship in 1893 based on empirical data.

Frequency form

In terms of frequency f (in hertz), Wien’s displacement law becomes fmax =

αk h T

≈ (5.879 × 1010 Hz/K) · T

where

α ≈ 2.821439... is a constant resulting from the numerical solution of the maximization equation,

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k is Boltzmann’s constant, h is Planck’s constant, and

T is temperature (in kelvin).

Because the spectrum resulting from →Planck’s law of black body radiation takes a different shape in the frequency domain from that of the wavelength domain, the frequency location of the peak emission does not correspond to the peak wavelength using the simple relationship between frequency, wavelength, and the speed of light.

Derivation

Wilhelm Wien formulated this law, in 1893, based entirely on empirical observations, prior to the development of →Planck’s law of black body radiation. With the benefit of hindsight, however, it is now possible to derive Wien’s law as a direct consequence of Planck’s more general expression. From Planck’s law, we know that the spectrum of black body radiation is u(λ) =

1 8πhc λ5 ehc/λkT −1

The value of λ for which this function is maximized is sought. To find it, we differentiate u(λ) with respect to λ and set it equal to zero Wien’s displacement law

655 = 8πhc

hc 1 λkT 1−e−hc/λkT

hc ehc/λkT kT λ7 (ehc/λkT −1)2

hc λkT

then x 1−e−x

1 5 λ6 ehc/λkT −1

 =0

−5=0

If we define x≡

−

FT

∂u ∂λ



−5=0

This equation cannot be solved in terms of elementary functions. It can be solved in terms of Lambert’s Product Log function but an exact solution is not important in this derivation. One can easily find the numerical value of x x = 4.965114231744276 . . .

(dimensionless)

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Solving for the wavelength λ in units of nanometers, and using units of kelvins for the temperature yields: λmax =

hc 1 kx T

=

2.89776829...×106 nm·K . T

The frequency form of Wien’s displacement law is derived using similar methods, but starting with Planck’s law in terms of frequency instead of wavelength.

External links

• •

Eric Weisstein’s World of Physics 390 PlanetPhysics 391

Source: http://en.wikipedia.org/wiki/Wien%27s_displacement_law Principal Authors: Metacomet, Lir, PAR, Vsmith, Patrick

390 http://scienceworld.wolfram.com/physics/WiensDisplacementLaw.html 391 http://planetphysics.org/?op=getobj&from=objects&id=20

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Wigner-Eckart theorem

FT

The Wigner-Eckart theorem is a theorem of representation theory and quantum mechanics allowing operators to be transformed from one basis to another. These transformations involve the use of →Clebsch-Gordan coefficients.

References • •

Eric W. Weisstein, Wigner-Eckart theorem 392 at MathWorld. Wigner-Eckart theorem 393

Source: http://en.wikipedia.org/wiki/Wigner-Eckart_theorem

Principal Authors: Spiralhighway, Magnus Manske, Charles Matthews, Hillman, Oleg Alexandrov

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Wigner quasi-probability distribution See also Wigner distribution, a disambiguation page.

A topic in Quantum theory <> 394

The Wigner quasi-probability distribution was introduced by Eugene Wigner in 1932 to study quantum corrections to classical statistical mechanics. The goal was to replace the wavefunction that appears in Schrödinger’s equation with a probability distribution in phase space. It was independently derived by Hermann Weyl in 1931 as the symbol of the density matrix in representation theory in mathematics. It was once again derived by J. Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal. It is also known as the "Wigner function," "Wigner-Weyl transformation" or the "Wigner-Ville distribution". It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields such as electrical engineering seismology, biology, and engine design. A classical particle has a definite position and momentum and hence, is represented by a point in phase space. When one has a collection (ensemble)

392 http://mathworld.wolfram.com/Wigner-EckartTheorem.html 393 http://electron6.phys.utk.edu/qm2/modules/m4/wigner.htm

Wigner quasi-probability distribution

657

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of particles, the probability of finding a particle at a certain position in phase space is given by a probability distribution. This is not true for a quantum particle due to the uncertainty principle. Instead, one can create a quasi-probability distribution, which necessarily does not satisfy all the properties of a normal probability distribution. For instance, the Wigner distribution can go negative for states which have no classical model (and hence, it can be used to identify non-classical states). The Wigner distribution P (q, p) is defined as: R∞ 1 ∗ 2ipy/~ P (x, p) = π~ −∞ dy ψ (x + y)ψ(x − y)e

where ψ is the wavefunction and x and p are position and momentum but could be any conjugate variable pair. (ie. real and imaginary parts of the electric field or frequency and time of a signal). It is symmetric in x and p: R∞ 1 ∗ −2ixq/~ P (x, p) = π~ −∞ dq φ (p + q)φ(p − q)e where φ is the Fourier transform of ψ.

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In the case of a mixed state: R∞ 1 P (x, p) = π~ ρ|x + yie2ipy/~ −∞ dy hx − y|ˆ where ρ is the density matrix.

Mathematical properties 1. P (x, p) is real

2. The x and p probability distributions are given by the marginals:

• Typically the trace of ρ is equal to 1. • 1. and 2. imply the P(x,p) is negative somewhere, with the exception of the coherent state (and mixtures of coherent states) and the squeezed vacuum state. 3. P (x, p) has the following reflection symmetries: • •

Time symmetry: ψ(x) → ψ(x)∗ ⇒ P (x, p) → P (x, −p) Space symmetry: ψ(x) → ψ(−x) ⇒ P (x, p) → P (−x, −p)

4. P (x, p) is Galilei-invariant: •

It is not Lorentz invariant.

5. The equation of motion for each point in the phase space is classical in the absence of forces: 6. State overlap is calculated as: Wigner quasi-probability distribution

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Figure 71 Figure 1: The Wigner quasiprobability distribution for a) the vacuuum b) An n = 1 →Fock state (e.g. a single photon) c) An n = 5 Fock state.

7. Operators and expectation values (averages) are calculated as follows: 8. In order that P (x, p) represent physical (positive) density matrices: where |θ> is a pure state.

Uses of the Wigner function outside quantum mechanics •

In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple ray tracing and the full wave analysis of the system. Here p/~ is replaced with k =|k |sinθ≈|k |θ in the small angle (paraxial) approximation. In this context, the Wigner function is the closest one can get to describing the system in terms of rays at position x and angle θ while still including the effects of interference. If it becomes negative at any point then simple ray-tracing will not suffice to model the system. Wigner quasi-probability distribution

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659

Figure 72 Figure 2: A contour plot of the Wigner-Ville distribution for a chirped pulse of light. The plot makes it obvious that the frequency is a linear function of time.

•

In signal analysis, a time-varying electrical signal, mechanical vibration, or sound wave are represented by a Wigner function. Here, x is replaced with the time and p/~ is replaced with the angular frequency ω=2πf, where f is the regular frequency.

•

In ultrafast optics, short laser pulses are characterized with the Wigner function using the same f and t substitutions as above. Pulse defects such as chirp (the change in frequency with time) can be visualized with the Wigner function. See Figure 2.

•

In quantum optics, x and p/~ are replaced with the X and P quadratures, the real and imaginary components of the electric field (see coherent state). The plots in Figure 1 are of quantum states of light.

Measurements of the Wigner function • • •

Tomography Homodyne detection FROG Frequency-resolved optical gating

Wigner quasi-probability distribution

660

Other related quasi-probability distributions

• •

Glauber P representation Husimi Q representation

Historical note

FT

The Wigner distribution was the first quasi-probability distribution but many more followed with various advantages:

As the introduction shows, the formula for the Wigner function was independently derived many times in different contexts. In fact, apparently Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac. However, they missed its significance as they believed it was only an approximation to the true quantum description of a system such as the atom. Incidentally, Dirac would later become Wigner’s brother-in-law. See references.

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See also •

Heisenberg group

References •

• • •

•

• •

•

E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", Phys. Rev. 40 (June 1932) 749-759. H. Weyl, Z. Phys. 46, 1 (1927). H. Weyl, Gruppentheorie und Quantenmechanik (Leipzig: Hirzel)(1928). H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931). J. Ville, "Théorie et Applications de la Notion de Signal Analytique", Cables et Transmission, 2A: (1948) 61-74. W. Heisenberg, "Über die inkohärente Streuung von Röntgenstrahlen", Physik. Zeitschr. 32, 737-740 (1931). P.A.M. Dirac, "Note on exchange phenomena in the Thomas atom", Proc. Camb. Phil. Soc. 26, 376-395 (1930). C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space ( World Scientific, Singapore, 2005).

Source: http://en.wikipedia.org/wiki/Wigner_quasi-probability_distribution Principal Authors: J S Lundeen, Michael Hardy, Linas, Btyner, Mjb

Wigner quasi-probability distribution

661

Work function

FT

The work function is the minimum energy (usually measured in electron volts) needed to remove an electron from a solid to a point immediately outside the solid surface. Here "immediately" means that the final electron position is far from the surface on the atomic scale but still close to the solid on the macroscopic scale. Work function is an important property of metal. The magnitude of work function is usually about a half of the ionization energy of a free atom of the metal.

Work Function and Surface Effect

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Work function W of a metal is closely related to its Fermi energy level F yet the two quantities are not exactly the same. This is due to the surface effect of a real-world solid: a real-world solid is not infinitely extended with electrons and ions repeatedly filling every primitive cell over all Bravais lattice sites. Neither can one simply take a set of Bravais lattice sites {R} inside the geometrical region V which the solid occupies and then fill undistorted charge distribution basis into every primitive cells of {R} . Indeed, the charge distribution in those cells near the surface will be distorted significantly from that in a cell of an ideal infinite solid, resulting in an effective surface dipole distribution, or, sometimes both a surface dipole distribution and a surface charge distribution. It can be proved that if we define work function as the minimum energy needed to remove an electron to a point immediately out of the solid, the effect of the surface charge distribution can be neglected, leaving only the surface dipole distribution. Let the potential energy difference across the surface due to effective surface dipole be WS . And let F be the →Fermi energy calculated for the finite solid without considering surface distortion effect, when taking the convention that the potential at r → ∞ is zero. Then, the correct formula for work function is: W = −F + WS

Where F is negative, which means that electrons are bound in the solid.

Example

For example, Caesium has ionization energy 3.9 eV and work function 1.9 eV.

Work function

662

Photoelectric work function

Photoelectric work function: φ=hf 0,

FT

The work function is the minimum energy that must be given to an electron to liberate it from the surface of a particular metal. In the photoelectric effect if a photon with an energy greater than the work function is incident on a metal photoelectric emission occurs. Any excess energy is given to the electron as kinetic energy.

where h is Planck’s constant and f 0 is the minimum (threshold) frequency of the photon required for photoelectric emission.

Thermionic work function

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The work function is also important in the theory of thermionic emission. Here the electron gains its energy from heat rather than photons. In this case, as for an electron escaping from the heated negatively-charged filament of a vacuum tube, the work function may be called the thermionic work function. Tungsten is a very common metal for vacuum tube elements, with a work function of approximately 4.5 eV.

The thermionic work function depends on the orientation of the crystal and will tend to be smaller for metals with an open lattice, larger for metals in which the atoms are closely packed. The range is about 1.5–6 V. It is somewhat higher on dense crystal faces than open ones.

Applications

In electronics the work function is important for design of the metalsemiconductor junction in Schottky diodes and for design of vacuum tubes. The work function is also important in the theory of thermionic emission, here the electron gains its energy from heat rather than photons. In this case, as for example that of an electron escaping from the heated negatively-charged filament of a vacuum tube, the work function may be called the thermionic work function. Tungsten is a very common metal for vacuum tube elements, with a work function of approximately 4.5 eV. It depends on the orientation of the crystal and will tend to be smaller for metals with an open lattice, larger for metals in which the atoms are closely packed. The range is about 1.5–6 eV. It is somewhat higher on dense crystal faces than open ones.

Work function

663

See also

•

free energy for the Helmholtz free energy equation, which is the thermodynamic work, note that this work is not related to electron emission and is thus not directly related to the work function. Electron affinity. See NEA cathode 395 for an application to condensed matter.

Reference

FT

•

Solid State Physics, by Ashcroft and Mermin. Thomson Learning, Inc, 1976

External links • • •

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•

Work functions of common metals 396 Work functions of various metals for the photoelectric effect 397 Contains some work functions for metals 398 and the corresponding heat required to eject them by thermionic emission Some work functions 399

Source: http://en.wikipedia.org/wiki/Work_function

Principal Authors: Omegatron, Jonathunder, Arnero, RTC, Pacaro

Zeeman effect

The Zeeman effect (IPA [ze m n]) is the splitting of a spectral line into several components in the presence of a magnetic field. It is analogous to the →Stark effect, the splitting of a spectral line into several components in the presence of an electric field.

395 http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-pub-8355.pdf 396 http://www.pulsedpower.net/Info/WorkFunctions.htm

397 http://hyperphysics.phy-astr.gsu.edu/hbase/tables/photoelec.html 398 http://www.fnrf.science.cmu.ac.th/theory/linac/Linac%20Basic%20Concepts.html 399 http://www.physchem.co.za/Light/Particles.htm#Work%20function

Zeeman effect

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Introduction

FT

In most atoms, there exist several electronic configurations that have the same energy, so that transitions between different pairs of configurations correspond to a single line.

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The presence of a magnetic field breaks the degeneracy, since it interacts in a different way with electrons with different quantum numbers, slightly modifying their energies. The result is that, where there were several configurations with the same energy, now there are different energies, which give rise to several very close spectral lines.

Without a magnetic field, configurations a, b and c have the same energy, as do d, e and f. The presence of a magnetic field splits the energy levels. A line produced by a transition from a, b or c to d, e or f now will be several lines between different combinations of a, b, c and d, e, f. Not all transitions will be possible – see transition rules. Since the distance between the Zeeman sub-levels is proportional with the magnetic field, this effect was used by astronomers to measure the magnetic field of the Sun and other stars. There is also an anomalous Zeeman effect that appears on transitions where the net spin of the electrons is not 0, the number of Zeeman sub-levels being even instead of odd if there’s an uneven number of electrons involved. It was called "anomalous" because the electron spin had not yet been discovered, and so there was no good explanation for it, at the time that Zeeman observed the effect.

Zeeman effect

665 If the magnetic field strength is too high, the effect is no longer linear; at even higher field strength, electron coupling is disturbed and the spectral lines rearrange. This is called Paschen-Back effect.

FT

The Zeeman effect is named after the Dutch physicist Pieter Zeeman.

Theoretical presentation

The total Hamiltonian of an atom in a magnetic field is: P ~ ·S ~ − P µ~α · B ~ H = H0 + H1 = H0 + ξ(r~α )L α

α

where H0 is the unperturbed Hamiltonian of the atom, and the sums over α are sums over the electrons in the atom. The term ~ ·S ~ ξ(r~α )L

is the LS-coupling for each electron (indexed by α) in the atom. The sum vanishes if there is only one electron. The magnetic coupling ~ = µB (gL L ~ + gS S) ~ ·B ~ µ~α · B

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~

is the energy due to the magnetic moment µ of the α-th electron. It can be written as sum of contributions of the orbital angular momentum and of spin angular momentum, with each multiplied by the gyroscopic or Landé g-factor. By projecting the vector quantities onto the z-axis, the Hamiltonian may be written as ~ ·S ~ + µB (gL Lz + gs Sz )Bz ≈ Hat + µB (Jz + Sz )Bz H = H0 + ξ(r)L ~

where the approximation results from taking the g-factors are gL = 1 and gS ≈ 2. The summation over the electrons was omitted for readability. Here, Jz = Lz + Sz is the total angular momentum, and the LS-coupling term has been folded into H0 . The size of the interaction term H ’ is not always small, and can induce large effects on the system. In the Paschen-Back effect, described below, H ’ cannot be treated as a perturbation, as its magnitude is comparable to or larger than the unperturbed system Hat . The H ’ term does not commute with Hat . In particular, Sz doesn’t commute with the spin-orbit interaction in Hat .

Zeeman effect

666

Strong Field (Paschen-Back effect)

FT

To simplify the solution, it is useful to assume that [Hat , Sz ] = 0, so that Lz and Sz have a set of common eigenfunctions with respect to Hat . This allows the expectation values of Lz and Sz to be easily evaluated on a general state |Ai:   Hat + Bz~µB (Lz + 2Sz ) |Ai = (Eat + Bz µB (ml + 2ms )|Ai The above may be read as implying that the LS-coupling is completely broken by the external field. The system re-arranges substantially according to the Bz field. The ml and ms are still "good" quantum numbers. This implies that the selection rules obtained from ∆S = 0, ∆L = ±1 are still very likely for the system. In particular, apart from the line splittings one might normally expect, only three spectral lines will be visible, corresponding to the ∆m = ±1 transition rule. The splitting depends upon the l level being considered. The spectral lines depend on the transition frequencies, that is, on the difference of energy.

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See also •

→Stark effect

References Historical •

Condon, E. U.; G. H. Shortley (1935). The Theory of Atomic Spectra. Cambridge University Press. ISBN 521-09209-4. (Chapter 16 provides a comprehensive treatment, as of 1935.) • Zeeman, P. (1897). "(Title unknown)". Phil.Mag. 43: 226. • Zeeman, P. (11 February 1897). "The Effect of Magnetisation on the Nature of Light Emitted by a Substance" 400. Nature 55: 347.

Modern •

Forman, Paul (1970). "Alfred Landé and the anomalous Zeeman Effect, 1919-1921". Historical Studies in the Physical Sciences 2: 153—261. • Griffiths, David J. (2004). Introduction to Quantum Mechanics (2nd ed.). Prentice Hall. ISBN 013805326X. • Liboff, Richard L. (2002). Introductory Quantum Mechanics. AddisonWesley. ISBN 0805387145.

400 http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Zeeman-effect.html

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667

Source: http://en.wikipedia.org/wiki/Zeeman_effect

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Principal Authors: Linas, AstroNomer, Bogdangiusca, Voyajer, Laurascudder, Nsh

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GNU Free Documentation License Version 1.2, November 2002

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Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

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Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License "or any later version" applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

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List of Figures Please note that images have various different licenses and usage constraints.

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License details for a specific image can be found at:

http://en.wikipedia.org/wiki/Image:IMAGENAME.JPG

For each image we list: description, file name, licence, user who added the image (if available), url for further information (if available). Note that license names are abbreviated like GFDL, PD, fairuse, etc. Please have a look at the corresponding page at Wikipedia for further information about the images and details of the associated licences.

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aharonov-bohm.png Bohratommodel.png, Bose_Einstein_condensate.png, PD-USGov-NIST BoseEinsteinGas1.png, PD-self Coherent_noise_compare3.jpg, GFDL coherent_state_wavepacket.jpg wigner_function_coherent_state.jpg photon_numbers_coherent_state.jpg Coherent_state2.png, GFDL Compton scattering diagram.png, PD-self Renormalized-vertex.png, GFDL Fringespos.png, PD FD_e_mu.jpg, PD-ineligible FD_kT_e.jpg, PD-ineligible FD_e_kT.jpg, PD-ineligible Atom.png, GFDL Vortex.jpg, fairusereview Holography-reconstruct.png, GFDL Human brain NIH.jpg, PD-USGov-HHS-NIH Epithelial-cells.jpg, GFDL Common clownfish.jpg, FormerFeaturedPicture Constructive_and_Destructive_Interference.gif, self2|GFDL|cc-by-sa-2.5,2.0,1.0 NASA_Hydrogen_spectrum.jpg, PD-USGov-NASA Gallery_SineWave_Generation.jpg, NoRightsReserved Bohr-planetary-atom-model.jpg, GFDL-self Bohr_atomic_wave.jpg, GFDL-self Josephson_junction_IV.png, PD-USGov Bohratommodel.png, 1D normal modes.gif, PD-user|Régis Lachaume|Lachaume parity_1drep.png 1-D Box.png Three_paths_from_A_to_B.png, GFDL Penrose_Interpretation.jpg, PD Photoelectric_effect.png, GFDL bbs.jpg Plum_pudding_atom.svg, cc-sa-1.0 state_discrimination_proj.png state_discrimination_POVM.png

5 14 23 37 55 56 57 58 59 68 76 106 131 132 134 197 198 201 202 204 205 227 228 231 233 237 255 270 307 325 333 349 360 372 381 388 395 397

674 431 443 444 460 533 537 566 567 568 569 570 571 572 573 574 575 575 576 576 577 577 578 579 579 580 582 583 584 586 588 616 653 658 659

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Witten.jpg, Promotional Schr-harmonic.png, Cc-by-sa-2.0 QHarmonicOscillator.png, GFDL HAtomOrbitals.png, GFDL-en Schroedingerscat2.jpg, GFDL-self Schrödinger’s cat - BLINK tag.gif, NowCommonsThis noise_squeezed_states.jpg wave_packet_squeezed_states.jpg wigner_function_squeezed_states.jpg noise_squeezed_states.jpg wave_packet_squeezed_states.jpg wigner_function_squeezed_states.jpg noise_squeezed_states.jpg wave_packet_squeezed_states.jpg wigner_function_squeezed_states.jpg photon_numbers_squeezed_coherent_states_subpoisson.jpg phase_distribution_squeezed_coherent_states_subpoisson.jpg photon_numbers_squeezed_coherent_states_subpoisson.jpg phase_distribution_squeezed_coherent_states_subpoisson.jpg photon_numbers_squeezed_coherent_states_subpoisson.jpg phase_distribution_squeezed_coherent_states_subpoisson.jpg photon_numbers_squeezed_vacuum.jpg Squid prototype2.jpg, NoRightsReserved Squid prototype.jpg, NoRightsReserved Stark splitting in hydrogen.png, PD-self Spin_up.gif, GFDL Stern-Gerlach experiment.PNG, GFDL Quantum_projection_of_S_onto_z_for_spin_half_particles.PNG, GFDL SternGerlach2.jpg, GFDL Atom.png, GFDL Interfering Electron Wave Packets animated.gif, GFDL bbs.jpg Wigner_functions.jpg, PD Chirpedpulse.jpg, PD

675

Index 645, 647 alfred clebsch 52 alfred lande 481, 626 alfred landé 666 algebra of physical space 98, 102 algebra representation of a hopf algebra 595 algebra representation of a lie superalgebra 595 alice and bob 455 alkali metal 187 alloy 572 alpha decay 498 alpha particle 10, 163, 199, 312, 517, 522 aluminium 163 aluminium arsenide 504 american gods 540 american physical society 552, 554 ammonia 370 amplitude 412, 641, 647 analog model of gravity 85 analysis 196 analytic 510 ancient india 9 ancilla 398 angle 174 angular frequency 87, 156 angular momentum 11, 16, 52, 54, 67, 170, 190, 241, 259, 260, 260, 262, 262, 263, 327, 344, 344, 345, 460, 463, 466, 558, 582, 597 angular momentum coupling 54, 546 angular momentum operator 182, 183 animal 571 anime 543 annalen der physik 373, 387 annihilate 438 annihilation operator 56, 259 anode 154 anomaly (physics) 356 ansatz 632 antibaryon 120 anticommutator 82, 597, 599 anticommute 239 antiderivation 356 antilinear 81 antimatter 119 antiparticle 118, 301, 353, 438, 589, 590 antisymmetric 640 antisymmetric tensor 262 antiunitary 315, 613

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1/f noise 409, 409 11 february 666 14 october 651 19th century 8, 15, 645 20th century 15, 104, 430, 458, 460, 641, 645 21 cm line 192 25 april 319 3-body problem 270 3-sphere 607 4-vector 264 5th century bc 9 6-j symbol 54 6th century bc 9 ˇcech cohomology 17 647

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a a. elgart 3 a. s. wightman 280 abelian 300, 613 abelian group 323, 324 abraham-lorentz force 434 a brief history of time 194 absolute value 177, 334, 403, 468 absolute zero 22, 138, 486, 581 absorption spectrum 459 abstract 197 abuse of notation 44 accuracy 620 accurate 619 acoustic metric 85 acoustic resonance 509 acoustics 412, 522 action 160 action-angle coordinates 16 action principle 468 activity 33, 137, 288 adhesion 23 adjoint 79, 81, 445 adjoint representation 78 aether 438 affine structure 354 afshar experiment 66, 222, 298, 651 ah! my goddess 543 aharonov-bohm effect 50, 595 albert abraham michelson 192 albert einstein 10, 11, 23, 34, 62, 213, 226, 230, 247, 372, 375, 382, 386, 451, 460, 462, 471, 477, 534, 591, 625, 643, 645,

676 b b-bbar oscillation 303 b. d. josephson 571 baidyanaith misra 505 balmer’s formula 235 balmer series 128 banach space 41, 124, 175, 180 bandgap 504 baryon 117, 120, 129, 312, 327, 591 baryon number 607 baryons 584 basic quantum mechanics 284, 530 basics of quantum mechanics 628 basilar membrane 509 basis function 533 bayesian network 561 bay of fundy 509 bcs theory 421 becquerel 312 behaviour 196 bell’s inequalities 273, 454 bell’s inequality 414 bell’s theorem 218, 222, 484, 651 bell inequality 627 bell labs 86, 552, 571, 648 bell state 593 bell test experiments 207, 415, 454 benzene 514 berlin 405 berry phase 6 bertrand’s theorem 262 bessel function 345 beta decay 93, 311, 591 bethe 434 big bang 71, 194, 225, 590 big crunch 195 billiard balls 522 binomial coefficient 32, 137 binomial theorem 321 biochemistry 8 biological 410 biological molecule 555 biology 656 biopotential 552 black-body 229 black-body radiation 375 black body 380, 380, 472, 617, 652 black body radiation 473, 591, 618 blackbody spectrum 269 black hole 21, 84, 92, 93, 379, 548 bloch’s theorem 339 bloch sphere 404, 453 bloch wave 139

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antonino lo surdo 577 applet 379 approximation 144, 533 arago spot 651 arnold sommerfeld 11, 131 aromatic 514 arthur compton 86, 87, 645 arthur holly compton 67 arxiv 507 arxiv:hep-th/9912205 435 asher peres 215, 363 as of 2005 591, 648 as of 2006 213 associated legendre function 189 associated legendre polynomials 54 astrodynamics 264 astronomy 256 astrophysics 509 asymptote 521 asymptotic freedom 78 atom 8, 12, 14, 15, 67, 104, 117, 127, 153, 163, 182, 187, 191, 248, 262, 311, 382, 388, 405, 421, 458, 459, 459, 460, 461, 464, 467, 474, 517, 519, 522, 568, 586, 588, 591, 645, 645, 664 atomic clock 193 atomic coherence 27 atomic nucleus 312, 464, 516, 520, 532, 588, 591 atomic number 188, 520, 589 atomic orbital 187, 377, 586 atomic physics 14, 191, 312, 516, 518, 576 atomic theory 645 atomic units 299 atomism 8, 9 atoms 124, 197, 225, 407 attachment 495 audio system measurements 409 auger 12 auger electron 211 auger electron spectroscopy 13 augustin fresnel 226 autocatalytic set 205 automorphism 315 automorphism group 180 avlis 193 avogadro’s number 424 avoided crossing 98 axiom 411, 430 axion 590 azimuth 188, 344 azimuthal quantum number 241, 248, 530

677 brillouin zone 139 brownian motion 10, 158, 162, 348 brst 356 bruno ganz 543 bryce dewitt 474 buckyball 648 buckyballs 514 buddhism 203, 206 buddhist 9 bijective 178, 315

FT

bob the angry flower 544, 545 bogolibov transformation 84 bohm interpretation 199, 219, 222, 295, 650 bohr 191, 472 bohr-einstein debates 222, 465, 471, 481, 625 bohr-sommerfeld quantization 413 bohr atom 516 bohr model 11, 88, 163, 164, 187, 190, 389, 586 bohr model of the atom 153, 578 bohr radius 16, 73, 183, 189 boltzmann’s constant 31, 32, 34, 90, 132, 137, 166, 285, 288, 407, 601, 654 boltzmann constant 139, 140, 419 boltzmann distribution 165, 287 borel functional calculus 169, 419, 491 borel set 390 boris podolsky 455 born-oppenheimer approximation 98, 115, 116, 299, 391 bose-einstein 290 bose-einstein condensate 30, 34, 35, 38, 61, 535 bose-einstein statistics 24, 34, 138, 291, 383, 601 bose gas 24, 27, 141, 601 bosenova 26 boson 30, 55, 62, 73, 80, 81, 118, 118, 120, 122, 424, 433, 589, 592, 640 bosons 22, 24, 27, 30, 34, 133, 286 bottom quark 301 boundary condition 148, 343, 513, 522 boundary conditions 607 boundary value problem 311 bounded operator 125 bounded set 179 bound particle 365 bound state 13, 169, 175, 345, 364, 488, 510, 523, 523 box 532 bra-ket 168, 584 bra-ket notation 52, 81, 113, 144, 174, 179, 211, 271, 281, 284, 298, 353, 415, 473, 488, 490, 524 brackett series 128 bragg 86, 87 brahman 206 brain 202, 495, 573 brane 122 bravais lattice 661 brian david josephson 253 brian greene 123

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c c*-algebra 363 c* algebra 272 c-parity 589 c-symmetry 301 c. k. ogden 203 caesium 193, 661 calculus 174 caltech 318 canonical angular momentum 49 canonical commutation relation 445 canonical commutation relations 277, 447 canonical coordinates 48, 74 canonical ensemble 289 canonical one-form 126 canonical partition function 493 canonical quantization 50, 77, 79, 269, 271, 326, 421, 425 canonical transformation technique 267 capacitor 145 car algebra 82 carbon 514 cargo cult science 318 carl d. anderson 591 carl sagan 192 carl wieman 22, 24 carol hill 541 cartesian coordinates 345 cartesian coordinate system 196, 282, 640 carver mead 650 casimir effect 438, 438 casimir invariant 263 cat 297, 532 categorical product 551 cathode 154 cathode ray 10, 645 cat state 537 cauchy sequence 175 ccr algebra 50, 81 cecil adams 538 center frequency 510, 511 central extension 323

678 classical physics 47, 173, 269, 283, 323, 557, 617 claude cohen-tannoudji 25 claude e. shannon 610 claus jönsson 104, 474 clebsch-gordan coefficients 656 clifford algebra 98, 103, 103 clinton davisson 86, 87, 474 clinton joseph davisson 648 closed graph theorem 180 closed system 267 closing 607 clyde l. cowan 474 coefficients 50 coherent 56 coherent state 450, 564, 565, 565, 657, 659 colatitude 188 cold cathode 145 cold emission 498 collapse of the wavefunction 272 collision 267, 523 color 653 color charge 119 colors of noise 409 colour confinement 120 commutation relation 259, 263 commutative 433 commutator 47, 47, 79, 80, 112, 173, 174, 283, 284, 422, 445, 597, 599, 620 commute 239 compact 390 compact lie group 52 complementarity 96 complementarity (physics) 456 complementary 565 completely positive 279 complete set of commuting operators 545 complex conjugate 42, 171, 176, 649 complex conjugation 613 complexity 488 complex number 42, 43, 147, 175, 176, 280, 332, 350, 403, 452, 461, 631 complex numbers 215 complex plane 194 complex pole 510 complex projective line 404 composite field 40 composite particle 119 compton effect 645 compton suppression 71 compton wavelength 69, 379 computational chemistry 150, 299, 459, 470 computer 365, 471

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central force 260, 260, 261, 262, 263 central force field 345 central potential 523 centripetal force 18 cern 122, 592, 616 cgs 5 chain rule 370 chair 297 chandrasekhar limit 92, 92, 93 chaos 409 chaos theory 412 characteristic equation 89 characteristic polynomial 308 characteristic x-ray 211 charge conjugation 301, 615 charge decay 376 charge density 271 chargino 122 charles galton darwin 483 checkerboard nightmare 544 chemical bond 117, 503 chemical bonding 187 chemical compound 10 chemical element 519, 520 chemical energy 411 chemical potential 25, 31, 32, 34, 131, 133, 137, 138, 140, 285, 288, 383 chemical reaction 391 chemical vapor deposition 504 chemistry 8, 8, 79, 312, 459, 459, 470, 485, 588, 589 chien-shiung wu 328 chiral spin state 604 chladni’s figures 460 chris fuchs 215 christiaan huygens 644, 646 christian huygens 226 christopher fuchs 223 christopher stasheff 542 church bells 51 circle 343, 512 ckm matrix 615 clairvoyance 496, 540 classical electrodynamics 430 classical electromagnetism 11, 459 classical electron radius 73 classical field theory 425 classical information channel 415 classical limit of quantum mechanics 272 classical mechanics 15, 16, 51, 75, 129, 174, 213, 260, 266, 267, 314, 330, 331, 348, 348, 410, 411, 412, 425, 459, 469, 497, 524, 640

679 coulomb potential 111, 364 countable 178 countably many 309 counterfactual definiteness 456 coupling constant 364 courant-hilbert 279 covalent bond 473 covariant derivative 76 cpt invariance 592, 614, 615 cpt symmetry 302, 303, 329, 589, 615 cpt violation 615 cp violation 303, 615, 615 creation and annihilation operators 80, 152, 259, 426, 450 creation operator 259 critical exponent 633 critical phenomena 633 cross product 260 cross section 320, 322, 630 crystal 117, 581, 603 crystal lattice 568 crystalline 87 crystallography 555 crystals 311 crystal structure 342 cubical atom 11 curie 312 curl 4 curvature form 17 curvilinear 196 cymbals 51

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computer code 129 condensed matter physics 138, 226, 421, 422, 424, 428, 434, 459, 603, 605 conduction electron 365, 428 configuration interaction 557 configuration space 343 confinement 40 conical intersection 98 conjugate quantities 619 conjugate transpose 41 connected correlation function 511 connection 17 consciousness 201, 495 consciousness causes collapse 219 conservation law 400, 529 conservation laws 438 conservation of energy 77, 320, 322, 436 conservation of probability 320 consistent histories 213, 214, 218, 222, 272, 536 conspiracy theory 542 constants of motion 261 constraint 74 constructive quantum field theory 272, 430 context 199 contextualism 204 continuity equation 399, 529 continuous spectrum 292, 390, 523 continuous symmetry 263 contradiction 198 convergent series 633 convex hull 125, 274 convex set 550 cooper pair 27, 93, 255, 365 coordinate 322, 343, 512 copenhagen interpretation 106, 107, 126, 213, 218, 222, 248, 282, 294, 296, 297, 298, 399, 471, 534, 626, 649 core hamiltonian 150 corpuscular theory 226, 646 correlation 414 correspondence limit 469 correspondence principle 51, 54, 113, 248, 248, 411, 444, 469, 628 cosmic radiation 509 cosmic ray 588, 589, 591 cosmological constant 84 cotangent bundle 126 coulomb 315, 517, 596 coulomb’s law 188, 311 coulomb collision 518 coulomb force 182 coulomb operator 151

d d’alembert operator 257 d. hestenes 98 daniel kleppner 587 dark energy 85 dark matter 85, 439 darmstadt 265 david bohm 195, 206, 219 david griffiths 474 david gross 77, 434 david hilbert 175, 245, 272, 279 david hume 478 david politzer 77, 434 davisson-germer experiment 87 deborah s. jin 27 de broglie 481 de broglie hypothesis 17, 647 de broglie wavelength 73, 504, 601 debye model 38 decay rate 364 december 12 652

680 diode 470 diode laser 504 dipole 546, 582 dirac 660 dirac’s constant 75, 87 dirac delta function 41, 428 dirac equation 11, 45, 45, 98, 157, 185, 246, 258, 299, 327, 422, 424, 469, 473, 528, 530, 649 dirac notation 636 dirac picture 276, 369 dirac sea 439 dirac string 5 direct product 433 direct sum 52, 151, 397 dirichlet problem 175 dirk gently’s holistic detective agency 538, 540 discrete 519 discrete delta-potential method 530 discrete spectrum 274, 293, 523, 631 discworld 542 dislocation 608 dispersion 617 dispersion relation 139 distance 174 distant anticipation 496 divergence theorem 401 doctoral thesis 348 doctor who 541 dolbeault operator 103 domain 157, 196 don misener 24 doppler 74 dot product 175 double-slit experiment 96, 214, 293, 473, 474, 489, 494, 539, 617, 645, 646, 649 double slit 643 double slit experiment 95 douglas adams 538, 540 driven harmonic motion 511 drop 522 dual space 41, 179, 354 duncan macinnes 552 duration 624 duru-kleinert transformation 351 dye laser 193 dynamics 273, 612 dyson series 371

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december 27 476 decoherence 114, 218, 272, 294, 358, 595, 605 deductive reasoning 478 deep inelastic scattering 120, 517 deflection 582 deformation quantization 47 degeneracy 31, 285, 604 degeneracy (disambiguation) 89 degeneracy pressure 141 degenerate 183, 579 degenerate dwarf 94 degenerate energy level 124 degenerate gases 246 degenerate star 94 degree of coherence 60, 61 delta function 13, 292 democritus 9, 10, 407 denmark 553 denominator 321 density 22, 309 density functional theory 114, 365, 530 density matrix 22, 266, 362, 393, 398, 417, 497, 550, 560, 656, 657 density of states 89, 141, 383, 504 density operator 274, 487, 489, 490 density state 594 de rham cohomology 17 derivation 356 derivative 48, 112, 307, 432 desy 121 detailed balance 169 detector 255 determinant 308, 322, 515, 556, 557 determinism 216, 293, 477, 478, 626 deuterium 129, 327 dewitt notation 354 dielectric constant 509 differential equation 147, 306, 311, 332, 523, 607, 645, 649 differential equations 280 differential geometry 6, 269 differential operator 102 diffraction 87, 322, 645, 646 diffraction pattern 86, 644 diffusion 200, 348 diffusion equation 352 digital devil saga 543 dimension 122, 303, 413, 499 dimensional analysis 73 dimensional regularization 78, 79 dimensionless 655 dinitrogen trioxide 10

e earthquake 573 eccentricity vector

264

681 electromagnetic potential 6 electromagnetic radiation 380, 405, 458 electromagnetic spectrum 128, 406 electromagnetic wave 11, 15, 55, 67, 461, 466, 646 electromagnetic waves 229 electromagnetism 213, 300, 328, 421, 461, 641 electron 4, 10, 12, 14, 15, 19, 30, 40, 45, 51, 67, 67, 78, 86, 89, 90, 104, 104, 115, 117, 118, 120, 128, 133, 139, 141, 142, 153, 182, 187, 211, 213, 234, 240, 266, 270, 271, 287, 297, 315, 327, 331, 344, 353, 360, 364, 371, 373, 375, 388, 405, 417, 458, 459, 460, 464, 467, 470, 473, 473, 474, 494, 498, 519, 557, 559, 580, 588, 588, 590, 591, 616, 645, 661, 664 electron-degenerate matter 94 electron affinity 663 electron charge 73 electron configuration 124, 187 electron correlation 150 electron cryomicroscope 555 electron diffraction 474, 647 electron hole 13 electronic configuration 89, 664 electronic molecular hamiltonian 115, 299 electronics 8, 30, 133, 140, 287, 470, 628, 662 electronic structure 591 electron mass 73 electron microscope 247, 470 electron microscopy 145, 651 electron neutrino 118 electrons 79, 87, 498, 582, 586, 605, 647 electron spin 185 electron volt 661 electroscope 376 electrostatic force 14 electrostatic levitation 377 electrostatics 260 electroweak force 121, 473 electroweak interaction 121 electroweak theory 329 elementary charge 116, 188, 440 elementary particle 10, 408, 619 elementary particles 88, 605 elitzur-vaidman bomb-testing problem 109, 298 ellipse 261 emanuel swedenborg 206 emission line 15, 128 emission spectrum 405, 459

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echolocation 522 edward condon 498 edward witten 431 effective action 548 effective field theory 432 effective theory 604 ehrenfest’s theorem 248 ehrenfest theorem 266 eigenequation 650 eigenfunction 280, 344, 650 eigenspace 631 eigenstate 2, 89, 116, 144, 187, 245, 293, 296, 310, 345, 423, 436, 442, 463, 494, 505, 565, 581, 596 eigenstates 182, 294 eigenvalue 2, 67, 89, 116, 168, 183, 259, 274, 280, 281, 292, 300, 306, 326, 343, 467, 513, 558, 565, 631, 650 eigenvalues 308, 362, 390, 424 eigenvector 280, 282, 292, 452, 467, 558, 624 eigenvectors 168, 169, 306, 308, 310, 579 eighteenth century 451 einstein 472, 480 einstein’s summation convention 103 einstein (disambiguation) 30 einstein-podolsky-rosen paradox 358 elastic 320 electric 315 electrical charge 407 electrical conductivity 140 electrical engineer 507 electrical engineering 140, 656 electrical potential 4, 429 electrical resistance 254, 440 electric charge 11, 15, 49, 75, 120, 300, 315, 327, 474, 527 electric current 376, 647 electric dipole moment 578, 613, 613 electric discharge 15 electric field 3, 120, 124, 145, 154, 316, 364, 576, 578 electricity 459 electric potential 115, 364 electrochemistry 552, 588 electrode 315, 572 electrodynamics 324, 522 electromagnetically induced transparency 26, 27 electromagnetic field 48, 61, 186, 314, 338, 421, 429, 432, 434, 473, 611 electromagnetic force 75 electromagnetic interaction 328

682 everett interpretation 295 everett many-worlds interpretation 97, 471 evolution operator 302, 630, 630 exactly solvable model 524, 608 exchange operator 151 excited state 40, 117, 581 exciton 392 excitons 580 exemplar 461 existence of god 480 exotic atom 190 exotic baryon 129 exotic meson 129 expectation value 112, 173, 283, 594, 631 expected value 170, 274, 366, 368 experiment 104, 196, 314, 315, 361, 619 exponential function 31, 286 extension 595 extensive variable 165 extra dimensions 122 extreme point 274

FT

empiricism 478 energy 12, 15, 23, 31, 39, 42, 67, 127, 138, 139, 142, 146, 191, 268, 275, 285, 292, 293, 295, 310, 332, 343, 344, 360, 363, 381, 436, 443, 460, 463, 466, 486, 494, 513, 519, 524, 581, 621, 624, 647, 661, 664 energy density 381 energy eigenfunctions 182 energy level 11, 19, 89, 124, 193, 300 energy levels 182, 312, 586 energy level splitting 89 energy operator 150 energy spectrum 39 englert 62 englert-greenberger duality 109 enrico fermi 130, 131, 591 ensemble 656 entangled 60, 465 entangled state 471 entanglement 114, 214, 293, 534, 537, 650 entanglement of formation 563 entanglement witness 362, 420, 551 entities 196, 197 entropy 165, 291, 391, 486, 610, 611 epicurean 407 epistemology 479 eponym 253 epr paradox 214, 217, 247, 293, 298, 298, 414, 416, 456, 471, 621 equation 129, 647 equations of state 92, 93 equilibrium 254, 309 equipartition theorem 386, 617 equivalence class 403 equivalence principle 126 equivalent 594 eric cornell 24 ernest marsden 163, 517 ernest rutherford 10, 15, 163, 312, 388, 474, 516, 517, 588, 591 ernst chladni 50 ernst pringsheim 405 erwin schroedinger 477 erwin schrödinger 11, 54, 57, 226, 240, 271, 271, 365, 405, 460, 462, 494, 524, 532, 540, 596, 624 estermann 648 euclidean quantum gravity 195 euclidean space 174, 175 eugene wigner 175, 422, 613, 656 euler-lagrange equation 48, 350, 354 ev 139 event horizon 84

DR A

f f. david peat 206 f. kurlbaum 405 f. paschen 405 factorial 424 fano resonance 510 faraday isolator 612 faster-than-light 644 fate 626 fault tolerant 605 feminist science fiction 541 fermi’s golden rule 371 fermi-dirac 290 fermi-dirac statistics 33, 90, 138, 139, 140, 141, 291, 601 fermi energy 92, 131, 132, 135, 141, 661 fermi gas 130, 140, 142, 601 fermilab 616 fermi liquid theory 141 fermion 62, 73, 80, 90, 118, 118, 118, 122, 130, 138, 138, 142, 246, 327, 327, 328, 424, 433, 556, 589, 592, 640, 648 fermionic condensate 27, 27 fermions 27, 30, 133, 141, 286, 312, 556, 557, 557, 559 fermi surface 138, 139 fermi temperature 135, 141 feshbach-fano partioning 130 feshbach resonance 25, 510 feynman 468 feynman diagram 186, 353, 364, 371

683 freezing 486 frequencies 407 frequency 50, 86, 87, 309, 373, 381, 509, 621, 647, 654 frequency spectrum 310 friction 8, 609 fritz london 266, 473 fubini-study metric 404, 417, 420 fullerenes 88, 648 functional 560 functional analysis 174, 181, 268, 271, 280, 393, 430, 619 functional calculus 169 functional derivative 354 functional derivative operator 356 functional integral 348, 351, 353, 355 functional integration (neurobiology) 157 functional integration (sociology) 157 functional measure 354, 355 functional space 174 functions 282 function space 177 fundamental field 511 fundamental force 118, 470 fundamental forces 615 fundamental group 607 fundamental interaction 311, 433 fundamental particle 121, 511, 522 fundamental particles 196 fundamental representation 78 futurama 541

DR A

FT

feynman path integral 162, 272, 425, 434 feynman slash notation 103 fiction 644 field emission 498 field emission display 145 field equation 257 field strength 6 fine-structure constant 73, 75 fine structure 20, 192 fine structure constant 19, 79, 184, 191, 440 finite potential well 338, 529 first-order phase transition 486 flash memory 145, 498 flatland 541 flatterland 541, 542 fluorescence 458 flutter 511 flux 197 fluxon 149 flux pinning 150 focal length 65 focal plane 64 fock space 153, 426, 521 fock state 56, 57, 60, 151, 152, 658 fokker-planck equation 597 folklore 279 force 155, 257, 517, 524 force constant 442 formalism 272 formant 511 fortune teller 496 fourier analysis 176, 311 fourier series 174, 232, 240, 344, 622 fourier transform 74, 174, 280, 355, 428, 642, 657 fowler-nordheim equation 145 fractional statistics 603 frames of reference 315 franck-hertz experiment 20 frank drake 192 frankfurt 585 frank jewett 552 frank wilczek 77, 434 fraunhofer diffraction 64, 645 frederick reines 474 frederik pohl 542 free electron gas 139 free electron model 338, 342 free energy 663 free field 80, 82 free field theory 421 free particle 257, 337, 463, 529 free states 169

g g-parity 301 g. johnstone stoney 588 galilean relativity 315 gallium arsenide 441, 504 gamma function 35, 166, 602 gamma ray 67, 211, 312 gamma spectroscopy 71 garrett birkhoff 219 gas 8, 15 gas giants 509 gas in a box 27, 34, 35, 38, 141, 331, 382 gas in a harmonic trap 35, 450 gauge anomaly 432 gauge boson 117, 118, 432, 433 gauge field 432 gauge group 433 gauge invariance 440 gauge invariant 49 gauge symmetry 356, 432 gauge theory 257, 432, 434

684 gravitation 63 gravitational coupling constant 79 gravitational field 63 gravitational singularities 194 gravitational wave 568 gravitino 590 graviton 118, 118, 122, 590 gravity 14, 23, 84, 118, 122, 141, 195, 260, 328, 392, 433, 439, 462, 470, 590, 648 gravity probe b 574 gravity wave 574 greatest lower bound 631 greebo 540 greek philosophers 9 greg egan 541 gregory breit 45 grid 646 ground state 40, 117, 127, 138, 185, 312, 423, 424, 486, 604, 631, 632 group 323 group representation 116 group velocity 87 gurps 544 gustav ludwig hertz 153 göttingen university 279

DR A

FT

gauge transform 404 gauge transformation 432, 433 gauss’ law 74 gaussian 56, 528 gaussian distribution 622 gaussian integral 160, 160 gautama buddha 9 geiger counter 540 generalized coordinate 16 generalized laguerre polynomials 183 generalized momentum 11, 16 general relativity 85, 118, 122, 126, 226, 360, 433, 459, 470, 574 general semantics 538 general theory of relativity 462 generating functional 355 geometric algebra 98, 102 geometrical optics 430 geometric quantization 17, 272 geophysical 522 george alec effinger 542 george chapline 84, 84 george gamow 231, 498 george mackey 475 george paget thomson 648 george sudarshan 505 geothermal energy 573 gerard ’t hooft 434 germanium whisker 608 german physical society 405 germany 265, 585 gev 77, 78 ghz 293 gibbs paradox 30, 133, 286 gigabyte 604 gilbert n. lewis 11 glauber p representation 660 global minimum 632 global symmetry 432 gluon 78, 118, 119, 120, 120, 364, 433, 470, 590 gluonic vacuum field 511 gluons 584 glycerine 200 gns construction 490 gold 10, 163, 517, 522, 572 google 203 gradient 528 graffiti 628 grand canonical ensemble 290 grand partition function 34 grand unification theory 121 gravastar 84, 85

h haag’s theorem 80, 211, 276 haag-kastler 430 hadron 76, 117, 119, 120, 129, 329, 591, 591 hadronic showers 265 hagen kleinert 111, 358, 475, 633 hahn-banach theorem 125, 362 half-integer 139, 557 half-iterate 102 halides 459 hall effect 440 hamel basis 178 hamilton-jacobi equation 169, 271 hamilton-jacobi equations 127 hamiltonian 49, 89, 266, 423, 578, 581, 665 hamiltonian mechanics 74, 112, 171, 172, 173, 283, 309 hanbury-brown and twiss effect 651 handbells 51 hanns johst 537 han purple 413 hans bethe 321, 476 hans geiger 163, 517 harmonic analysis 181, 280, 344 harmonic function 391 harmonic oscillator 153, 310, 511, 596

685 hilbert-schmidt 125 hilbert space 39, 41, 42, 48, 74, 81, 114, 151, 168, 213, 215, 268, 271, 273, 273, 280, 314, 323, 348, 393, 393, 394, 403, 415, 422, 426, 452, 467, 490, 497, 521, 523, 524, 549, 559, 564, 609, 630, 631, 638, 650 hill potential 499 hindu 9 history of quantum field theory 434 history of science 8 history of thermodynamics 12 hole 580 hole theory 422 holism 196 hologram 200 holographic principle 206 holomorphic functional calculus 169 holomovement 197 holonomy 6 homodyne detection 55, 659 homogeneous space 403 homotopy group 607 hopf algebra 595 howard percy robertson 624 html 537 hugh everett 474 husimi q representation 660 hydrocyanic acid 540 hydrogen 15, 182, 186, 262, 345, 518, 519, 578, 589, 596 hydrogen-like atom 183, 263, 529 hydrogen atom 16, 17, 40, 67, 128, 182, 184, 187, 260, 267, 270, 364, 460, 468, 469, 472, 488, 523, 529, 581 hydrogen molecule 473 hypercube 166 hypercylinder 166 hyperfine structure 11, 20, 586 hypernucleus 312 hyperon 312 hyperspace theory 439 hypersphere 166 hysteresis 255 hückel’s rule 514

DR A

FT

harmonics 310 harmonic series (music) 310 hartle-hawking state 194 hartree-fock 11, 151, 268, 530, 557 hartree-fock method 632 hartree product 556 hartree theory 556 hawking radiation 85, 439, 470, 568 healer 496 heat 405 heat death 611 heat kernel 103, 437 heinrich hertz 226, 373 heinrich rubens 405 heisenberg 47, 277, 348, 565, 640, 660 heisenberg’s microscope 621 heisenberg’s uncertainty principle 166 heisenberg compensator 628 heisenberg group 48, 660 heisenberg picture 112, 210, 275, 423, 532, 599 heisenberg uncertainty principle 96, 207 heisenbug 628 helium 17, 142, 376, 468, 517, 573 hellsing 538, 542 hemt 504 henri becquerel 473 henri poincaré 297, 472 hera collider 121 hermann weyl 47, 175, 271, 271, 656 hermitean 302 hermite polynomials 443 hermitian 17, 57, 113, 173, 281, 282, 283, 292, 393, 416, 467, 494, 558, 630 hermitian adjoint 259 hermitian conjugate 41 hermitian operator 168, 259, 365, 515, 631 hertz 654 heterostructure 331, 441 hidden-variable 273 hidden variable 199 hidden variables 207, 248, 650 hidden variable theory 414 hideki yukawa 591 hierarchy problem 121 higgs boson 118, 118, 120, 121, 121, 590, 591 higgs mechanism 121 high-energy physics 102, 596 high energy physics 589 high frequency approximation 548 high temperature superconductivity 604 hilary putnam 215

i i. a. richards 203 ian mcewan 542 ideal gas 34, 38, 139, 141, 165, 165, 291, 601 identical particles 24, 43, 424 image 197, 200

686 456, 471 interstellar medium 192 intertwiner 595 introduction to quantum mechanics 459 invariant 325, 595 invariant theory 52 inverse-square law 517 inversely proportional 86, 620 inverse scattering problem 522 inverse scattering transform 524, 608 inversion layer 440 inversion of series 37 ion 91, 128, 187, 193, 338, 568 ion-trap quantum computing 193 ionization energy 661 ionosphere 2 iron 163 irreducible representation 52, 325 irreducible representations 323 ir spectroscopy 503 isaac newton 641, 644, 646 isidor rabi 586 islam 9 islamic golden age 9 is logic empirical? 215 isomorphic 176 isospin 312, 607 isotope 25, 26, 142 isotropic 182

DR A

FT

imaginary number 189, 194 imaginary unit 47 immanuel kant 204, 206, 477, 478 impact ionization 13 impedance 511 index of refraction 321 indicator function 390 indium 572 inductive reasoning 478 industrial revolution 8 inequivalent 594 inertia 439 inertial frame 185 inert pair effect 20 infimum 560 infinite divisibility 10, 375 infinite potential well 146, 149 infinitesimal 433 influence 200 information 414, 467, 470 information retrieval 204 information theory 562 infrared 255, 405 infrared imaging 256 inhomogeneous 583 initial condition 308 inner product 40, 41, 42, 69, 468, 639 inner product space 174 inside joke 541 instanton 548 institute for advanced study 552 insulator 254, 486 integer 5, 24, 263, 440 integrable 490 integral 157, 399, 649 integrate 321 integration 158 intensity 321 intentionality 203 interacting boson model 313 interaction 75, 129, 521 interaction energy 39 interaction picture 77, 80, 174, 276, 284, 423, 431, 532 interference 25, 25, 26, 104, 226, 309, 350, 468, 489, 494, 506, 571, 616, 645, 646 interferometer 63 internal conversion (chemistry) 211 internal conversion coefficient 212, 212 international space station 319 interpretation of quantum mechanics 22, 249, 294, 357, 360, 625 interpretations of quantum mechanics 361,

j j. e. evron 3 j. j. thomson 388, 588, 591 j.j. thomson 10, 517, 645 j. robert oppenheimer 552 jack steinberger 327 jacobi identity 174 jahn-teller effect 124 jain 9 james chadwick 11, 589, 591 james clerk maxwell 226, 374, 375, 610 james edward zimmerman 571 james franck 153 james jeans 386, 618 january 17 554 jean perrin 10 johannes stark 577 john archibald wheeler 348, 553 john c. slater 556 john cramer 219, 650 john dalton 8, 9, 12 johnny mnemonic 574 john searle 203

687 landau pole 77 lande interval rule 192 landé g-factor 665 laplace 262 laplace operator 528 laplacian 102, 299 large hadron collider 122, 433, 590, 592 laser 56, 65, 193, 369, 470, 568 laser cooling 25 latin 404, 460 lattice 603 law of cosines 68 law of multiple proportions 10 laws 196 lax pair 285 lead 163, 572 lebesgue measure 168, 390 leiden university 23 lenz 262 leo kadanoff 434 leon lederman 328 leon rosenfeld 422 lepton 117, 118, 327, 590 leptons 119 leptoquark 591 lester germer 86, 87, 474 lester halbert germer 648 leucippus 9, 407 lev davidovich landau 142 levi-civita symbol 50, 262 lev landau 77 lhc 121 lie algebra 50, 264, 595 lie bracket 262, 264 lie derivative 50 lie superalgebra 595, 597 lifetime 510 light 104, 376, 459, 522, 617, 644, 646, 653 light bulb 55 lightbulb 654 light emission 518 light quantum 404 light wave 565 limits to computing 615 lindblad equation 112 linear 81, 527 linear algebra 43, 178, 259, 271, 280, 306, 310, 314 linear combination 187, 558 linear combination of atomic orbitals molecular orbital method 187 linear functional 40, 41 linear operator 42, 173, 272, 280, 283, 426,

FT

johnson-nyquist noise 409 john stewart bell 484 john von neumann 175, 219, 271, 271, 272, 284, 452, 467, 473 joseph john thomson 374, 473 josephson constant 253 josephson effect 502 josephson junction 571 joule 620 joule second 230 julian schwinger 226, 545, 552 june 15 413 june 5 24

DR A

k k. k. darrow 552 kanada 9, 407 kaon 301, 302, 303, 329, 591, 630 karl popper 203 karl pribram 201 keith mayes 110 kek 615 kekulé 514 kelvin 141, 255, 285, 413, 654 kenneth wilson 434 kinematics 612 kinetic energy 13, 51, 86, 115, 373, 374, 391, 469, 508, 517, 527 kinetic theory 8, 12 kingdom of loathing 544 klaus von klitzing 440 klein-gordon equation 157, 424, 469, 528, 649 klein-nishina formula 68 klystrode 145 klystron 145 knowledge 204 kochen-specker theorem 273 korteweg-de vries equation 608 kristofer straub 544 kronecker delta 48, 426 l ladder operators 79, 84 lagrange’s formula 260 lagrange multipliers 32, 137, 288 lagrangian 48, 75, 76, 122, 349, 356, 425 lagrangian mechanics 309 laguerre polynomials 183, 488, 596 lake 392 lamb-rutherford experiment 186 lamb shift 186 laminar flow 206

688 m macromolecules 88, 495 macroscopic 8, 297, 410, 461 magnet 603 magnetic dipole 191 magnetic evaporative cooling 25 magnetic field 3, 20, 50, 124, 149, 185, 257, 266, 440, 486, 504, 571, 576, 582, 663 magnetic flux 4, 254 magnetic flux quantum 150, 253, 408 magnetic moment 185, 241, 246, 266, 266, 364, 558 magnetic monopole 5, 546 magnetic quantum number 249, 530 magnetic resonance imaging 587 magneto-optic effect 612 magnetoencephalography 573 magnetometer 571 magnetoreception 411 magnon 25 majorana 327 manga 538, 542 manhattan project 552 manifold 523 many-body theory 79, 422 many-minds interpretation 222 many-worlds interpretation 218, 222, 279, 298, 538, 544, 644 many-worlds interpretation of quantum mechanics 536 many worlds 248 many worlds interpretation 107, 649 marshall stone 452 martin gutzwiller 412 martinus veltman 434 maser 587 mass 72, 120, 163, 306, 332, 424, 517, 528, 589, 601 mass gap 521 mass number 589 mass renormalization 302, 432 mass shell 77 master equation 112, 266 master of mosquiton 543 material science 8 mathematical 74, 196, 459 mathematical analysis 181 mathematical formulation of quantum mechanics 48, 168, 175, 298, 314, 398, 415, 442, 467, 524 mathematical physics 279, 522 mathematician 507 mathematics 40, 79, 102, 157, 174, 403,

DR A

FT

494 linear span 177 line broadening 369 line bundle 6, 17 line integral 348 linus pauling 473 liouville equation 266 liquid 8, 9, 142, 603 liquid drop model 313 lisa mason 542 lise meitner 13 list of astronomical topics 71 list of famous experiments 164 list of mathematical topics in quantum theory 280 list of noise topics 409 list of particles 123, 592 list of physics topics 71, 222 list of quantum field theories 434 lithium 20 local hidden variable theory 218, 471 locality 293 local maximum 392 local minimum 391 local realism 414, 627, 650 local symmetry 432 locc 563 logarithmic derivative 347 logic 478 london force 267 london moment 266 loop expansion 548 lord rayleigh 386, 618 lords and ladies 539 lorentz covariant 423 lorentz factor 509 lorentz force 611 lorentz force law 49 lorentz group 595 lorentzian function 510 lorentz invariance 173, 283 lorentz invariant 422, 657 lorentz transformation 45, 264 loschmidt’s paradox 611, 615 louis-victor de broglie 237, 647 louis de broglie 86, 219, 226, 645 lower bound 619 ls-coupling 665 ls coupling 547 lsz formalism 521 ludwig boltzmann 21, 385 lyman series 20, 128, 519 léon brillouin 552

689 middle ages 9 mind 495 mind’s eye 206 minimum 506 minkowski space 102 mit opencourseware 476 mixed state 112, 456, 594 mixing paradox 165 modes of vibration 50 molecular beam epitaxy 504 molecular geometry 98, 116, 299 molecular hamiltonian 98, 116 molecular physics 459 molecule 8, 117, 127, 182, 191, 267, 299, 522, 555 molecules 27, 124, 407, 604 moment 266 momentum 42, 44, 67, 72, 81, 86, 87, 138, 139, 139, 142, 163, 165, 180, 240, 242, 260, 268, 292, 442, 463, 528, 565, 619, 619, 640, 647 momentum operator 113 momentum space 139, 488, 636 monochromatic 376 moon 377 morse potential 391, 450 mosfet 440 mossbauer effect 313 mott insulator 26 multiplicative quantum number 614 muon 118, 119, 590, 591 muon neutrino 118 murray gell-mann 226 music 310, 407 musical instruments 509 mutual information 560

DR A

FT

522, 523, 619, 630, 633 mathworld 656 matrix mechanics 172, 174, 213, 240, 249, 271, 279, 462, 472, 622 matrix population models 285 matter 8, 10, 22, 86, 90, 118, 118, 201, 225, 269, 405, 470, 644 max born 2, 22, 213, 225, 226, 244, 267, 271, 281, 400, 421, 460, 471, 498, 585 max planck 11, 33, 226, 229, 372, 375, 380, 381, 405, 460, 472, 591, 618 max tegmark 537 maxwell 485 maxwell’s demon 542, 610, 615 maxwell’s equations 609 maxwell-boltzmann distribution 137 maxwell-boltzmann statistics 32, 33, 138, 601 maxwell equation 49 maxwell equations 528 may 31 413 may 6 652 mcscf 557 mean 282, 624 measurable space 394 measure 161 measurement 214, 277, 414, 619 measurement in quantum mechanics 310, 465, 582 measurement in quantum theory 437 measurement problem 222, 294, 298, 314 measurement theory 320, 548 measure space 177 medical imaging 313, 522 meissner effect 149 mellin transform 35 memories 202 mems 145 mendeleev 312 meson 117, 120, 129, 129, 311, 591 metal 139, 141, 142, 254, 266, 315, 662 metallic hydrogen 90 metaphysical 201, 297, 644 meter 652 michael berry 412 michael fisher 434 michael pepper 440 michel houellebecq 117 michio kaku 439 micro black hole 379 microscope 163, 316 microscopic 297 microscopy 555 microwave 193, 193

n n-body problem 267 nanometer 648, 653 nanotesla 571 nathan rosen 455 national academy of sciences 552 national bureau of standards 328 natural unit 597 nearly-free electron model 342 neil gaiman 540 nethack 539, 543 neutralino 122 neutrino 119, 301, 311, 327, 474, 557, 588, 590, 591 neutrino experiment 474 neutrino mass 615

690 normalization factor 556 normalized wavefunction 403 normal mode 425 norman f. ramsey 587 noumenon 206 november 5 374, 519 np-hard 417 nuclear compton scattering 67 nuclear engineering 313 nuclear fission 313 nuclear force 313 nuclear fusion 91, 313 nuclear magnetic dipole 191 nuclear magnetic resonance 46, 313 nuclear medicine 313 nuclear physics 129, 422, 459, 498, 516, 588, 589 nuclear power 313, 313 nuclear reaction 313 nuclear spin 192 nuclear structure 313 nuclear technology 313 nuclear units 312 nuclear weapon 313 nuclei 115 nucleon 40, 120, 589, 591, 614 nucleus 11, 211 nukees 544 numb3rs 543 number operator 152, 327 numerical 404

DR A

FT

neutrino oscillation 303 neutron 11, 40, 63, 90, 117, 141, 142, 301, 311, 312, 327, 327, 470, 557, 584, 588, 589, 591, 591, 648 neutron-degenerate matter 94 neutron interferometer 648 neutron interferometry 63 neutronium 90 neutron matter 94 neutron star 8, 92, 92, 93, 94, 141 new scientist 253, 298, 476 newton 226, 478 newton’s laws 477 newton’s laws of motion 8, 330, 612 newton’s rings 108 newton’s second law 113, 464 newtonian physics 231, 494 new york academy of sciences 552 new york times 319 nickel 86, 87 niels bohr 11, 15, 15, 51, 62, 153, 164, 182, 190, 226, 234, 237, 294, 297, 422, 460, 471, 477, 480, 496, 553, 619, 625 niels bohr institute 553 night court 628 nikola tesla 374 niobium 572 nist 506 nitrogen dioxide 10 nitrous oxide 10 nobel prize 315, 375, 400 nobel prize for physics 86, 648 nobel prize in physics 25, 67, 153, 405, 440, 647 nobel prizes for physics 62 no cloning theorem 415, 416 nodal line 185 nodal surface 185 noether’s theorem 263, 300, 356 noise-equivalent power 409 noise level 409 noise power 409 non-linear optics 568 noncommutative 239, 494 noncommutative algebra 75 nondestructive testing 522 nonholonomic mapping 352 nonlinear sigma model 354, 355 nonlocal 257 nonlocality 358 norm 432 normalisable wavefunction 640 normalization constant 189

o o.r. lummer 405 observable 168, 210, 215, 245, 273, 274, 293, 390, 436, 451, 463, 594, 619 observables 200, 281 observation 534, 619, 620 observer 196, 297, 315 observer effect 619, 627 obsolete scientific theory 15 occam’s razor 594 oil 315 oil-drop experiment 474 old quantum theory 269 one-dimensional periodic case 344 one-loop feynman diagram 548 on shell 354, 356 ontological 196 open ball 175 open quantum system 548 operator 48, 67, 79, 150, 259, 355, 393, 422, 467, 564, 630, 656

691 particle detector 591 particle in a box 156, 295, 338, 343, 345, 364, 382, 402, 428, 466, 504, 529, 620 particle in a one-dimensional lattice (periodic potential) 529 particle in a ring 331, 338, 529 particle in a spherically symmetric potential 331, 529 particle number operator 436 particle physics 117, 123, 129, 226, 301, 312, 329, 365, 421, 422, 434, 459, 522, 588, 589, 592, 607, 615 particles 142, 151, 197, 522 particle statistics 130 particle zoo 123 partite 167 partition function 286, 288 paschen-back effect 665, 665 paschen series 128 pascual jordan 281, 421 path integral formulation 107, 468, 649 patient 496 pattern 646 patterns 197 paul dirac 40, 83, 102, 130, 131, 213, 226, 241, 246, 271, 271, 348, 350, 368, 421, 445, 452, 460, 462, 467, 472, 480, 488 paul ehrenfest 112, 617 paul gordan 52 pauli 192 pauli exclusion principle 27, 30, 90, 91, 94, 130, 133, 136, 139, 152, 246, 286, 422, 427, 556, 557, 648 pauli matrices 99, 103, 302, 558, 559, 593 pauli principle 141, 556, 557 pbs 123, 512 penning trap 506 penrose interpretation 222, 494 peres-horodecki criterion 551 pericenter 261 periodic function 304, 339 periodic table 187, 459 periodic table of elements 312 permittivity 369 permittivity of free space 75 permutation 424, 640 perpendicular 190 perturbation 89, 191 perturbation theory 16, 45, 423, 578 peter debye 71 petrochemical 8 pharmaceutical 8 phase distribution 567

DR A

FT

operator algebra 181 operator ordering problem 351 operators 282, 619 optical aberration 498 optical fiber 580 optical field 57 optical mode 398 optical society of america 523 optical theorem 432, 630 optics 55, 306, 310, 321, 412, 656 orbit 261, 264 orbital 117, 236 orbital angular momentum 185, 665 orbital elements 264 orbital resonance 509 ordinary differential equation 527 ordinary differential equations 280 organic chemistry 514 orthodox 62 orthogonal 282, 433 orthogonal group 404 orthogonality 174 orthogonal polynomials 174 orthonormal 168, 292 orthonormal basis 176, 177, 310, 452, 490, 639 osama bin laden 538 oscillator 282, 380 oskar klein 256, 258 otto stern 474, 582, 582, 585, 648 outer product 43 overlap matrix 515 oxygen 10 p p. a. m. dirac 474 parable of the cave 206 parachronics 544 paradigm 196 paradox 298, 645 parastatistics 33, 138 paravector 102 paravectors 98 parity (physics) 300, 613 partial derivative 171, 524 partial differential equation 43, 126, 175, 177, 188, 269, 370, 522, 526, 527 partial differential equations 157, 280, 522 partial trace 315, 418, 561 particle 39, 113, 153, 303, 314, 453, 494, 497, 582, 588, 608, 619, 644 particle accelerator 509, 588, 589, 591 particle collider 122

692 planck’s constant 16, 17, 47, 51, 69, 73, 87, 173, 249, 269, 283, 350, 373, 380, 407, 440, 455, 463, 466, 499, 597, 601, 613, 618, 620, 646, 647, 654, 662 planck’s law 618 planck’s law of black body radiation 33, 654, 654 planck length 73, 88, 379, 438, 649 planck mass 73, 379, 649 planck time 438, 439 planetary mechanics 16 plane wave 156, 175, 257, 320, 321, 401, 463, 529 plasma physics 1 plastic 267 plato 206 plum 388 plum-pudding model 517 plum pudding 389 plum pudding model 10, 163, 516 poet laureate 537 poincare group 301 poincare symmetry 589, 592 poison 532 poisson bracket 47, 112, 173, 262, 263, 283, 348 poisson brackets 262 poissonian 57, 60 poissonian distribution 58 polar coordinates 343, 512 polarization 108, 369, 396, 421, 432, 585 polonium 10 polyatomic molecule 115 polylogarithm 35, 384 polynomial 354, 391 polynomially bounded 353 pontryagin duality 74 position 72, 463, 619, 619 position manifold 126 position operator 292 positive linear functional 594 positive matrix 417 positron 40, 119, 311, 353, 424, 473 positronium 40, 190 possibility 199 post hartree-fock 530 postulate 349, 380 potential 13, 113, 129, 188, 295, 310, 342, 345, 424, 498, 506 potential difference 154, 316 potential energy 330, 391, 468, 517, 527, 596 potential energy surface 98, 116, 116, 299,

DR A

FT

phase noise 409 phase shift 3 phases of matter 8 phase space 17, 33, 47, 58, 165, 268, 270, 273, 611, 656 phase transition 486, 603, 607 phase transitions 434 phase velocity 310 phenomenon 266 philosophical interpretation of classical physics 222 phonon 127, 365, 428, 450, 568 phosphorescence 458 photoelectric 230 photoelectric effect 68, 71, 106, 211, 269, 382, 386, 472, 473, 645, 647, 662 photoelectric emission 662 photographic 200 photon 13, 16, 19, 23, 31, 40, 67, 73, 118, 119, 120, 121, 127, 128, 211, 230, 297, 353, 364, 371, 373, 375, 382, 386, 396, 404, 422, 424, 432, 433, 438, 458, 461, 469, 472, 519, 588, 590, 591, 618, 644, 645, 647, 662 photon number distribution 567 photons 33, 104, 106, 229, 230, 269, 312, 508, 535, 584, 605, 641, 648 photosynthesis 411 physical cosmology 194, 548 physical limits to computing 611 physical review letters a 409 physical system 634 physicist 84, 84, 213, 519, 524 physicists 87 physics 8, 12, 39, 47, 52, 62, 67, 74, 75, 79, 86, 89, 124, 126, 130, 138, 153, 155, 157, 167, 172, 178, 179, 311, 314, 315, 320, 320, 322, 380, 382, 388, 404, 404, 410, 411, 434, 458, 485, 486, 486, 509, 514, 517, 522, 524, 546, 548, 555, 564, 601, 603, 607, 612, 630, 641, 644, 652 physics experiment 153 physics today 535 physics world 105 pi 283 picometer 648 pier giorgio merli 104 pierre victor auger 12, 13 pieter zeeman 665 pilot wave 273 pin group 327 pion 311, 327, 591 pioneer plaque 192

693 proton collapse experiment 474 proton conductor 267 proton decay 121 pseudoscalar 323, 328 pseudoscience 651 psychic healing 496 psychokinesis 496 psychology 628 pudding 388 pure mathematics 268 pure state 215, 393, 456, 491, 492, 549 purification of quantum state 393 pythagoreans 407

FT

392 potential theory 391 potential well 149, 504 pound-rebka falling photon experiment 648 power series 364, 443, 499, 633 power spectrum 310 precision 619, 619 precognition 496 preon 94, 122 preon-degenerate matter 94 preon matter 94 preon star 94 pressure 90, 309 primitive cell 661 princeton university press 474 principal quantum number 16, 19, 241, 248, 530 principle 196 principle of complementarity 62, 237 principle of emergence 603 principle of locality 415 private investigator 541 probability 22, 146, 157, 310, 349, 400, 488, 529, 546, 630, 640, 646, 649 probability amplitude 41, 52, 103, 468, 471, 488, 490, 630 probability current 399, 528 probability density 184, 400, 460, 528, 529, 635 probability density function 304, 399 probability distribution 213, 215, 243, 451, 463, 471, 481, 619, 622 probability flux 399, 528 probability measure 274 probability theory 269 proca equation 528 processes 197 process physics 439 product rule 113, 173, 284 professor walter ernhart-plank 474 projection 67 projection-valued measure 278, 390, 398, 452 projection operator 43, 216, 418 projective hilbert space 416, 551 projective representation 323, 403 projective space 403 proof of the existence of god 644 propagator 353, 511 proquest 554 prospecting 573 proton 11, 40, 67, 88, 117, 142, 182, 270, 311, 312, 312, 327, 470, 520, 557, 584, 588, 589, 591, 591, 648

DR A

q qcd 522 qed 522 q factor 193, 510, 511 qft 355 quadratic form 272 quadrupole 546 quant-ph 507 quanta 11, 80, 230, 281, 375, 422, 460 quantization 48, 273, 277, 408 quantization condition 16 quantization of constrained systems 277 quantization of gauge theories 277 quantized 586 quantum 459, 486, 656 quantum-mechanical 266 quantum-mechanical circuit 253 quantum-well intermixing 504 quantum annealing 438 quantum chaos 468 quantum chemistry 12, 79, 98, 114, 115, 186, 226, 365, 391, 459, 470, 473, 522, 522, 656 quantum chromodynamics 78, 79, 159, 364, 430, 433, 434, 470, 473, 614 quantum coherence 61, 109, 294 quantum coin-flipping 395 quantum communication 550 quantum computation 494, 605 quantum computer 254, 256, 408, 470 quantum computing 40, 536, 537, 605, 611, 615 quantum cosmology 218, 320 quantum cryptography 395, 408, 414, 470, 536 quantum cybernetics 650 quantum decoherence 293, 296, 297, 472, 535, 541 quantum dot 392, 504

694 quantum mechanics, philosophy and controversy 249 quantum memory 605 quantum metaphysics 222 quantum mind 456 quantum money 395 quantum monte carlo 530 quantum noise 565 quantum number 128, 182, 187, 189, 248, 300, 301, 408, 462, 487, 530, 582, 589 quantum numbers 193, 241 quantum operation 214, 217, 279, 314, 398 quantum optics 60, 61, 273, 320, 404, 569, 656 quantum particle 392 quantum physics 85, 104, 144, 314, 392, 436, 451, 519, 544, 619 quantum process tomography 497 quantum state 22, 40, 57, 91, 112, 117, 127, 128, 139, 151, 167, 167, 186, 187, 241, 293, 302, 344, 408, 414, 421, 424, 463, 495, 497, 514, 521, 558, 603, 609, 614 quantum states 24, 196, 324, 348, 613 quantum stationary hamilton-jacobi equation 126 quantum statistical mechanics 489, 624 quantum suicide 537, 539 quantum superposition 471 quantum system 114 quantum teleportation 414, 415, 471 quantum theory 11, 51, 62, 86, 89, 310, 405, 495, 588, 591, 608 quantum trajectory representation theories of quantum mechanics 126 quantum tunneling 145 quantum tunnelling 149 quantum well 392 quantum wire 392, 504 quark 40, 78, 78, 94, 117, 118, 119, 120, 129, 316, 364, 470, 588, 590 quark-degenerate matter 94 quark matter 94 quark model 129, 591, 591, 592 quarks 118, 557, 584 quark star 93, 94 quartz 374 quasar 509 quasiparticle 25, 142, 422, 603 qubit 193, 254, 404, 490, 537 qubits 563

DR A

FT

quantum electrodynamics 55, 75, 77, 79, 158, 186, 326, 353, 364, 432, 433, 434, 434, 440, 469, 473 quantum electronics 408 quantum entanglement 66, 247, 456, 462, 467, 493, 549, 560, 603, 605 quantum eraser experiment 95, 95, 109 quantum field theory 2, 11, 39, 48, 61, 61, 72, 75, 76, 77, 77, 78, 79, 83, 186, 258, 272, 276, 295, 300, 302, 348, 352, 353, 353, 420, 432, 445, 462, 469, 475, 510, 522, 545, 548, 568, 581, 589, 590, 596, 614, 614, 615 quantum field theory in curved spacetime 272 quantum fluctuation 438 quantum gravity 122, 360, 444, 462, 470, 548 quantum gyroscope 254, 256 quantum hall effect 504 quantum harmonic oscillator 54, 79, 80, 83, 259, 266, 277, 364, 426, 468, 469, 488, 490, 503, 529, 564 quantum immortality 408 quantum indeterminacy 66, 213, 222, 396, 628 quantum information 393, 395, 507, 551 quantum information theory 124, 167, 420, 420, 592 quantum jump 16, 19 quantum leap 11, 270, 331 quantum logic 219, 272, 398, 467, 490 quantum magnet 486 quantum measurement 272, 398, 451, 456, 650 quantum mechanic 506, 543 quantum mechanical 345 quantum mechanics 2, 13, 15, 22, 23, 39, 40, 41, 45, 51, 52, 54, 67, 72, 73, 74, 84, 85, 87, 97, 98, 102, 104, 109, 114, 115, 122, 126, 127, 146, 150, 151, 153, 153, 157, 172, 172, 174, 174, 176, 180, 182, 186, 194, 207, 210, 212, 222, 224, 248, 259, 260, 264, 266, 267, 281, 283, 284, 291, 296, 298, 302, 303, 306, 306, 310, 312, 320, 324, 324, 330, 338, 343, 345, 348, 348, 352, 363, 372, 382, 386, 390, 391, 399, 400, 403, 404, 405, 408, 409, 410, 411, 412, 421, 423, 425, 438, 451, 456, 459, 487, 489, 494, 507, 512, 522, 531, 532, 540, 546, 548, 549, 556, 557, 557, 559, 564, 581, 582, 589, 594, 596, 596, 612, 613, 614, 617, 619, 621, 630, 631, 634, 640, 640, 644, 645, 649, 656

r r. a. fairthorne

203

695 relativistic mass 509 relativistic plasma 509 relativistic quantum field theory 295 relativistic wave equations 509, 528 relativity 415 relativity theory 45 relic particles 590 renaissance 407 renninger negative-result experiment 298 renormalization 79, 430, 437 renormalization group 348, 434 rené descartes 478 representation of a hopf algebra 595 representation theory 52, 656 reproducing kernel hilbert space 181 residue theorem 387 resolution 293 resolution of the identity 367, 527 resonance 39, 40, 127, 130, 399 resonance (disambiguation) 509 resonator 511 rest energy 19 rest mass 87, 508 reversible computing 611, 615 richard courant 279 richard feynman 63, 107, 111, 213, 226, 318, 348, 552 richard p. feynman 474, 475, 484 riemannian manifold 102 riemann zeta function 35 riesz representation theorem 41, 125, 179, 179 rigged hilbert space 44, 175, 181, 277 right triangle 407 ring wave guide 529 ripple tank 65, 104 ritz method 632 robert a. heinlein 541 robert andrews millikan 375 robert anton wilson 538, 542 robert b. laughlin 441 robert b. leighton 474 robert brown 10 robert h. scanlan 511 robert millikan 315, 474 robert oppenheimer 422 robertson-schrödinger relation 624 rockefeller foundation 553 roleplaying game 544 ronald gurney 498 roothaan equations 150, 151 rotation 266, 322, 323 rotational symmetry 183

DR A

FT

r.e. siday 3 r. f. streater 280 r. neill graham 474 radar 74 radar ambiguity function 74 radian 232 radiant energy 374 radiation 269 radiation therapy 71 radioactive 498 radioactive decay 211, 313 radioactive nuclei 39 radioactivity 312, 313, 473 radiobiology 71 radio drama 541 radio wave 522 radius 317 rain 522 rainbow 522, 646 ralph kronig 246, 342 raman transition 193 random 465 random variables 560 range criterion 551 rationalism 478 ray 168 rayleigh-jeans law 617 rayleigh scattering 653 ray tracing 658 reality 196, 469 real line 635 real number 168, 194, 213, 525 reciprocal 354 recorded 203 redshift 71 reduced mass 189, 442 reduced planck constant 189, 275 reducible 594 reduction criterion 551 reductionism 196, 470 redundancy 145 reeh-schlieder theorem 420 reference frame 626 refraction 646 reginald cahill 439 region 200 relationship between string theory and quantum field theory 434 relative state interpretation 279, 315, 465 relativistic 39, 173, 283 relativistic beaming 509 relativistic breit-wigner distribution 511 relativistic jet 509

696 schwinger-dyson equation 354, 355, 356, 434 science 8 science fiction 542, 574, 628 scientific notation 230 scientific rigor 644 screen 200 screw dislocation 608 sea level 392 sea water 522 second 193, 620 second law of thermodynamics 609, 610, 615 second quantization 79, 422 secular equation 515 segre embedding 417, 549, 551 segre mapping 404 seismology 656 selenium 373 self-adjoint 453 self-adjoint operator 48, 168, 216, 274, 314, 393, 452, 524, 525, 594, 595 self adjoint operator 273, 292, 390 semantics 217 semiclassical 320 semiclassical gravity 548 semiconductor 8, 13, 140, 141, 392, 498, 662 semidefinite programming 296 seminar 498 separability 563 separable 422 separable states 420 separation of variables 188, 280, 335, 336 sequence 175, 175 sequence space 176 set 176 shamanism 203 shannon entropy 418 shell model 313 shin megami tensei 543 shot noise 60, 409, 441 si 4, 315, 380 signal-to-noise ratio 409 signals 198 simple harmonic motion 511 simultaneous equation 366 sinclair ql 458 sine-gordon equation 608 sine curve 232 singularity 84 sir roger penrose 360 skyrmion 607 slac 615 slater-type orbital 187 slater determinant 114, 151

FT

rotation group 263, 264 rotation operator 170 royal society of london 106 roy j. glauber 55, 60 rubidium 22, 25, 193 rudolf grimm 27 rule of thumb 112 rumor 538 rutherford 474, 645 rutherford backscattering 517 rutherford model 388 rutherford scattering 522 rydberg-ritz combination principle 518, 521 rydberg constant 235, 519, 520 rydberg formula 15, 19, 128

DR A

s s-matrix 39, 510, 630 sackur-tetrode equation 38, 165, 167 saint-germain 543 samsara 206 saturnine martial & lunatic 543 satyendra nath bose 23, 33, 34, 386 scalar 321, 323, 528 scalar field 78, 126, 526 scalar potential 49 scale anomaly 77 scanning squid microscope 573 scanning tunnelling microscope 498, 502 scattering 467, 522, 522 scattering amplitude 320, 321 scattering theory 320, 521, 651 schottky diode 662 schroedinbug 539 schroedinger equation 49, 522, 523, 645, 649 schrödinger 16, 348, 352 schrödinger’s cat 297, 298, 494, 540 schrödinger’s cat in fiction 539 schrödinger’s cat trilogy 542 schrödinger’s equation 171, 246, 249, 656 schrödinger equation 6, 11, 20, 43, 54, 80, 112, 127, 156, 169, 173, 174, 182, 188, 210, 211, 214, 218, 256, 260, 275, 277, 283, 284, 294, 295, 296, 298, 299, 303, 306, 310, 310, 330, 331, 339, 343, 345, 350, 364, 399, 402, 423, 443, 464, 467, 499, 512, 521, 596, 640, 640 schrödinger picture 172, 173, 210, 275, 283, 369, 423, 530 schumann resonance 511 schwarzian derivative 126, 127 schwarzschild 578 schwarzschild radius 73, 379

697 spherical harmonics 54, 182, 488 spin 45, 52, 396 spin-dirac operator 103 spin-orbital 556 spin-orbit coupling 185 spin-orbit interaction 46, 665 spin angular momentum 665 spinning 266 spinor 45, 102, 323, 327, 558, 559, 613 spin quantum number 246, 249, 530 spin singlet 455 spin statistics theorem 557, 592, 614 spiral galaxy 439 spontaneous parametric down conversion 96 spontaneous symmetry breaking 121 spooky action at a distance 650 square-integrable 467 square integrable 41, 399 square potential 501 squark 122 squashed entanglement 420 squeezed coherent state 58 squeezing operator 567 squid 254, 502 squid (disambiguation) 571 stability 506 stable 39 standard deviation 242, 624 standard model 117, 118, 121, 123, 158, 249, 327, 328, 329, 421, 422, 434, 522, 590, 591, 592, 615, 615 standing wave 231, 309 standing waves 407 stanford encyclopedia of philosophy 220, 220 star 92 stargate sg-1 538, 541, 628 stark broadening 577 stark effect 16, 20, 124, 364, 663, 666 star trek 628 state 22 state vector 172, 210, 283, 494 static 196 stationary point 350 statistical 196 statistical ensemble 314, 490 statistical mechanics 23, 30, 130, 165, 269, 285, 320, 348, 352, 353, 355, 383, 391, 470, 529, 597, 617, 656 statistics 30 stefan-boltzmann law 385 stellar mass black hole 85 stephen donaldson 628 stephen hawking 194, 537

DR A

FT

slepton 122 sliders 538, 541 slow glass 27 slow light 27 s matrix 521, 523, 523 smoke detector 313 so(3) 50, 52, 263 so(3,1) 264 so(4) 264 sobolev space 177 social science 628 solar cell 376 solar mass 92, 92 solar power 376 solar system 14, 509 solenoid 3, 4 solid 8, 30, 90, 133, 287, 428, 450, 603 solid helium 486 solid state physics 450, 608 soliton 364, 607 solvay conference 553 sommerfeld-wilson-ishiwara quantization 269, 270 sound 410 sound wave 522, 621 source field 354 space 467 space-like 422 space-time 196, 353, 432 spacecraft 377 space quantization 586 spacetime 22, 122, 194, 285, 425 spallation neutron source 574 spark gap 373 sparticle 118, 122 special orthogonal group 323 special relativity 51, 72, 102, 185, 256, 315, 424, 462, 467, 469, 473, 508, 509, 589, 625 special unitary group 323 spectral line 71, 182, 364, 369, 472, 576, 663 spectral measure 169, 456, 490 spectral theorem 292, 416, 417, 452 spectral theory 272, 274, 280 spectrometer 459 spectroscopy 19, 229, 270, 281, 412 spectrum 168, 523 speculative fiction 644 speed 508, 509 speed of light 49, 69, 72, 75, 87, 185, 192, 414, 508, 509, 654 sphere 139, 266 spherical coordinates 183, 188 spherical harmonic 52, 183, 190, 345, 345

698 superconductors 253 superdense coding 420 superfield 600 superfluid 24, 27, 85, 142, 410, 428, 461, 603 superfluidity 26, 248 supergravity 433, 590 supernova 26 superposition 114, 293, 308, 309, 310, 360, 469, 505, 537 superposition principle 494 superpotential 598 superselection 114 superselection sector 275 supersolid 27 supersymmetry 122, 122, 274, 356, 433, 590 surveyor program 377 switch 145 symmetric 188, 640 symmetry 77, 89, 345 symmetry of second derivatives 386 symplectic form 17 symposium 410 synchronicity 496 synchrotron radiation 15 synonym 462 system 414

DR A

FT

stephen notley 544 stern-gerlach experiment 474, 558 steve martin 542 steven chu 25 steven weinberg 327, 535 stiffness 306 stimulated emission 458 stirling’s approximation 32, 137, 166, 288 stochastic 495 stochastic process 175 stokes’ law 317 stokes theorem 4, 6 stone’s theorem 275 stone’s theorem on one-parameter unitary groups 275 stone-von neumann theorem 48, 173, 277, 283 strange matter 90, 93 strangeness 312 strange quark 93 stream 197 string-net condensation 603, 605 string theory 61, 78, 122, 285, 433, 462, 470 strong cp problem 615 strong cp violation 614 strong force 590, 591 strong interaction 76, 119, 120, 121, 129, 329 strong interactions 328, 614 strong nuclear force 469 structures 197 stuart kauffman 205 sturm-liouville theory 280, 311 su(2) 433, 607 su(3) 78, 433 sub-poissonian 567 subalgebra 594, 595 subatomic particle 21, 123, 129, 379, 408 subatomic particles 10, 225 submarine 522 subspace 180 substance 200 sudarshan-glauber p representation 61 sun 92, 406, 588, 653 sunlight 522 sunyaev zel’dovich effect 71 super-consciousness 644 super-kamiokande 121 supercommutator 599 superconducting 571 superconductive 266 superconductivity 27, 365, 421 superconductor 93, 266, 486, 498

t t-symmetry 301 table of clebsch-gordan coefficients 53 tacoma narrows bridge 511 talbot lau interferometer 648 tangent bundle 159 taoism 206 target manifold 356 target space 355 tau lepton 119, 590 tau neutrino 118 taylor series 356 tears for fears 539, 543 technology 470 telepathy 496 teleportation 644 temperature 22, 31, 32, 34, 132, 137, 140, 141, 142, 286, 288, 413, 440, 486, 581, 601, 652 tensor 323 tensor category 603 tensor product 43, 52, 151, 274, 415, 549 terminal velocity 317 terry pratchett 539, 540, 542 tetraphenylporphyrin 648

699 tomography 659 tonks-girardeau gas 26, 27 topological defect 607 topological entropy 603 topological invariant 607 topological order 441, 604 topological quantum field theory 604 topological vector space 175, 176 topologies on the set of operators on a hilbert space 181 torus 333 trace-class 124 trace class 168, 274, 417, 550 trajectory 645 transactional interpretation 219, 222, 298, 650 transduction 509 transformation law 315 transistor 470, 605 transition radiation 509 transition radiation detector 509 transition rate 11 transition rule 664 translational invariance 355 translationally invariant 354 transmission electron microscopy 555 trap 568 tuned circuit 511 tungsten 662 tunnel diode 502 tunneling time 364 twentieth century 412 two-photon generation 567 two lumps 544 type ii superconductor 149 type i superconductor 149

DR A

FT

the bohr model 182 the cat who walks through walls 541 the coming of the quantum cats 542 the compass rose 541 the elegant universe 123 the feynman lectures on physics 474 the homing pigeons 542 the last hero 539, 542 the meaning of meaning 203 theorem 344, 619, 656 theoretical physics 74, 285, 434, 459, 596, 630 theory 8, 198, 269 theory of everything 196 theory of relativity 226, 480 thermal de broglie wavelength 30, 132, 286, 291 thermal equilibrium 30, 136, 285 thermal light 56 thermal noise 255, 409 thermal wavelength 35 thermionic emission 145, 662, 663 thermodynamics 8, 140, 420 the trick top hat 542 the universe next door 542 thin film 486 things 197 thinkgeek 539 third law of thermodynamics 581 thomas-fermi approximation 34 thomas kuhn 205 thomas young 226 thomson scattering 67, 71, 73 thought 196 thought experiment 269, 532, 540, 610, 625 three-body force 129 tidal force 392 tidal resonance 509 time 169, 188, 310, 523, 531, 621 time-resolved spectroscopy 510 time evolution 214, 217, 273, 275, 279 timeline of chemical element discovery 12 timeline of cosmic microwave background astronomy 71 timeline of quantum mechanics, molecular physics, atomic physics, nuclear physics, and particle physics 12 timeline of thermodynamics, statistical mechanics, and random processes 12 time ordered 353 time ordering 355 time reversal invariance 301 tolman-oppenheimer-volkoff limit 93

u u(1) 433 ultraviolet 506 ultraviolet catastrophe 269, 386, 641 ultraviolet divergence 617 ultraviolet light 374 ultraviolet radiation 371, 374 umklapp scattering 143 unbounded operator 169 uncertainty 619 uncertainty principle 47, 51, 63, 77, 94, 190, 242, 248, 275, 283, 335, 348, 396, 425, 436, 438, 461, 462, 463, 466, 468, 472, 480, 494, 564, 624, 640, 657 uncertainty relation 57, 77, 565 uncountable set 429

700 vector space 175, 461 velocity 87, 257, 497 vernacular 458 vertex renormalization 432 vibrational spectroscopy 310 vienna 88 virial theorem 18 virtual particle 353, 436, 438, 438, 582, 624 virtual particles 77, 438 viscosity 24, 317 viscous 200 vladimir fock 422 voigt 578 volt 254, 377 volume 381 von-karman boundary condition 337 von klitzing constant 440 von neumann 294, 475, 620 von neumann entropy 419 vortex 197 voyager golden record 192 vulgar fraction 440

DR A

FT

undergraduate 557 unit 315, 594 unital 594 unitarity 630 unitary group 404 unitary operator 48, 169, 214, 324, 391, 419, 581 unitary representation 594 unitary representation of a star lie superalgebra 595 unitary transformation 98 unit vector 524 universe 21, 117, 194, 498 university of aberdeen 648 university of colorado at boulder 22, 27 university of frankfurt 585 university of innsbruck 27 university of manchester 163 university of tübingen 104 university of vienna 648 unobservables 206 unruh effect 568 unsolved problems in physics 222, 469, 615 unstable 39 unstable particle 510 upper bound 13 upthrust 317 ursula k. le guin 541 uv 376 v v. a. fock 2, 151, 153 vacuum 316, 376, 519, 520 vacuum energy 439 vacuum expectation value 61, 353 vacuum fluctuation 186 vacuum state 61, 82, 326, 426, 581, 582 vacuum tube 459, 662, 662 vaisheshika 407 valence shell 187 valentine bargmann 13 vapour pressure 316 variance 58, 162 variational method 514 variational method (quantum mechanics) 633 variational parameters 633 variational perturbation theory 364 variational principle 530, 631 vector 262, 282 vector boson 326 vector bundle 102 vector mesons 584 vector potential 49, 429, 598

w walter gerlach 474 walter heitler 473 walter ritz 514 walther bothe 71 walther gerlach 582, 582, 585 w and z bosons 118, 120, 120, 584 ward-takahashi identity 356 water 104 wave 67, 86, 104, 225, 310, 320, 321, 376, 463, 494, 511, 620, 644 wave-function renormalization 431 wave-particle duality 26, 56, 62, 66, 86, 86, 96, 104, 237, 271, 348, 371, 461, 462, 463, 467, 532, 608, 617, 620 wave equation 257, 522, 641 wave function 63, 71, 98, 271, 281, 284, 296, 303, 312, 314, 343, 399, 400, 403, 432, 512, 524, 565, 645 wavefunction 22, 30, 44, 79, 80, 114, 128, 132, 152, 175, 183, 189, 195, 211, 213, 214, 286, 297, 306, 310, 310, 332, 460, 487, 488, 490, 494, 505, 526, 533, 556, 559, 656 wavefunction collapse 218, 222, 245, 293, 294, 295, 297, 298, 464, 469, 533, 620 wave functions 461 wavelength 67, 86, 87, 93, 108, 170, 381, 406, 463, 495, 498, 519, 519, 520, 647, 652 wavelet 176

701 wigner function 57 wiki 507 wild arms 3 543 wilhelm wien 381, 654, 654 william chinowsky 327 william d. phillips 25 willoughby smith 373 winding number 607, 608 wiretap 534 wkb approximation 365, 502, 530, 548, 597 wojciech h. zurek 114 wolfgang ketterle 25, 27 wolfgang pauli 11, 226, 246, 263, 422, 460, 552, 591 work function 371, 372, 373 wormhole 438, 439 w state 167

FT

wave mechanics 16, 263, 271, 281, 462, 472 wavenumber 87 wave packet 56, 399, 464, 494, 565, 640 wavepacket 565 waves 522 wave vector 156 w boson 590 weak decay 615 weak force 590 weak gauge boson 121 weak interaction 121, 329, 591 weak interactions 328 weak measurement 297 weak nuclear force 469, 470, 609 webcomic 544 well-behaved 176 wendell furry 422 werner ehrenberg 3 werner heisenberg 62, 226, 238, 271, 271, 281, 281, 320, 405, 421, 436, 460, 462, 480, 619 wess-zumino-witten model 608 weyl 475 weyl quantization 47 wheeler’s delayed choice experiment 95 white 653 white dwarf 30, 90, 91, 92, 94, 133, 141, 286 white dwarf material 94 white noise 409 whole number 10 wick rotation 195, 350, 352, 353, 354, 356 wiener measure 158 wigner’s friend 220, 537 wigner 3-j symbol 54 wigner distribution 656

DR A

x x-ray 13, 67, 86, 255, 376 x-rays 371 x boson 121 x ray 316 y ybco 573 yukawa interaction

73

z z boson 77, 590 zeeman effect 20, 50, 124, 228, 557, 576 zero-point energy 331, 438, 439, 444 zig-zag 351 zinc sulfide 163 zorn’s lemma 178

FT

DR A