2006-7 Quantum Theory Slides Lecture 5

Quantum theory and atomic spectroscopy Lecture 5 Wave-particle duality

Old quantum theory • Old quantum theory very good at describing the microscopic world • Energy levels of atoms quantized • Electrons in atoms can be labelled by four quantum numbers n, l, ml and ms • n, l and ml take integer values, ms halfinteger • Add Pauli exclusion principle, Hund’s rule etc • BUT …

The BIG question is…… • Why oh why is this true? Why do we have quantization? Why do we have these quantum numbers?

Classical physics • Waves- light, sound • Mechanical disturbance of medium • For light- the aether • Maxwell equations • No aether found! • Special relativity (1905) • Matter- particles, obey Newton’s laws

The photoelectric effect

• The kinetic energy depends on wavelength • The kinetic energy doesn’t depend on intensity • As soon as light strikes the surface, electrons appear so long as a minimum frequency is exceeded

The workfunction • We must exceed a minimum frequency (energy) that is metal dependent which is called the workfunction

Enter Einstein • Albert simply asked the question: what would happen if light itself was quantised? • Final part of his paper he suggested it would be a neat solution to photoelectric effect • Imagine individual photons colliding with electrons!

Young’s slits • One problem with the theory was that it didn’t seem to explain the interference of light • Some thought maybe this was a statistical effect • G I Taylor showed that interference worked with very weak light (one “photon”) in 1909!

This is still not well understood in 2006!

So people took a bit of convincing… • Milliken checked the experimental data- nothing wrong there • Bohr so desperate to keep wave picture he wanted to drop energy conservation! • Bothe and Geiger (1924) finally demonstrated Einstein probably right.

Light pressure • This had been understood for a long time • Kepler used the idea to explain comet tails • However, the momentum of light is tiny (easily exceeded by solar wind)

De- Broglie wavelength If light can have momentum… Can particles have a wavelength?

E p = c

h λ= p

hν h p = = c λ This is the mathematical description of WAVE-PARTICLE DUALITY

An example…. • Consider a snooker ball, mass 0.1 kg and velocity 10 ms-1

h 6.626 x10 −34 Js -34 m λ = = = 6.626 x 10 −1 p 1kgms

But if you work this out for ELECTRONS, the wavelength is similar to X-RAYS

X- rays • X- rays discovered by Rontgen • 1912 – von Laue suggested they could be used to investigate crystal structure – X-ray diffraction If electrons have a wavelength like xrays, can we use them in the same way?

Can electrons diffract? • Yes they can! • Diffraction pattern similar to that with X rays • Electrons easier to use than X-rays

A wonderful coincidence… • Electron is particle – J.J. Thomson (NP 1906) • Electron is wave – G. Thomson (NP 1937)

• Father and son!!!!

Wave properties • What is a perfect sine-wave? • It has a fixed wavelength (momentum) • But it is infinite in extent! • Add more wavelengths- get beat notes

Momentum- position complementarity • Add more and more waves to get a wavepacket • With enough waves, get a “spike” localised in space • But how many waves have you added? An infinite number! • Know the position, don’t know the momentum

Heisenberg’s Uncertainty Principle • Werner Heisenberg suggested this behaviour would affect material properties • “It is impossible to specify simultaneously, with arbitrary precision, both the position and conjugate momentum of a quantum object”

h ∆p x ∆x ≥ 4π

Enter Schrödinger • He wanted to understand what deBroglie’s concept would mean for atomic levels • He got an idea from an electrical engineer: find a wave-equation • In about six weeks he had changed science forever!!

What he introduced was a new, mathematical description of small objects he called the WAVEFUNCTION

Schrödinger’s wave-equation In this communication I wish to show that the usual rules of quantisation can be replaced by another postulate, in which there occurs no mention of whole numbers. Instead, the introduction of integers arises in the same natural way as, for example, in a vibrating spring.

• He was picturing the hydrogen atom as a series of standing waves! • Similar to a vibrating guitar string! (But in 3-D)

Standing waves for dummies • Create a standing wave by imposing a boundary condition e.g. a wall, a pin or the finger of a guitarist • Quantization of frequencies!! • For hydrogen, solve its Schrödinger equation: do we get quantization?

Answers to the earlier questions.. • The quantum behaviour of electrons is due to their wave- particle duality • This leads to fundamental changes in behaviour e.g. Heisenberg Uncertainty Principle • The Schrödinger equation becomes the fundamental description of nature!

Alright then…… • What is the Schrödinger equation and how do we use it? • Next time, we will solve it for a free-particle and meet this new description of objects: the wavefunction