Pc Chapter 41

Chapter 41 Quantum Mechanics

Probability – A Particle Interpretation 

From the particle point of view, the probability per unit volume of finding a photon in a given region of space at an instant of time is proportional to the number N of photons per unit volume at that time and to the intensity Probability N  I V V

Probability – A Wave Interpretation 

From the point of view of a wave, the intensity of electromagnetic radiation is proportional to the square of the electric field amplitude, E

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2view gives Combining the points of IE

Probability  E2 V

Probability – Interpretation Summary 

For electromagnetic radiation, the probability per unit volume of finding a particle associated with this radiation is proportional to the square of the amplitude of the associated em wave 

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The particle is the photon

The amplitude of the wave associated with the particle is called the probability amplitude or the wave function 

The symbol is ψ

Wave Function 

The complete wave function ψ for a system depends on the positions of all the particles in the system and on time 

The function can be written as ψ(r1, r2, … rj…., t) = ψ(rj)e-iωt 

rj is the position of the jth particle in the system

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ω = 2πƒ is the angular frequency

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i  1

Wave Function, cont.  

The wave function is often complex-valued The absolute square |ψ|2 = ψ∗ψ is always real and positive  

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ψ* is the complete conjugate of ψ It is proportional to the probability per unit volume of finding a particle at a given point at some instant

The wave function contains within it all the information that can be known about the particle

Wave Function, General Comments, Final 

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The probabilistic interpretation of the wave function was first suggested by Max Born Erwin Schrödinger proposed a wave equation that describes the manner in which the wave function changes in space and time 

This Schrödinger wave equation represents a key element in quantum mechanics

Wave Function of a Free Particle 

The wave function of a free particle moving along the x-axis can be written as ψ(x) = Aeikx 

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k = 2π/λ is the angular wave number of the wave representing the particle A is the constant amplitude

If ψ represents a single particle, |ψ|2 is the relative probability per unit volume that the particle will be found at any given point in the volume 

|ψ|2 is called the probability density

Wave Function of a Free Particle, cont. 

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In general, the probability of finding the particle in a volume dV is |ψ|2 dV With one-dimensional analysis, this becomes |ψ| 2 dx The probability of finding the particle in the arbitrary interval a ≤ x ≤ b is b

Pψ ab  dx 

2

a

and is the area under the curve

Wave Function of a Free Particle, Final 

Because the particle must be somewhere along the x axis, the sum of all the probabilities over all values of x must be 1 

Pψ ab  dx 



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2

1

Any wave function satisfying this equation is said to be normalized Normalization is simply a statement that the particle exists at some point in space

Expectation Values  

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ψ is not a measurable quantity Measurable quantities of a particle can be derived from ψ The average position is called the expectation value of x and is defined as  xψ xψdx* 

Expectation Values, cont. 

The expectation value of any function of x can also be found 

f  xψ *   f  x ψdx 

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

The expectation values are analogous to weighted averages

Summary of Mathematical Features of a Wave Function 

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ψ(x) may be a complex function or a real function, depending on the system ψ(x) must be defined at all points in space and be single-valued ψ(x) must be normalized ψ(x) must be continuous in space 

There must be no discontinuous jumps in the value of the wave function at any point

Particle in a Box 

A particle is confined to a one-dimensional region of space 

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The “box” is onedimensional

The particle is bouncing elastically back and forth between two impenetrable walls separated by L

Potential Energy for a Particle in a Box 

As long as the particle is inside the box, the potential energy does not depend on its location 

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We can choose this energy value to be zero

The energy is infinitely large if the particle is outside the box 

This ensures that the wave function is zero outside the box

Wave Function for the Particle in a Box 

Since the walls are impenetrable, there is zero probability of finding the particle outside the box 

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ψ(x) = 0 for x < 0 and x > L

The wave function must also be 0 at the walls  

The function must be continuous ψ(0) = 0 and ψ(L) = 0

Wave Function of a Particle in a Box – Mathematical 

The wave function can be expressed as a real, sinusoidal function  2πx  ψ (x )  A sin   λ  

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Applying the boundary conditions and using the de Broglie wavelength  nπx  ψ(x )  A sin   L  

Graphical Representations for a Particle in a Box

Active Figure 41.4

(SLIDESHOW MODE ONLY)

Wave Function of the Particle in a Box, cont. 

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Only certain wavelengths for the particle are allowed |ψ|2 is zero at the boundaries |ψ|2 is zero at other locations as well, depending on the values of n The number of zero points increases by one each time the quantum number increases by one

Momentum of the Particle in a Box 

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Remember the wavelengths are restricted to specific values Therefore, the momentum values are also restricted h nh p  λ 2L

Energy of a Particle in a Box 

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We chose the potential energy of the particle to be zero inside the box Therefore, the energy of the particle is just its kinetic energy 

h2  2 En   n 2   8mL  

n  1, 2, 3 ,K

The energy of the particle is quantized

Energy Level Diagram – Particle in a Box 

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The lowest allowed energy corresponds to the ground state En = n2E1 are called excited states E = 0 is not an allowed state The particle can never be at rest

Active Figure 41.5

(SLIDESHOW MODE ONLY)

Boundary Conditions 

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Boundary conditions are applied to determine the allowed states of the system In the model of a particle under boundary conditions, an interaction of a particle with its environment represents one or more boundary conditions and, if the interaction restricts the particle to a finite region of space, results in quantization of the energy of the system In general, boundary conditions are related to the coordinates describing the problem

Erwin Schrödinger  

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1887 – 1961 Best known as one of the creators of quantum mechanics His approach was shown to be equivalent to Heisenberg’s Also worked with:   

statistical mechanics color vision general relativity

Schrödinger Equation 

The Schrödinger equation as it applies to a particle of mass m confined to moving along the x axis and interacting with its environment through a potential energy function U(x) is 2

2

h dψ   Uψ  Eψ 2 2m dx 

This is called the time-independent Schrödinger equation

Schrödinger Equation, cont. 

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Both for a free particle and a particle in a box, the first term in the Schrödinger equation reduces to the kinetic energy of the particle multiplied by the wave function Solutions to the Schrödinger equation in different regions must join smoothly at the boundaries

Schrödinger Equation, final  

ψ(x) must be continuous ψ(x) must approach zero as x approaches ±∞ 

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This is needed so that ψ(x) obeys the normalization condition

dψ/dx must also be continuous for finite values of the potential energy

Solutions of the Schrödinger Equation 

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Solutions of the Schrödinger equation may be very difficult The Schrödinger equation has been extremely successful in explaining the behavior of atomic and nuclear systems 

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Classical physics failed to explain this behavior

When quantum mechanics is applied to macroscopic objects, the results agree with classical physics

Potential Wells 

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A potential well is a graphical representation of energy The well is the upward-facing region of the curve in a potential energy diagram The particle in a box is sometimes said to be in a square well 

Due to the shape of the potential energy diagram

Schrödinger Equation Applied to a Particle in a Box 

In the region 0 < x < L, where U = 0, the Schrödinger equation can be expressed in the form 2 dψ 2mE 2   ψ   k ψ 2 2 dx h

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The most general solution to the equation is ψ(x) = A sin kx + B cos kx 

A and B are constants determined by the boundary and normalization conditions

Schrödinger Equation Applied to a Particle in a Box, cont. 

Solving for the allowed energies gives 

h2  2 En   n 2   8mL  

The allowed wave functions are given by 2  nπx   nπx  ψn (x )  A sin  sin    L  L   L  

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The second expression is the normalized wave function These match the original results for the particle in a box

Finite Potential Well 

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A finite potential well is pictured The energy is zero when the particle is 0 < x
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In region II

The energy has a finite value outside this region 

Regions I and III

Finite Potential Well – Region II   

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U=0 The allowed wave functions are sinusoidal The boundary conditions no longer require that ψ be zero at the ends of the well The general solution will be ψII(x) = F sin kx + G cos kx 

where F and G are constants

Finite Potential Well – Regions I and III 

The Schrödinger equation for these regions may be written as 2 2m  U  E  dψ 2  ψ  C ψ 2 2 dx h

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The general solution of this equation is Cx Cx ψ  Ae  Be 

A and B are constants

Finite Potential Well – Regions I and III, cont. 

In region I, B = 0 

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This is necessary to avoid an infinite value for ψ for large negative values of x

In region III, A = 0 

This is necessary to avoid an infinite value for ψ for large positive values of x

Finite Potential Well – Graphical Results for ψ 

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The wave functions for various states are shown Outside the potential well, classical physics forbids the presence of the particle Quantum mechanics shows the wave function decays exponentially to approach zero

Finite Potential Well – Graphical Results for ψ2 

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The probability densities for the lowest three states are shown The functions are smooth at the boundaries

Active Figure 41.8

(SLIDESHOW MODE ONLY)

Finite Potential Well – Determining the Constants 

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The constants in the equations can be determined by the boundary conditions and the normalization condition The boundary conditions are ψI  ψII ψII  ψIII

dψI dψII and  at x  0 dx dx dψII dψIII and  at x  L dx dx

Application – Nanotechnology 

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Nanotechnology refers to the design and application of devices having dimensions ranging from 1 to 100 nm Nanotechnology uses the idea of trapping particles in potential wells One area of nanotechnology of interest to researchers is the quantum dot 

A quantum dot is a small region that is grown in a silicon crystal that acts as a potential well

Tunneling 

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The potential energy has a constant value U in the region of width L and zero in all other regions This a called a square barrier U is the called the barrier height

Tunneling, cont. 

Classically, the particle is reflected by the barrier 

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Regions II and III would be forbidden

According to quantum mechanics, all regions are accessible to the particle 

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The probability of the particle being in a classically forbidden region is low, but not zero According to the uncertainty principle, the particle can be inside the barrier as long as the time interval is short and consistent with the principle

Tunneling, final 

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The curve in the diagram represents a full solution to the Schrödinger equation Movement of the particle to the far side of the barrier is called tunneling or barrier penetration The probability of tunneling can be described with a transmission coefficient, T, and a reflection coefficient, R

Tunneling Coefficients 

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The transmission coefficient represents the probability that the particle penetrates to the other side of the barrier The reflection coefficient represents the probability that the particle is reflected by the barrier T+R=1  

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The particle must be either transmitted or reflected T ≈ e-2CL and can be nonzero

Tunneling is observed and provides evidence of the principles of quantum mechanics

Applications of Tunneling 

Alpha decay 

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In order for the alpha particle to escape from the nucleus, it must penetrate a barrier whose energy is several times greater than the energy of the nucleusalpha particle system

Nuclear fusion 

Protons can tunnel through the barrier caused by their mutual electrostatic repulsion

More Applications of Tunneling – Resonant Tunneling Device

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Electrons travel in the gallium arsenide semiconductor They strike the barrier of the quantum dot from the left The electrons can tunnel through the barrier and produce a current in the device

Active Figure 41.10

(SLIDESHOW MODE ONLY)

More Applications of Tunneling – Scanning Tunneling Microscope 

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An electrically conducting probe with a very sharp edge is brought near the surface to be studied The empty space between the tip and the surface represents the “barrier” The tip and the surface are two walls of the “potential well”

Scanning Tunneling Microscope 

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The STM allows highly detailed images of surfaces with resolutions comparable to the size of a single atom At right is the surface of graphite “viewed” with the STM

Scanning Tunneling Microscope, final 

The STM is very sensitive to the distance from the tip to the surface 

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This is the thickness of the barrier

STM has one very serious limitation 

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Its operation is dependent on the electrical conductivity of the sample and the tip Most materials are not electrically conductive at their surfaces The atomic force microscope overcomes this limitation

Simple Harmonic Oscillator 

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Reconsider black body radiation as vibrating charges acting as simple harmonic oscillators The potential energy is U = ½ kx2 = ½ mω2x2 Its total energy is K + U = ½ kA2 = ½ mω2A2

Simple Harmonic Oscillator, 2 

The Schrödinger equation for this problem is 2 h2 dψ 1 2 2   mω x ψ  Eψ 2 2m dx 2

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The solution of this equation gives the wave function of the ground state as

ψ  Be

 mω 2 h x 2

Simple Harmonic Oscillator, 3 

The remaining solutions that describe the excited states all include the exponential function 2

e

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Cx

The energy levels of the oscillator are quantized The energy for an arbitrary quantum number n is En = (n + ½) ω where n = 0, 1, 2,…

Energy Level Diagrams – Simple Harmonic Oscillator 

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The brown curves represent probability densities for the first three states The blue curves represent the classical probability densities corresponding to the same energies

Energy Levels in the Harmonic Oscillator 

The state n = 0 corresponds to the ground state 

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The energy is Eo = ½ ω

The state n = 1 is the first excited state The separations between adjacent levels are equal and are given by ∆E = ω As n increases, the agreement between the classical and the quantum-mechanical results improve