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Chapter 41

- Quantum Mechanics

Quantum Mechanics

- The theory of quantum mechanics was developed in

the 1920s - By Erwin Schrödinger, Werner Heisenberg and

others - Enables use to understand various phenomena

involving - Atoms, molecules, nuclei and solids

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 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 - Combining the points of view gives

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 - 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 ?

Wave Function

- The complete wave function ? for a system depends

on the positions of all the particles in the

system and on time - The function can be written as
- rj is the position of the jth particle in the

system - ? 2pƒ is the angular frequency

Wave Function, cont.

- The wave function is often complex-valued
- The absolute square ?2 ?? is always real and

positive - ? is the complete conjugate of ?
- 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 Interpretation Single Particle

- Y cannot be measured
- Y2 is real and can be measured
- Y2 is also called the probability density
- The relative probability per unit volume that the

particle will be found at any given point in the

volume - If dV is a small volume element surrounding some

point, the probability of finding the particle in

that volume element is - P(x, y, z) dV Y 2 dV

Wave Function, General Comments, Final

- 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 ?(x) Aeikx - A is the constant amplitude
- k 2p/? is the angular wave number of the wave

representing the particle - Although the wave function is often associated

with the particle, it is more properly determined

by the particle and its interaction with its

environment - Think of the system wave function instead of the

particle wave function

Wave Function of a Free Particle, cont.

- In general, the probability of finding the

particle in a volume dV is ?2 dV - With one-dimensional analysis, this becomes ?2

dx - The probability of finding the particle in the

arbitrary interval a x b is - 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 - 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

- Measurable quantities of a particle can be

derived from ? - Remember, ? is not a measurable quantity
- Once the wave function is known, it is possible

to calculate the average position you would

expect to find the particle after many

measurements - The average position is called the expectation

value of x and is defined as

Expectation Values, cont.

- The expectation value of any function of x can

also be found - The expectation values are analogous to weighted

averages

Summary of Mathematical Features of a Wave

Function

- ?(x) may be a complex function or a real

function, depending on the system - ?(x) must be defined at all points in space and

be single-valued - ?(x) must be normalized
- ?(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 - The box is one- dimensional
- The particle is bouncing elastically back and

forth between two impenetrable walls separated by

L

Please replace with fig. 41.3 a

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 - 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 - ?(x) 0 for x lt 0 and x gt L
- The wave function must also be 0 at the walls
- The function must be continuous
- ?(0) 0 and ?(L) 0

Wave Function of a Particle in a Box

Mathematical

- The wave function can be expressed as a real,

sinusoidal function - Applying the boundary conditions and using the de

Broglie wavelength

Graphical Representations for a Particle in a Box

Active Figure 41.4

- Use the active figure to measure the probability

of a particle being between two points for three

quantum states

PLAY ACTIVE FIGURE

Wave Function of the Particle in a Box, cont.

- Only certain wavelengths for the particle are

allowed - ?2 is zero at the boundaries
- ?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

- Remember the wavelengths are restricted to

specific values - l 2 L / n
- Therefore, the momentum values are also

restricted

Energy of a Particle in a Box

- 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 - The energy of the particle is quantized

Energy Level Diagram Particle in a Box

- 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

- Use the active figure to vary
- The length of the box
- The mass of the particle
- Observe the effects on the energy level diagram

PLAY ACTIVE FIGURE

Boundary Conditions

- 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

- 1887 1961
- American physicist
- Best known as one of the creators of quantum

mechanics - His approach was shown to be equivalent to

Heisenbergs - 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 - This is called the time-independent Schrödinger

equation

Schrödinger Equation, cont.

- 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

- ?(x) must be continuous
- d?/dx must also be continuous for finite values

of the potential energy

Solutions of the Schrödinger Equation

- 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 - Classical physics failed to explain this behavior
- When quantum mechanics is applied to macroscopic

objects, the results agree with classical physics

Potential Wells

- 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 lt x lt L, where U 0, the

Schrödinger equation can be expressed in the form

- The most general solution to the equation is ?(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
- The allowed wave functions are given by
- These match the original results for the particle

in a box

Finite Potential Well

- A finite potential well is pictured
- The energy is zero when the particle is 0 lt x lt L
- In region II
- The energy has a finite value outside this region

- Regions I and III

Classical vs. Quantum Interpretation

- According to Classical Mechanics
- If the total energy E of the system is less than

U, the particle is permanently bound in the

potential well - If the particle were outside the well, its

kinetic energy would be negative - An impossibility
- According to Quantum Mechanics
- A finite probability exists that the particle can

be found outside the well even if E lt U - The uncertainty principle allows the particle to

be outside the well as long as the apparent

violation of conservation of energy does not

exist in any measurable way

Finite Potential Well Region II

- U 0
- The allowed wave functions are sinusoidal
- The boundary conditions no longer require that ?

be zero at the ends of the well - The general solution will be
- ?II(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 - The general solution of this equation is
- A and B are constants

Finite Potential Well Regions I and III, cont.

- In region I, B 0
- This is necessary to avoid an infinite value for

? for large negative values of x - In region III, A 0
- This is necessary to avoid an infinite value for

? for large positive values of x - The solutions of the wave equation become

Finite Potential Well Graphical Results for ?

- 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 ?2

- The probability densities for the lowest three

states are shown - The functions are smooth at the boundaries

Active Figure 41.7

- Use the active figure to adjust the length of the

box - See the effect on the quantized states

PLAY ACTIVE FIGURE

Finite Potential Well Determining the Constants

- The constants in the equations can be determined

by the boundary conditions and the normalization

condition - The boundary conditions are

Application Nanotechnology

- 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

- 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 - Regions II and III would be forbidden
- According to quantum mechanics, all regions are

accessible to the particle - 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

- 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

- 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
- 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
- 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 nucleus-alpha particle system - Nuclear fusion
- Protons can tunnel through the barrier caused by

their mutual electrostatic repulsion

More Applications of Tunneling Scanning

Tunneling Microscope

- 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

- 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 - This is the thickness of the barrier
- STM has one very serious limitation
- 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

More Applications of Tunneling Resonant

Tunneling Device

- The gallium arsenide in the center is a quantum

dot - It is located between two barriers formed by the

thin extensions of aluminum arsenide

Resonance Tunneling Devices, cont

- Figure b shows the potential barriers and the

energy levels in the quantum dot - The electron with the energy shown encounters the

first barrier, it has no energy levels available

on the right side of the barrier - This greatly reduces the probability of tunneling

Resonance Tunneling Devices, final

- Applying a voltage decreases the potential with

position - The deformation of the potential barrier results

in an energy level in the quantum dot - The resonance of energies gives the device its

name

Active Figure 41.11

- Use the active figure to vary the voltage

PLAY ACTIVE FIGURE

Resonant Tunneling Transistor

- This adds a gate electrode at the top of the

resonant tunneling device over the quantum dot - It is now a resonant tunneling transistor
- There is no resonance
- Applying a small voltage reestablishes resonance

Simple Harmonic Oscillator

- Reconsider black body radiation as vibrating

charges acting as simple harmonic oscillators - The potential energy is
- U ½ kx2 ½ m?2x2
- Its total energy is
- E K U ½ kA2 ½ m?2A2

Simple Harmonic Oscillator, 2

- The Schrödinger equation for this problem is
- The solution of this equation gives the wave

function of the ground state as

Simple Harmonic Oscillator, 3

- The remaining solutions that describe the excited

states all include the exponential function - The energy levels of the oscillator are quantized
- The energy for an arbitrary quantum number n is

En (n ½)?w where n 0, 1, 2,

Energy Level Diagrams Simple Harmonic Oscillator

- The separation between adjacent levels are equal

and equal to DE ?w - The energy levels are equally spaced
- The state n 0 corresponds to the ground state
- The energy is Eo ½ h?
- Agrees with Plancks original equations