Chapter 41

Quantum Mechanics

Quantum Mechanics

The theory of quantum mechanics was developed in the 1920s.

By Erwin Schrödinger, Werner Heisenberg and others

Enables us to understand various phenomena involving

Atoms, molecules, nuclei and solids

The discussion will follow from the quantum particle model and will incorporate some of the features of waves under boundary conditions.

Introduction

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.

ProbabilityI

N

V V∝ ∝

Section 41.1

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

2ProbabilityE

V∝

2I E∝

Section 41.1

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 ψ

Section 41.1

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

rj is the position of the jth particle in the system

ω = 2πƒ is the angular frequency

1i = −

( ) ( ), −Ψ = + + + + = Ψr r r r r

K K1 2 3i ù t

j jt er r r r r

Section 41.1

Wave Function, cont.

The wave function is often complex-valued.

The absolute square |ψ|2 = ψ*ψ is always real and positive.

ψ* is the complex 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.

Section 41.1

Wave Function Interpretation – Single Particle

cannot be measured.

2 is real and can be measured.

2 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 = ||2 dV

Section 41.1

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.

The wave function is most properly associated with a system.

The function is determined by the particle and its interaction with its environment.

Since, in many cases, the particle is the only part of the system that experiences a change, common language associates the wave function with the particle.

There are some examples in which it is important to think of the system wave function instead of the particle wave function.

Section 41.1

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 .

A is the constant amplitude.

k = 2πλ is the angular wave number of the wave representing the particle.

Section 41.1

Wave Function of a Free Particle, cont.

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

and is the area under the curve.

2b

ab aP ø dx=∫

Section 41.1

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.

21abP ø dx

∞

−∞= =∫

Section 41.1

Expectation Values

Measurable quantities of a particle can be derived from ψ.

Remember, ψ 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

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

The expectation values are analogous to weighted averages.

x ø xødx∞

−∞≡∫ *

( ) ( )*f x ø f x ødx∞

−∞= ∫

Section 41.1

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.

Classically, it can be modeled as a particle under constant speed.

If the particle’s speed is constant, so are its kinetic energy and its momentum.

Classical physics places no restrictions on these values.

Section 41.2

Potential Energy for a Particle in a Box

The quantum-mechanical approach requires finding the appropriate wave function consistent with the conditions of the situation.

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.

Section 41.2

Wave Function for the Particle in a Box – Boundary Conditions

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

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

Only wave functions that satisfy these boundary conditions are allowed.

Section 41.2

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

2( ) sin

∂ xø x A

ë⎛ ⎞= ⎜ ⎟⎝ ⎠

( ) sinn∂ x

ø x AL

⎛ ⎞= ⎜ ⎟⎝ ⎠

Section 41.2

Graphical Representations for a Particle in a Box

Section 41.2

Wave Function of the Particle in a Box, cont.

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.

Section 41.2

Momentum of the Particle in a Box

Remember the wavelengths are restricted to specific values

= 2 L / n

Therefore, the momentum values are also restricted.

.2

h nhp

ë L= =

Section 41.2

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.

22

21 2 3

8, , ,n

hE n n

mL

⎛ ⎞= =⎜ ⎟⎝ ⎠

K

Section 41.2

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.

Section 41.2

Boundary Conditions – General

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.

Section 41.2

Erwin Schrödinger

1887 – 1961

American physicist

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

Section 41.3

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.

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.

2 2

22

d øUø Eø

m dx− + =h

Section 41.3

Schrödinger Equation, final

Once you have a preliminary solution to the Schrödinger equation, the following condition can be imposed to find the exact solution and the allowed energies.

ψ(x) must be normalized.

ψ(x) must go to 0 as x → ±∞ and remain finite as x → 0.

ψ(x) must be continuous in x and be single-valued everywhere.

Solutions in different regions must join smoothly at the boundaries between the regions.

dψ/dx must be finite, continuous, and single-valued everywhere for finite values of the potential energy.

Section 41.3

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.

Section 41.3

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.

A particle moves only horizontally at a fixed vertical position in a potential energy diagram, representing the conserved energy of the system of the particle and its environment.

The particle in a box is sometimes said to be in a square well.

Due to the shape of the potential energy diagram.

Section 41.3

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:

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.

Applying the boundary conditions gives B = 0 and k L = n π where n is an integer.

22

2 2

2 2d ø mE mEø k ø where k

dx=− =− =

h h

Section 41.3

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.

22

28n

hE n

mL

⎛ ⎞=⎜ ⎟⎝ ⎠

( ) sinn

n∂ xø x A

L⎛ ⎞= ⎜ ⎟⎝ ⎠

Section 41.3

Finite Potential Well

A finite potential well is pictured.

The energy is zero when the particle is 0 < x < L

In region II

The energy has a finite value outside this region.

Regions I and III

Section 41.4

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 < 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.

Section 41.4

Finite Potential Well – Region II

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

Section 41.4

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

( )2

2 2

2m U Ed øø

dx

−=

h

Cx Cxø Ae Be−= +

Section 41.4

Finite Potential Well – Regions I and III, cont.

In region I, B = 0

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.

The solutions of the wave equation become

0Cx CxI IIIAe for x and Be for x L− = < = >

Section 41.4

Finite Potential Well – Graphical Results for ψ

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.

Section 41.4

Finite Potential Well – Graphical Results for ψ2

The probability densities for the lowest three states are shown.

The functions are smooth at the boundaries.

Section 41.4

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

I III II

II IIIII III

and at 0

and at

dø døø ø x

dx dxdø dø

ø ø x Ldx dx

= = =

= = =

Section 41.4

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.

The storage of binary information using quantum dots is an active field of research.

Section 41.4

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.

Section 41.5

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.

If the barrier is relatively narrow, this short time interval can allow the particle to pass through the barrier.

Section 41.5

Tunneling, final

The curve in fig. 41.8 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.

Section 41.5

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.

Section 41.5

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.

Section 41.6

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”.

Section 41.6

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.

Section 41.6

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.

Section 41.6

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.

Section 41.6

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.

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.

Section 41.6

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.

Section 41.6

Simple Harmonic Oscillator

Classically, a particle that is subject to a linear force F = - k x is said to be in simple harmonic motion.

The potential energy is

U = ½ kx2 = ½ mω2x2

Its total energy is

E = K + U = ½ kA2 = ½ mω2A2

Any value of E is allowed, including E = 0.

Section 41.7

Simple Harmonic Oscillator, 2

The simple harmonic oscillator can also be treated from a quantum mechanical point of view.

The Schrödinger equation for this problem is

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

2 22 2

2

1

2 2

d ømù x ø Eø

m dx− + =h

( ) 22mù xø Be−= h

Section 41.7

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) where n = 0, 1, 2,…

2Cxe−

Energy Level Diagrams – Simple Harmonic Oscillator

The separations between adjacent levels are equal and equal to E =

The energy levels are equally spaced.

The state n = 0 corresponds to the ground state.

The energy is Eo = ½ ω.

Agrees with Planck’s original equations

Section 41.7