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Copyright © 2012 Pearson Education Inc.

PowerPoint® Lectures for

University Physics, Thirteenth Edition

– Hugh D. Young and Roger A. Freedman

Chapter 40

Quantum Mechanics


Goals for Chapter 40

• To introduce and interpret the Schrödinger wave equation

for quantum-mechanical waves

• To solve the Schrödinger equation for a one-dimensional

“particle in a box”

• To study the behavior of a quantum-mechanical particle in

a finite potential well

• To examine tunneling, in which quantum mechanics allows

a particle to travel through a region that would be forbidden

by Newtonian physics

• To consider the quantum-mechanical harmonic oscillator, a

model for molecular vibrations


Introduction

• Just as we use the wave equation to analyze waves on a string or sound waves in a pipe, we can use a related equation—the Schrödinger equation—to analyze the behavior of matter from a quantum-mechanical perspective.

• Microscopic beads of different sizes fluoresce under ultraviolet light (see the figure at right); the smaller the beads, the shorter the wavelength of visible light they emit. The Schrödinger equation will help us understand why.


The Schrödinger equation in 1-D

• In a one-dimensional model, a quantum-mechanical particle is described

by a wave function (x, t).

• The one-dimensional Schrödinger equation for a free particle of mass m

is

• The presence of i (the square root of –1) in the Schrödinger equation

means that wave functions are always complex functions.

• The square of the absolute value of the wave function, |(x, t)|2, is

called the probability distribution function. It tells us about the

probability of finding the particle near position x at time t.

h2

2m

2 x,t x2

ih x,t

t


The Schrödinger equation in 1-D: A free particle

• If a free particle has definite

momentum p and definite

energy E, its wave function

(see Figure 40.3 at right) is

• Such a particle is not

localized at all: The wave

function extends to infinity.

x,t Aeikxeit

where p hk and E h.


The Schrödinger equation in 1-D: Wave packets

• A free-particle wave packet

(see Figure 40.6 at right) is a

superposition of states of

definite momentum and

energy.

• To make the wave packet

more localized in space, the

greater the range of momenta

and energies it must include.

• Follow Example 40.1.


The Schrödinger equation in 1-D: Stationary states

• If a particle of mass m moves in the presence of a potential energy function U(x), the one-dimensional Schrödinger equation for the particle is

• If the particle has a definite energy E, the wave function (x, t) is a product of a time-independent wave function (x) and a factor that depends on time t but not position. For such a stationary state the probability distribution function |(x, t)|2 = |(x)|2 does not depend on time.

• The time-independent one-dimensional Schrödinger equation for a stationary state of energy E is


22

2

, ,,

2

x t x tU x x t i

m x t

/, iEtx t x e

22

22

d xU x x E x

m dx


Particle in a box

• A “particle in a box” is a particle of mass m confined to a region

between x = 0 and x = L (see Figure 40.8 at lower left). The potential

energy is zero inside the “box” and infinite outside (see Figure 40.9 at

lower right).


Particle in a box: Wave functions, energy levels

• Figure 40.11 (below) illustrates the energy levels and associated

stationary-state wave functions (x) for a particle in a box.



Particle in a box: Probability and normalization

• Figure 40.12 at right shows the first three stationary-state wave functions (x) for a particle in a box (top) and the associated probability distribution functions |(x)|2 (bottom). There are locations where there is zero probability of finding the particle.

• Wave functions must be normalized so that the integral of |(x)|2 over all x equals 1 (which means there is 100% probability of finding the particle somewhere).



Q40.3

A. least for n = 1.

B. least for n = 2 and n = 4.

C. least for n = 5.

D. the same (and nonzero)

for n = 1, 2, 3, 4, and 5.

E. zero for n = 1, 2, 3, 4,

and 5.

The first five wave

functions for a particle in

a box are shown. The

probability of finding the

particle near x = L/2 is


A40.3

A. least for n = 1.

B. least for n = 2 and n = 4.

C. least for n = 5.


for n = 1, 2, 3, 4, and 5.

E. zero for n = 1, 2, 3, 4,

and 5.

The first five wave



probability of finding the

particle near x = L/2 is


Q40.4

The first five wave



average value of the x-

component of momentum

is A. least for n = 1.

B. least for n = 5.

C. the same (and

nonzero) for n = 1 and

n = 5.

D. zero for both n = 1

and n = 5.


A40.4

The first five wave



average value of the x-

component of momentum

is A. least for n = 1.

B. least for n = 5.

C. the same (and

nonzero) for n = 1 and n

= 5.

D. zero for both n = 1

and n = 5.


Q40.5

The first five wave


a box are shown.

Compared to the n = 1

wave function, the n = 5

wave function has

A. the same kinetic energy

(KE).

B. 5 times more KE.

C. 25 times more KE.

D. 125 times more KE.

E. none of the above


A40.5

The first five wave


a box are shown.

Compared to the n = 1

wave function, the n = 5

wave function has

A. the same kinetic energy

(KE).

B. 5 times more KE.

C. 25 times more KE.

D. 125 times more KE.

E. none of the above


Particle in a finite potential well I

• A finite potential well is a region where potential energy U(x) is

lower than outside the well, but U(x) is not infinite outside the

well (see Figure 40.13 below).

• In Newtonian physics, a particle whose energy E is less than the

“height” of the well can never escape the well. In quantum

mechanics the wave function of such a particle extends beyond the

well, so it is possible to find

the particle outside the well.



Particle in a finite potential well II

• Figure 40.15 (below left) shows the stationary-state wave functions (x) and corresponding energies for one particular finite well.

• Figure 40.16 (below right) shows the corresponding probability distribution functions |(x)|2.



Q40.6

The first three wave

functions for a finite

square well are shown.

The probability of finding

the particle at x > L is

A. least for n = 1.

B. least for n = 2.

C. least for n = 3.


for n = 1, 2, and 3.

E. zero for n = 1, 2, and 3.


A40.6

The first three wave

functions for a finite

square well are shown.

The probability of finding

the particle at x > L is

A. least for n = 1.

B. least for n = 2.

C. least for n = 3.


for n = 1, 2, and 3.

E. zero for n = 1, 2, and 3.


Potential barriers and tunneling

• Figure 40.19 (below left) shows a potential barrier. In Newtonian physics, a particle whose energy E is less than the barrier height U0 cannot pass from the left-hand side of the barrier to the right-hand side.

• Figure 40.20 (below right) shows the wave function (x) for such a particle. The wave function is nonzero to the right of the barrier, so it is possible for the particle to “tunnel” from the left-hand side to the right-hand side.



Q40.7

A potential-energy

function is shown. If a

quantum-mechanical

particle has energy E <

U0, it is impossible to

find the particle in the

region

A. x < 0.

B. 0 < x < L.

C. x > L.

D. misleading question—

the particle can be found at

any x


A40.7

A potential-energy

function is shown. If a

quantum-mechanical

particle has energy E <

U0, it is impossible to

find the particle in the

region

A. x < 0.

B. 0 < x < L.

C. x > L.

D. misleading question—

the particle can be found at

any x


Applications of tunneling

• A scanning tunneling microscope measures the atomic topography of a surface. It does this by measuring the current of electrons tunneling between the surface and a probe with a sharp tip (see Figure 40.21 below).

• An alpha particle inside an unstable nucleus can only escape via tunneling (see Figure 40.22 at right).


A comparison of Newtonian and quantum oscillators

• Figure 40.26 (below, top) shows the first four stationary-state wave

functions (x) for the harmonic oscillator. A is the amplitude of oscillation in Newtonian physics.

• Figure 40.27 (below, bottom) shows the corresponding probability

distribution functions |(x)|2. The blue curves are the Newtonian probability distributions.


Q40.8

The figure shows the first

six energy levels of a

quantum-mechanical

harmonic oscillator. The

corresponding wave

functions

A. are nonzero outside the region

allowed by Newtonian mechanics.

B. do not have a definite

wavelength.

C. are all equal to zero at x = 0.

D. Both A. and B. are true.

E. All of A., B., and C. are true.


A40.8

The figure shows the first

six energy levels of a

quantum-mechanical

harmonic oscillator. The

corresponding wave

functions

A. are nonzero outside the region

allowed by Newtonian mechanics.

B. do not have a definite

wavelength.

C. are all equal to zero at x = 0.

D. Both A. and B. are true.

E. All of A., B., and C. are true.


A particle in a potential well emits a photon when it drops

from the n = 3 energy level to the n = 2 energy level. The

particle then emits a second photon when it drops from the

n = 2 energy level to the n = 1 energy level. The first photon

has the same energy as the second photon. What kind of

potential well could this be?

Q40.9

A. an infinitely deep square potential well (particle in a

box)

B. a harmonic oscillator

C. either A. or B.

D. neither A. nor B.


A particle in a potential well emits a photon when it drops

from the n = 3 energy level to the n = 2 energy level. The

particle then emits a second photon when it drops from the

n = 2 energy level to the n = 1 energy level. The first photon

has the same energy as the second photon. What kind of

potential well could this be?

A40.9

A. an infinitely deep square potential well (particle in a

box)

B. a harmonic oscillator

C. either A. or B.

D. neither A. nor B.

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