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A MODER N APPROACH TO
QUANTUM MECHA NICS

John S. Townsend
Professor of Physics
Harvey Mudd College
Claremont, California

University Science Books
Sausalito, California

University Science Books
55D Gate Five Road
Sausalito, CA 94965
Fax: (415) 332-5393
www.uscibooks.com

This book is printed on acid-free paper.

Copyright© 2000 by University Science Books
ISBN 10: 1-891389-13-0
ISBN 13: 978-1-891389-13-9

Reproduction or translation of any part of this work beyond
that permitted by Section 107 or 108 of the 1976 United States
Copyright Act without the permission of the copyright owner
is unlawful. Requests for permission or further information
should be addressed to the Permissions Department,
University Science Books.

Library of Congress Cataloging-in-Publication Data
Townsend, John S.
A modem approach to quantum mechanics I John S. Townsend
p. cm
Includes bibliographic references and index.
ISBN 1-891389-13-0 (acid-free paper)
I. Quantum theory. I. Title
QC174.12.T69 2000
530. l 2-dc21

Printed in the United States of America
10 9 8 7 6 5 4 3

99-058197

ABOUT THE AUTHOR

John S. Townsend was born in Boston, Massachusetts, in 1946 and grew up in Chambersburg, Pennsylvania. He received his B.S. degree summa cum laude in physics from
Duke University and his Ph.D. in physics from Johns Hopkins University, where he held
an NSF Graduate Fellowship. After postdoctoral positions at Johns Hopkins and the
Stanford Linear Accelerator Center, he joined the physics department at Harvey Mudd
College, the science and engineering college of The Claremont Colleges, where he has
taught since 1975. Professor Townsend's primary research area is in theoretical particle
physics. On a recent sabbatical leave, he was a Science Fellow at the center for International Security and Arms Control at Stanford University, and a visiting professor at Duke
University. Professor Townsend is married, has two daughters, and enjoys playing tennis.

v

CONTENTS

Preface

xi

1 Stem-Gerlach Experiments
1.1
l.2

1.3
1.4
1.5
1.6

The Original Stem-Gerlach Experiment
Four Experiments
The Quantum State Vector
Analysis of Experiment 3
Experiment 5
Summary

2 Rotation of Basis States and Matrix Mechanics
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8

3
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8

1

5
9
13
15
18
24

The Beginnings of Matrix Mechanics
Rotation Operators
The Identity and Projection Operators
Matrix Representations of Operators
Changing Representations
Expectation Values
Photon Polarization and the Spin of the Photon
Summary

24
28
36
41
45
50
51
56

Angular Momentum

64

Rotations Do Not Commute and Neither Do the Generators
Commuting Operators
The Eigenvalues and Eigenstates of Angular Momentum
The Matrix Elements of the Raising and Lowering Operators
Uncertainty Relations and Angular Momentum
The Spin-t Eigenvalue Problem
A Stem-Gerlach Experiment with Spin- I Particles
Summary

64
69
70
77
78
80
85
88

viii

CONTENTS

4
4.1
4.2
4.3
4.4
4.5
4.6
4.7

5
5.1
5.2
5.3
5.4
5.5
5.6

6
6.1
6.2
6.3
6.4
6.5
6.6

6.7
6.8
6.9
6.10
6.11

7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11

8
8.1
8.2

Time Evolution
The Hamiltonian and the SchrOdinger Equation
Time Dependence of Expectation Values
Precession of a Spin-! Particle in a Magnetic Field
Magnetic Resonance
The Ammonia Molecule and the Ammonia Maser
The Energy-Time Uncertainty Relation
Summary

A System of 1Wo Spin-! Particles
The Basis States for a System of Tho Spin-~ Particles
The Hyperfine Splitting of the Ground State of Hydrogen
The Addition of Angular Momenta for Tho Spin-! Particles
The Einstein-Podolsky-Rosen Paradox
A Nonquantum Model and the Bell Inequalities
Summary

Wave Mechanics in One Dimension
Position Eigenstates and the Wave Function
The Translation Operator
The Generator of Translations
The Momentum Operator in the Position Basis
Momentum Space
A Gaussian Wave Packet
The Heisenberg Uncertainty Principle
General Properties of Solutions to the Schrooinger Equation
in Position Space
The Particle in a Box
Scattering in One Dimension
Summary

The One-Dimensional Harmonic Oscillator
The Importance of the Harmonic Oscillator
Operator Methods
An Example: Torsional Oscillations of the Ethylene Molecule
Matrix Elements of the Raising and Lowering Operators
Position-Space Wave Functions
The Zero-Point Energy
The Classical Limit
Time Dependence
Solving the SchrOdinger Equation in Position Space
Inversion Symmetry and the Parity Operator
Summary

Path Integrals
The Multislit, Multiscreen Experiment
The Transition Amplitude

93
93
96
97
104
108
115
116
120
120
122
126
131
134
143
147
147
151
153
156
158
160

164
166
171
177
185
194
194
196
199
201
202
205
207
208
209
212
213
216
216
218

CONTENTS

8.3
8.4
8.5
8.6
8.7
8.8

Evaluating the Transition Amplitude for Short Time Intervals
The Path Integral
Evaluation of the Path Integral for a Free Particle
Why Some Particles Follow the Path of Least Action
Quantum Interference Due to Gravity
Summary

ix

219
221
224
226
231

233

9 Translational and Rotational Symmetry
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10

10
IO.I
10.2
10.3
10.4
10.5
10.6

11
11.1
I 1.2
11.3
11.4
I 1.5
I 1.6
11.7
11.8
11.9

12
12.1
12.2
12.3
12.4
12.5

in the Two-Body Problem

237

The Elements of Wave Mechanics in Three Dimensions
Translational Invariance and Conservation of Linear Momentum
Relative and Center-of-Mass Coordinates
Estimating Ground-State Energies Using the Uncertainty
Principle
Rotational Invariance and Conservation of Angular Momentum
A Complete Set of Commuting Observables
Vibrations and Rotations of a Diatomic Molecule
Position-Space Representations of L in Spherical Coordinates
Orbital Angular Momentum Eigenfunctions
Summary

237
241
244
246
248
250
254
260
263
268

Bound States of Central Potentials

274

The Behavior of the Radial Wave Function Near the Origin
The Coulomb Potential and the Hydrogen Atom
The Finite Spherical Well and the Deuteron
The Infinite Spherical Well
The Three-Dimensional Isotropic Harmonic Oscillator
Conclusion

274
277
288
292
296
302

Time-Independent Perturbations

306
306
31 l
314
316
319
322

Nondegenerate Perturbation Theory
An Example Involving the One-Dimensional Harmonic Oscillator
Degenerate Perturbation Theory
The Stark Effect in Hydrogen
The Ammonia Molecule in an External Electric Field Revisited
Relativistic Perturbations to the Hydrogen Atom
The Energy Levels of Hydrogen, Including Fine Structure,
the Lamb Shift, and Hyperfine Splitting
The Zeeman Effect in Hydrogen
Summary

Identical Particles
Indistinguishable Particles in Quantum Mechanics
The Helium Atom
Multielectmn Atoms and the Periodic Table
Covalent Bonding
Conclusion

331

334
335
341
341
345
355

360
366

x

CONTE!IITS

13 Scattering
13. l
13.2
13.3
13.4
13.5
13.6

The Asymptotic Wave Function and the Differential Cross Section
The Born Approximation
An Example of the Born Approximation: The Yukawa Potential
The Partial Wave Expansion
Examples of Phase-Shift Analysis
Summary

14 Photons and Atoms
14.1
14.2
14.3
14.4
14.5
14.6
14.7
14.8
14.9

The Aharonov-Bohm Effect
The Hamiltonian for the Electromagnetic Field
Quantizing the Radiation Field
The Properties of Photons
The Hamiltonian of the Atom and the Electromagnetic Field
Time-Dependent Perturbation Theory
Fermi's Golden Rule
Spontaneous Emission
Higher-Order Processes and Feynman Diagrams

368
368
375
379
381
385
393
399
399
404

409
410
414
417
425
430
437

Appendixes
A
B

c

D
E
F
G

444

Electromagnetic Units
The Addition of Angular Momenta
Dirac Delta Functions
Gaussian Integrals
The Lagrangian for a Charge q in a Magnetic Field
Values of Physical Constants
Answers to Selected Problems

449
453
457
460
463
465

Index

467

PREFACE

There have been two revolutions in the way we view the physical world in the
twentieth century: relativity and quantum mechanics. In quantum mechanics the
revolution has been both profound-requiring a dramatic revision in the structure
of the laws of mechanics that govern the behavior of all particles, be they electrons
or photons-and far-reaching in its impact-determining the stability of matter
itself, shaping the interactions of particles on the atomic, nuclear, and particle
physics level, and leading to macroscopic quantum effects ranging from lasers
and superconductivity to neutron stars and radiation from black holes. Moreover,
in a triumph for twentieth-century physics, special relativity and quantum mechanics have been joined together in the fonn of quantum field theory. Field theories
such as quantum electrodynamics have been tested with an extremely high precision, with agreement between theory and experiment verified to better than nine
significant figures. It should be emphasized that while our understanding of the
laws of physics is continually evolving, always being subjected to experimental scrutiny, so far no confirmed discrepancy between theory and experiment for
quantum mechanics has been detected.
This book is intended for an upper-division course in quantum mechanics.
The most likely audience for the book consists of students who have completed
a course in modem physics that includes an introduction to quantum mechanics in the form of one-dimensional wave mechanics. Such a modem physics
course is likely to emphasize the historical development of the subject. Rather
than continue with a similar approach in a second course, I have chosen to introduce the fundamentals of quantum mechanics through a detailed discussion of the
physics of intrinsic spin. Such an approach has a number of significant advantages. First, students find starting a course with something "new" such as intrinsic
spin both interesting and exciting, and they enjoy making connections with what
they have seen before. Second, spin systems provide us with many beautiful
but straightforward illustrations of the essential structure of quantum mechanics,
a structure that is not obscured by the mathematics of wave mechanics. Quanxi


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