Quantum Physics: A Fundamental Approach to Modern Physics

417 Pages·2010·83.57 MB·English

by  John Townsend

Quantum Physics
A Fundamental Approach to Modern Physics
John S. Townsend
HARVEY MUDD COLLEGE
-4>-
University Science Books
Sausalito, California
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Library of Congress Cataloging·in·Publieation Data
Townsend, John S.
Quantum physics: a fundamenlal <Ipproach to modern physics / John S. Townsend.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-891389-62-7 (alk. paper)
I. Quantum theory- Textbooks. 2. Physics- Textbooks. I. Title.
QCI74.12.T6942010
530.12- d022
2009022678
Primed in the United States of America
10 9 8 7 6 5 4 3 2 I
Contents in Brief
CHAPTER Light 1
CHAPTER 2 Wave Mechanics 51
CHAPTER 3 The Time-Independent Schrbdinger Equation 89
CHAPTER 4 One-Dimensional Potentials 113
CHAPTER 5 Principles of Quantum Mechanics 153
CHAPTER 6 Quantum Mechanics in Three Dimensions 177
CHAPTER 7 Identical Particles 211
CHAPTER 8 Solid-State Physics 257
CHAPTER 9 Nuclear Physics 277
CHAPTER 10 Particle Physics 317
APPENDIX A Special Relativity 367
APPENDIX B Power-Series Solutions 393
v
Contents
Preface xi
CHAPTER 1 Light
l.J Waves
1.2 Interference and Diffraction: The Wave Nature of Light 4
1.3 Photons: The Particle Nature of Light 7
1.4 Probability and the Quantum Nature of Light 16
1.5 Interference with Single Photons 23
1.6 The Double-Slit Experiment 32
1.7 Diffraction Gratings 36
1.8 The Principle of Least Time 38
1.9 Summary 43
CHAPTER 2 Wave Mechanics 51
2.1 Atom Interferometry 52
2.2 Crystal Diffraction 58
2.3 The Schrodinger Equation 62
2.4 The Physical Significance ortlle Wave Function 64
2.5 Conservation of Probability 66
2.6 Wave Packets and the Heisenberg Uncertainty Principle 67
2.7 Phase and Group Velocity 72
2.8 Expectation Values and Uncertainty 76
2.9 Ehrenfest's Theorem 79
2. 10 Summary 82
vii
viii Contents
CHAPTER 3 The Time-Independent Schrodinger Equation 89
3.1 Separation ofYariables 89
3.2 The Particle in a Box 91
3.3 Statistical Interpretation of Quantum Mechanics 98
3.4 The Energy Operator: Eigenvalues and Eigenfunctions 103
3.5 Summary lOS
CHAPTER 4 One-Dimensional Potentials 113
4.1 The Finite Square Well 113
4.2 Qualitative Features 119
4.3 The Simple Harmonic Oscillator 123
4.4 The Dirac Delta Function Potential 130
4.5 The Double Well and Molecular Binding 132
4.6 Scattering and the Step Potential 135
4.7 Tunneling and the Square Barrier 140
4.S Summary 145
CHAPTER 5 Principles of Quantum Mechanics 153
5.1 The Parity Operator 153
5.2 Observables and Hermitian Operators 155
5.3 Commuting Operators 160
5.4 Noncommuting Operators and Uncertainty Relations 162
5.5 Time Development 165
5.6 EPR, Sehrodinger's Cat, and All That 169
5.7 Summary 173
CHAPTER 6 Quantum Mechanics in Three Dimensions 177
6.1 The Three-Dimensional Box 177
6.2 Orbital Angular Momentum 179
6.3 The Hydrogen Atom 187
6.4 The Zeeman Effect 194
6.5 Intrinsic Spin 196
6.6 Summary 204
CHAPTER 7 Identical Particles 211
7.1 Multipartiele Systems 211
7.2 Identical Particles in Quantum Mechanics 212
7.3 Multielectron Atoms 215
7.4 The Fermi Energy 221
Contents ix
7.5 Quantum Statistics 226
7.6 Cavity Radiation 235
7.7 Bose-Einstein Condensation 242
7.8 Lasers 247
7.9 Summary 251
CHAPTER 8 Solid-State Physics 257
8.1 The Band Structure of Solids 257
8.2 Electrical Properties of Solids 263
8.3 The Silicon Revolution 268
8A Superconductivity 273
8.5 Summary 275
CHAPTER 9 Nuclear Physics 277
9.1 Nuclear Notation and Properties 278
9.2 The Curve of Binding Energy 281
9.3 Radioactivity 290
9A Nuclear Fission 298
9.5 Nuclear Fusion 303
9.6 Nuclear Weapons: History and Physics 306
9.7 Summary 312
CHAPTER 10 Particle Physics 317
10.1 Quantum Electrodynamics 317
10.2 Elementary Particles 326
10.3 Hadrons 330
lOA Quantum Chromodynamics 334
10.5 Quantum Flavor Dynamics 340
10.6 Mixing Angles 343
10.7 Symmetries and Conservation Laws 351
10.8 The Standard Model 358
10.9 Summary 363
APPENDtX A Special Relativity 367
A.I The Relativity Principle 367
A.2 The Postulates of Special Relativity 368
A.3 The Lorentz Transformation 378
AA Four-vectors 382
A.5 Momentum and Energy 383
x Contents
APPENDIX B Power-Series Solutions 393
B.I The Simple Harmonic Oscillator 393
8.2 Orbital Angular Momentum 395
B.3 The Hydrogen Atom 397
Answers to Odd-Numbered Problems 399
Index 405
Preface
As an undergraduate, my first course in quannun mechanics was in the spring semester of
my senior year. This was, frankly, too little, too late. As a sophomore, I had taken a fairly
standard course in modern physics. Despite the fact that I thought the professor teaching
the course did a good job, I was not pleased with the content. The course seemed to be
a summary of phenomenology, without giving me any understanding of the underlying
physics. To a budding physicist, this was not a satisfying experience.
Today, I am confident we do better by our students, at least in the upper-division
physics curriculum. This confidence is inspired by the quality and nature of the quantum
mechanics textbooks that we use there, one of which (A Moderl/ Approach to Qual/tulII
Mechal/ics) I am proud to have authored. At the introductory level the changes have
been, in my judgment. less clear cut. I believe that students deserve a serious introduction
to quantum mechanics, comparable to the introduction they receive to the subjects of
mechanics and electromagnetism, Moreover, with the appropriate grounding in quantum
mechanics, it is possible to give students real understanding and insight into an array of
topics that often fall under the rubric of modern physics. Students can see in quantum
mechanics a common thread that ties these topics together into a coherent picture of how
the world works, a picture that gives students confidence that quantum mechanics itself
really works, too.
While I have used the term "modern physics" to describe the material typically taught
in an introductory course, I believe this term has reached the end of its useful life, at least in
the way it is commonly used. Most ifnot all modern physics textbooks follow an historical
ordering of the material, with, in order of appearance, Planck, Einstein, Rutherford, Bohr,
and Schriidinger among the key characters in the story. Now I enjoy the history as much
as anyone, and I try to weave it into the text in a natural way. But I don't think following
the historical ordering so closely makes a lot of sense. After all, a story that starts in
the early 1900s does not sound modern to students learning physics in the twenty-first
century. Moreover, times have changed. We now have the advantage of truly modern
experiments, such as single-photon and single-atom interferometry experiments, that
have replaced the thought experiments that characterized much of the early discussions
of quantum mechanics. So why not slart with real experiments, which is what physics is
really based on, after all.
xi
xii Preface
Chapter I focuses on the quantum nature of light. While this chapter does include
discussion of the photoelectric effect (the key to understanding the operation of a
photodetcctor) and Compton scattering, the single-photon anticoincidence and inter
ference experiments carried out by Alain Aspect and coworkers in the late 1980s are at
the center of this chapter. Understanding the rcsults of these experiments leads us to the
concept of a probability amplitude and to the rules for multiplying and, in particular,
adding thcse probability amplitudes when there are multiple paths that a photon can
take betwecn the source and the detector. This is really the sum-aver-paths approach to
quantum mechanics pioneered by Richard Feynman. One of the advantages of this ap
proach is that students can see right away how quantum mechanics can explain everyday
phenomena such as the the law of reflection, Snell's law, and a diffraction grating (in, say,
the reflection of light from a CD) as straightforward extensions of the sum-aver-paths
approach from a few paths to many paths (leading naturally to Fermat's principle ofIeast
time). Although the approach that I follow in Chapter I is not the same as that given by
Feynman in his short series of lectures titled QED, it is inspired by these lectures.
Chaptcr 2 starts with the double-slit experiment, a topic that was discussed in Chap
ter I as an illustration of the sum-aver-paths approach to quantum mechanics. But in
Chapter 2, the key experiment is a double-slit experiment with helium atoms carried out
by Jiirgen Mlynek's group in the 1990s. This beautiful experiment really brings home to
students the strangeness of quantum mechanics. Since the sum-aver-paths approach is
not as useful for determining the behavior of particles such as electrons when they travel
microscopic distances, Chapter 2 moves naturally toward the Schrodinger equation. This
wave equation plays a similar role for nonrelativistic particles to that played by the wave
equation for light in Chapter I. Other topics in this chapter include wave packets, phase
and group velocities, expectation values and uncertainty, and Ehrenfest's equations.
Chapter 3 and Chapter 4 are all about solving the Schrodinger equation for a variety
of one-dimensional potentials. The centerpiece of Chapter 3 is the particle in a box, a
great laboratory for seeing many quantum effects. Chapter 4 includes discussion of the
finite square well, the harmonic oscillator, and the Dirac delta function potential, both
as a simple model for an atom and, more interestingly, as a double well that can be used
to capture the key features of molecular binding. Chapter 4 also includes a discussion of
scattering (and tunneling) in one-dimensional quantum mechanics. One relatively novel
feature of Chapter 3 at this level is the use of the particle in a box to illustrate the key
features of the energy eigenvalue equation, including the principles of superposition and
completeness and the way these principles are utilized to calculate the probability of
events for a wave function that is not an energy eigenfunction. These ideas are gener
alized in Chapter 5 to the more gencral class of Hermitian operators corresponding to
observables. Here the role that cOlllmuting and, in particular, noncommuting operators
and uncertainty relations play in quantum mechanics is emphasized.
Chapter 6 extends the discussion of quantum mechanics to three-dimensional sys
tems. Because the particle in a three-dimensional box, the orbital angular momentum
eigenvalue problem, and the hydrogen atom can be attacked by the technique of separation
of variables, these systems have much in common with the treatment of one-dimensional
potentials from the earlier chapters. I make an effort to keep the mathematical level
accessible to students. Some of the details, such as solving the hydrogen atom by a
power-series solution, are left to an appendix for the interested reader. But the simple,
direct way in which the eigenvalue problem for the z component of the orbital angular
momentum, for example, quantizes the eigenvalues to integral multiples of fl is a very