Quantum Mechanics: Foundations and Applications

350 Pages·2.668 MB·English

by  Donald Gary Swanson

Quantum
Mechanics
Foundations and Applications

Quantum
Mechanics
Foundations and Applications
D G Swanson
Auburn University, Alabama, USA
New York London
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Preface
This book grew out of a compilation of lecture notes from a variety of other
texts inModern Physics and Quantum Mechanics. Except for the part about
digital frequency counters related to the uncertainty principle, none of the
material is truly original, although virtually everything has been rewritten in
order to unify the presentation. Over the course of twenty years, I taught
the Quantum Mechanics off and on for physics majors in their junior or se-
nior year, and this same course filled the gap for incoming graduate students
who were not properly prepared in Quantum Mechanics to take our graduate
course. Originally, it was a full year course over three quarters, but when we
converted to semesters, the first half was required and the second half was
optional, but recommended for majors intending to go on to graduate work
in physics. In our present curriculum, the first five chapters comprise the
first semester and is designed for juniors, while the remainder is designed for
seniors. This leaves many of the later chapters optional for the instructor,
and some sections are marked as advanced by an asterisk and can be easily
deletedwithoutaffectingthecontinuity. Iliketousethegeneratingfunctions
in Appendix B to establish normalization constants and recursion formulas,
but they can be omitted completely. The 1-D scattering in Chapter 10 is
markedasadvanced,andmaybeskipped,buttheuniqueconnectionbetween
the1-DSchr¨odingerequationandtheKorteweg–deVriesequationforsolitons,
which is solved using the inverse scattering method, where a nonlinear classi-
cal problem is solved by the linear techniques of Quantum Mechanics I found
compelling.
MyownfirstcourseinQuantumMechanicswastaughtbyRobertLeighton
the first year his text came out, and his use of the Fourier transform pairs
forthewavefunctionsincoordinateandmomentumspacemadetheoperator
formalism transparent to me, and this feature, more than any other single
factor, provided the impetus for me to write this text, since this formalism is
uncommon in most of the modern texts. Some of the text follows Leighton,
buttherehavebeensomanyotherprimarytextsforourcourseovertheyears
(with my notes as an adjunct to the text) that the topics and problems come
from many sources, most of which are listed in the bibliography. In recent
years, I have expanded the notes to form a primary text which has been
used successfully in preparing our students for the graduate level courses in
Quantum Mechanics and Quantum Statistics.
This text may be especially appealing to Electrical Engineering students
(I taught the Quantum Mechanics for Engineers course at the University of
v
vi
Texas at Austin for many years) because they are already familiar with the
properties of Fourier transforms.
I would like to thank Leslie Lamport and Donald Knuth for their devel-
opment of LATEX and TEX respectively, without which I would never have
attemped to write this book. The figures have been set with TEXniques by
Michael J. Wichura.
If a typographical or other error is discovered in the text, please report it
to me at [email protected] and I will keep a downloadable errata
page on my webpage at www.physics.auburn.edu/∼swanson.
D. Gary Swanson
Contents
1 The Foundations of Quantum Physics 1
1.1 The Prelude to Quantum Mechanics . . . . . . . . . . . . . . 1
1.1.1 The Zeeman Effect . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Black-Body Radiation . . . . . . . . . . . . . . . . . . 2
1.1.3 Photoelectric Effect . . . . . . . . . . . . . . . . . . . 6
1.1.4 Atomic Structure, Bohr Theory . . . . . . . . . . . . . 7
1.2 Wave–Particle Duality and the Uncertainty Relation . . . . . 10
1.2.1 The Wave Properties of Particles . . . . . . . . . . . . 10
1.2.2 The Uncertainty Principle . . . . . . . . . . . . . . . . 11
1.2.3 Fourier Transforms . . . . . . . . . . . . . . . . . . . . 14
1.3 Fourier Transforms in Quantum Mechanics . . . . . . . . . . 19
1.3.1 The Quantum Mechanical Transform Pair . . . . . . . 19
1.3.2 Operators . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.4 The Postulatory Basis of Quantum Mechanics . . . . . . . . . 23
1.4.1 Postulate 1 . . . . . . . . . . . . . . . . . . . . . . . . 24
1.4.2 Postulate 2 . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.3 Postulate 3 . . . . . . . . . . . . . . . . . . . . . . . . 26
1.4.4 Postulate 4 . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5 Operators and the Mathematics of Quantum Mechanics . . . 27
1.5.1 Linearity. . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.5.2 Hermitian Operators . . . . . . . . . . . . . . . . . . . 28
1.5.3 Commutators . . . . . . . . . . . . . . . . . . . . . . . 29
1.5.4 Matrices as Operators . . . . . . . . . . . . . . . . . . 30
1.5.5 Dirac Notation . . . . . . . . . . . . . . . . . . . . . . 30
1.6 Properties of Quantum Mechanical Systems . . . . . . . . . . 31
1.6.1 Wave–Particle Duality . . . . . . . . . . . . . . . . . . 31
1.6.2 The Uncertainty Principle and Schwartz’s Inequality . 32
1.6.3 The Correspondence Principle . . . . . . . . . . . . . . 34
1.6.4 Eigenvalues and Eigenfunctions . . . . . . . . . . . . . 36
1.6.5 Conservation of Probability . . . . . . . . . . . . . . . 40
2 The Schr¨odinger Equation in One Dimension 41
2.1 The Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1.1 Boundary Conditions and Normalization . . . . . . . . 42
2.1.2 Wave Packets . . . . . . . . . . . . . . . . . . . . . . . 43
2.1.3 Transmission and Reflection at a Barrier . . . . . . . . 46
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viii
2.1.4 The Infinite Potential Well . . . . . . . . . . . . . . . 48
2.1.5 The Finite Potential Well . . . . . . . . . . . . . . . . 51
2.1.6 Transmission through a Barrier . . . . . . . . . . . . . 55
2.2 One-Dimensional Harmonic Oscillator . . . . . . . . . . . . . 56
2.2.1 The Schr¨odinger Equation . . . . . . . . . . . . . . . . 57
2.2.2 Changing to Dimensionless Variables . . . . . . . . . . 58
2.2.3 The Asymptotic Form . . . . . . . . . . . . . . . . . . 58
2.2.4 Factoring out the Asymptotic Behavior . . . . . . . . 59
2.2.5 Finding the Power Series Solution . . . . . . . . . . . 59
2.2.6 The Energy Eigenvalues . . . . . . . . . . . . . . . . . 60
2.2.7 Normalized Wave Functions . . . . . . . . . . . . . . . 60
2.3 Time Evolution and Completeness . . . . . . . . . . . . . . . 63
2.3.1 Completeness . . . . . . . . . . . . . . . . . . . . . . . 63
2.3.2 Time Evolution . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Operator Method . . . . . . . . . . . . . . . . . . . . . . . . . 65
2.4.1 Raising and Lowering Operators . . . . . . . . . . . . 65
2.4.2 Eigenfunctions and Eigenvalues . . . . . . . . . . . . . 67
2.4.3 Expectation Values . . . . . . . . . . . . . . . . . . . . 70
3 The Schr¨odinger Equation in Three Dimensions 73
3.1 The Free Particle in Three Dimensions . . . . . . . . . . . . . 73
3.2 Particle in a Three-Dimensional Box . . . . . . . . . . . . . . 74
3.3 The One-Electron Atom . . . . . . . . . . . . . . . . . . . . . 75
3.3.1 Separating Variables . . . . . . . . . . . . . . . . . . . 76
3.3.2 Solution of the Φ(φ) Equation . . . . . . . . . . . . . . 77
3.3.3 Orbital Angular Momentum . . . . . . . . . . . . . . . 77
3.3.4 Solving the Radial Equation . . . . . . . . . . . . . . . 87
3.3.5 Normalized Wave Functions . . . . . . . . . . . . . . . 91
3.4 Central Potentials . . . . . . . . . . . . . . . . . . . . . . . . 95
3.4.1 Nuclear Potentials . . . . . . . . . . . . . . . . . . . . 96
3.4.2 Quarks and Linear Potentials . . . . . . . . . . . . . . 98
3.4.3 Potentials for Diatomic Molecules∗ . . . . . . . . . . . 101
4 Total Angular Momentum 105
4.1 Orbital and Spin Angular Momentum . . . . . . . . . . . . . 105
4.1.1 Eigenfunctions of Jˆ and Jˆ . . . . . . . . . . . . . . . 106
x y
4.2 Half-Integral Spin Angular Momentum . . . . . . . . . . . . . 108
4.3 Addition of Angular Momenta . . . . . . . . . . . . . . . . . . 113
4.3.1 Adding Orbital Angular Momenta for Two Electrons . 113
4.3.2 Adding Orbital and Spin Angular Momenta . . . . . . 116
4.4 Interacting Spins for Two Particles . . . . . . . . . . . . . . . 119
4.4.1 Magnetic Moments . . . . . . . . . . . . . . . . . . . . 119
ix
5 Approximation Methods 123
5.1 Introduction — The Many-Electron Atom . . . . . . . . . . . 123
5.2 Nondegenerate Perturbation Theory . . . . . . . . . . . . . . 125
5.2.1 Nondegenerate First-Order Perturbation Theory . . . 126
5.2.2 Nondegenerate Second-Order Perturbation Theory . . 127
5.3 Perturbation Theory for Degenerate States . . . . . . . . . . 130
5.4 Time-Dependent Perturbation Theory . . . . . . . . . . . . . 133
5.4.1 Perturbations That Are Constant in Time . . . . . . . 134
5.4.2 Perturbations That Are Harmonic in Time . . . . . . 136
5.4.3 Adiabatic Approximation . . . . . . . . . . . . . . . . 138
5.4.4 Sudden Approximation. . . . . . . . . . . . . . . . . . 139
5.5 The Variational Method . . . . . . . . . . . . . . . . . . . . . 141
5.5.1 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.6 Wentzel, Kramers, and Brillouin Theory (WKB) . . . . . . . 144
5.6.1 WKB Approximation . . . . . . . . . . . . . . . . . . 144
5.6.2 WKB Connection Formulas . . . . . . . . . . . . . . . 145
5.6.3 Reflection . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6.4 Transmission through a Finite Barrier . . . . . . . . . 147
6 Atomic Spectroscopy 151
6.1 Effects of Symmetry . . . . . . . . . . . . . . . . . . . . . . . 151
6.1.1 Particle Exchange Symmetry . . . . . . . . . . . . . . 151
6.1.2 Exchange Degeneracy and Exchange Energy . . . . . . 157
6.2 Spin–Orbit Coupling in Multielectron Atoms . . . . . . . . . 159
6.2.1 Spin–Orbit Interaction . . . . . . . . . . . . . . . . . . 159
6.2.2 The Thomas Precession . . . . . . . . . . . . . . . . . 160
6.2.3 LS Coupling, or Russell–Saunders Coupling . . . . . . 161
6.2.4 Selection Rules for LS Coupling . . . . . . . . . . . . . 166
6.2.5 Zeeman Effect. . . . . . . . . . . . . . . . . . . . . . . 168
7 Quantum Statistics 175
7.1 Derivation of the Three Quantum Distribution Laws . . . . . 175
7.1.1 The Density of States . . . . . . . . . . . . . . . . . . 176
7.1.2 Identical, Distinguishable Particles . . . . . . . . . . . 179
7.1.3 Identical,IndistinguishableParticleswithHalf-Integral
Spin (Fermions) . . . . . . . . . . . . . . . . . . . . . . 180
7.1.4 Identical,IndistinguishableParticleswithIntegralSpin
(Bosons) . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.1.5 The Distribution Laws . . . . . . . . . . . . . . . . . . 181
7.1.6 Evaluation of the Constant Multipliers . . . . . . . . . 183
7.2 Applications of the Quantum Distribution Laws . . . . . . . . 185
7.2.1 General Features . . . . . . . . . . . . . . . . . . . . . 185
7.2.2 Applications of the Maxwell–Boltzmann Distribution
Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
7.2.3 Applications of the Fermi–Dirac Distribution Law . . 191