Concepts in quantum mechanics

615 Pages·2009·5.95 MB·English

by  Mathur

Series in PURE and APPLIED PHYSICS
Concepts in
Quantum
Mechanics
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Concepts in Quantum Mechanics
Vishu Swarup Mathur
Surendra Singh
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Series in PURE and APPLIED PHYSICS
Concepts in
Quantum
Mechanics
Vishnu Swarup Mathur
Surendra Singh
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
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Library of Congress Cataloging-in-Publication Data
Mathur, Vishnu S. (Vishnu Swarup), 1935-
Concepts in quantum mechanics / Vishnu S. Mathur, Surendra Singh.
p. cm. -- (CRC series in pure and applied physics)
Includes bibliographical references and index.
ISBN 978-1-4200-7872-5 (alk. paper)
1. Quantum theory. I. Singh, Surendra, 1953- II. Title. III. Series.
QC174.12.M3687 2008
530.12--dc22 2008044066
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Dedicated to the memory of
Professor P. A. M. Dirac
This page intentionally left blank
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
1 NEED FOR QUANTUM MECHANICS AND ITS PHYSICAL BASIS 1
1.1 Inadequacy of Classical Description for Small Systems . . . . . . . . . . . 1
1.1.1 Planck’s Formula for Energy Distribution in Black-body Radiation 1
1.1.2 de Broglie Relation and Wave Nature of Material Particles. . . . . 2
1.1.3 The Photo-electric Effect . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.4 The Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.5 Ritz Combination Principle . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Basis of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Principle of Superposition of States . . . . . . . . . . . . . . . . . . 9
1.2.2 Heisenberg Uncertainty Relations . . . . . . . . . . . . . . . . . . . 12
1.3 Representation of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Dual Vectors: Bra and Ket Vectors . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5.1 Properties of a Linear Operator . . . . . . . . . . . . . . . . . . . . 16
1.6 Adjoint of a Linear Operator . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Eigenvalues and Eigenvectors of a Linear Operator . . . . . . . . . . . . . 18
1.8 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.8.1 Physical Interpretation of Eigenstates and Eigenvalues . . . . . . . 20
1.8.2 Physical Meaning of the Orthogonality of States . . . . . . . . . . 21
1.9 Observables and Completeness Criterion . . . . . . . . . . . . . . . . . . . 21
1.10 Commutativity and Compatibility of Observables . . . . . . . . . . . . . . 23
1.11 Position and Momentum Commutation Relations . . . . . . . . . . . . . . 24
1.12 Commutation Relation and the Uncertainty Product . . . . . . . . . . . . 26
Appendix 1A1: Basic Concepts in Classical Mechanics . . . . . . . . . . . . . . 31
1A1.1 Lagrange Equations of Motion . . . . . . . . . . . . . . . . . . . . 31
1A1.2 Classical Dynamical Variables . . . . . . . . . . . . . . . . . . . . . 32
2 REPRESENTATION THEORY 35
2.1 Meaning of Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 How to Set up a Representation . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Representatives of a Linear Operator . . . . . . . . . . . . . . . . . . . . . 37
2.4 Change of Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Coordinate Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.1 Physical Interpretation of the Wave Function . . . . . . . . . . . . 44
2.6 Replacement of Momentum Observable pˆby −i(cid:126) d . . . . . . . . . . . . . 45
dqˆ
2.7 Integral Representation of Dirac Bracket (cid:104)A |Fˆ |A (cid:105) . . . . . . . . . . . 50
2 1
2.8 The Momentum Representation . . . . . . . . . . . . . . . . . . . . . . . . 52
2.8.1 Physical Interpretation of Φ(p ,p ,···p ) . . . . . . . . . . . . . . 52
1 2 f
2.9 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.9.1 Three-dimensional Delta Function . . . . . . . . . . . . . . . . . . 55
2.9.2 Normalization of a Plane Wave . . . . . . . . . . . . . . . . . . . . 56
2.10 Relation between the Coordinate and Momentum Representations . . . . 56
3 EQUATIONS OF MOTION 67
3.1 Schr¨odinger Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Schr¨odinger Equation in the Coordinate Representation . . . . . . . . . . 69
3.3 Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Time-independent Schr¨odinger Equation in the Coordinate Representation 72
3.6 Time-independent Schr¨odinger Equation in the Momentum Representation 74
3.6.1 Two-bodyBoundStateProblem(inMomentumRepresentation)for
Non-local Separable Potential . . . . . . . . . . . . . . . . . . . . . 76
3.7 Time-independent Schr¨odinger Equation in Matrix Form . . . . . . . . . . 77
3.8 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.9 The Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Appendix 3A1: Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3A1.1 Characteristic Equation of a Matrix . . . . . . . . . . . . . . . . . 86
3A1.2 Similarity (and Unitary) Transformation of Matrices . . . . . . . . 87
3A1.3 Diagonalization of a Matrix . . . . . . . . . . . . . . . . . . . . . . 87
4 PROBLEMS OF ONE-DIMENSIONAL POTENTIAL BARRIERS 89
4.1 Motion of a Particle across a Potential Step . . . . . . . . . . . . . . . . . 90
4.2 Passage of a Particle through a Potential Barrier of Finite Extent . . . . . 94
4.3 Tunneling of a Particle through a Potential Barrier . . . . . . . . . . . . . 99
4.4 Bound States in a One-dimensional Square Potential Well . . . . . . . . . 103
4.5 Motion of a Particle in a Periodic Potential . . . . . . . . . . . . . . . . . 107
5 BOUND STATES OF SIMPLE SYSTEMS 115
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2 Motion of a Particle in a Box . . . . . . . . . . . . . . . . . . . . . . . . . 115
5.2.1 Density of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.3 Simple Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.4 Operator Formulation of the Simple Harmonic Oscillator Problem . . . . 122
5.4.1 Physical Meaning of the Operators aˆ and aˆ† . . . . . . . . . . . . . 123
5.4.2 Occupation Number Representation (ONR) . . . . . . . . . . . . . 125
5.5 Bound State of a Two-particle System with Central Interaction . . . . . . 126
5.6 Bound States of Hydrogen (or Hydrogen-like) Atoms . . . . . . . . . . . . 131
5.7 The Deuteron Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.8 Energy Levels in a Three-dimensional Square Well: General Case . . . . . 144
5.9 Energy Levels in an Isotropic Harmonic Potential Well . . . . . . . . . . . 147
Appendix 5A1: Special Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5A1.1 Legendre and Associated Legendre Equations . . . . . . . . . . . . 156
5A1.2 Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5A1.3 Laguerre and Associated Laguerre Equations . . . . . . . . . . . . 162
5A1.4 Hermite Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
5A1.5 Bessel Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Appendix 5A2: Orthogonal Curvilinear Coordinate Systems . . . . . . . . . . . 174
5A2.1 Spherical Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . 174
5A2.2 Cylindrical Coordinates . . . . . . . . . . . . . . . . . . . . . . . . 175
5A2.3 Parabolic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 177
5A2.4 General Features of Orthogonal Curvilinear System of Coordinates 178
6 SYMMETRIES AND CONSERVATION LAWS 181
6.1 Symmetries and Their Group Properties . . . . . . . . . . . . . . . . . . . 181
6.2 Symmetries in a Quantum Mechanical System . . . . . . . . . . . . . . . . 182
6.3 Basic Symmetry Groups of the Hamiltonian and Conservation Laws . . . 183
6.3.1 Space Translation Symmetry . . . . . . . . . . . . . . . . . . . . . 184
6.3.2 Time Translation Symmetry . . . . . . . . . . . . . . . . . . . . . . 185
6.3.3 Spatial Rotation Symmetry . . . . . . . . . . . . . . . . . . . . . . 185
6.4 Lie Groups and Their Generators . . . . . . . . . . . . . . . . . . . . . . . 188
6.5 Examples of Lie Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6.5.1 Proper Rotation Group R(3) (or Special Orthogonal Group SO(3)) 191
6.5.2 The SU(2) Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
6.5.3 Isospin and SU(2) Symmetry . . . . . . . . . . . . . . . . . . . . . 194
Appendix 6A1: Groups and Representations . . . . . . . . . . . . . . . . . . . . 199
7 ANGULAR MOMENTUM IN QUANTUM MECHANICS 203
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
7.2 Raising and Lowering Operators . . . . . . . . . . . . . . . . . . . . . . . 206
7.3 Matrix Representation of Angular Momentum Operators . . . . . . . . . . 208
7.4 Matrix Representation of Eigenstates of Angular Momentum . . . . . . . 209
7.5 Coordinate Representation of Angular Momentum Operators and States . 212
7.6 General Rotation Group and Rotation Matrices . . . . . . . . . . . . . . . 214
7.6.1 Rotation Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.7 Coupling of Two Angular Momenta . . . . . . . . . . . . . . . . . . . . . . 218
7.8 Properties of Clebsch-Gordan Coefficients . . . . . . . . . . . . . . . . . . 219
7.8.1 The Vector Model of the Atom . . . . . . . . . . . . . . . . . . . . 221
7.8.2 Projection Theorem for Vector Operators . . . . . . . . . . . . . . 221
7.9 Coupling of Three Angular Momenta . . . . . . . . . . . . . . . . . . . . . 227
7.10 Coupling of Four Angular Momenta (L−S and j−j Coupling) . . . . . 228
8 APPROXIMATION METHODS 235
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.2 Non-degenerate Time-independent Perturbation Theory . . . . . . . . . . 236
8.3 Time-independent Degenerate Perturbation Theory . . . . . . . . . . . . . 242
8.4 The Zeeman Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
8.5 WKBJ Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254
8.6 Particle in a Potential Well . . . . . . . . . . . . . . . . . . . . . . . . . . 262
8.7 Application of WKBJ Approximation to α-decay . . . . . . . . . . . . . . 264
8.8 The Variational Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
8.9 The Problem of the Hydrogen Molecule . . . . . . . . . . . . . . . . . . . 270
8.10 System of n Identical Particles: Symmetric and Anti-symmetric States . . 274
8.11 Excited States of the Helium Atom . . . . . . . . . . . . . . . . . . . . . . 278
8.12 Statistical (Thomas-Fermi) Model of the Atom . . . . . . . . . . . . . . . 280
8.13 Hartree’s Self-consistent Field Method for Multi-electron Atoms . . . . . . 281
8.14 Hartree-Fock Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
8.15 Occupation Number Representation . . . . . . . . . . . . . . . . . . . . . 290
9 QUANTUM THEORY OF SCATTERING 299
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
9.2 Laboratory and Center-of-mass (CM) Reference Frames . . . . . . . . . . 300
9.2.1 Cross-sections in the CM and Laboratory Frames . . . . . . . . . . 302
9.3 Scattering Equation and the Scattering Amplitude . . . . . . . . . . . . . 303