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Series in PURE and APPLIED PHYSICS Concepts in Quantum Mechanics C7872_FM.indd 1 11/7/08 2:35:20 PM Handbook of Particle Physics M. K. Sundaresan High-Field Electrodynamics Frederic V. Hartemann Fundamentals and Applications of Ultrasonic Waves J. David N. Cheeke Introduction to Molecular Biophysics Jack A. Tuszynski Michal Kurzynski Practical Quantum Electrodynamics Douglas M. Gingrich Molecular and Cellular Biophysics Jack A. Tuszynski Concepts in Quantum Mechanics Vishu Swarup Mathur Surendra Singh C7872_FM.indd 2 11/7/08 2:35:20 PM 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 A CHAPMAN & HALL BOOK C7872_FM.indd 3 11/7/08 2:35:20 PM Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2009 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-13: 978-1-4200-7872-5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. 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QC174.12.M3687 2008 530.12--dc22 2008044066 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com C7872_FM.indd 4 11/7/08 2:35:20 PM 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 Schrödinger Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Schrödinger Equation in the Coordinate Representation . . . . . . . . . . 69 3.3 Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.5 Time-independent Schrödinger Equation in the Coordinate Representation 72 3.6 Time-independent Schrödinger Equation in the Momentum Representation 74 3.6.1 Two-bodyBoundStateProblem(inMomentumRepresentation)for Non-local Separable Potential . . . . . . . . . . . . . . . . . . . . . 76 3.7 Time-independent Schrödinger 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