Series in PURE and APPLIED PHYSICS Concepts in Quantum Mechanics www.indd 1 11/7/08 2:35:20 PM Handbook of Particle Physics M. Sundaresan High-Field Electrodynamics Frederic V. Hartemann Fundamentals and Applications of Ultrasonic Waves J. 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 www.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 www.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. 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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Mathur, Surendra Singh. -- (CRC series in pure and applied physics) Includes bibliographical references and index. Singh, Surendra, 1953- II.12--dc22 2008044066 Visit the Taylor & Francis Web site at http://www.com and the CRC Press Web site at http://www.indd 4 11/7/08 2:35:20 PM Dedicated to the memory of Professor P.com This page intentionally left blank www.com Contents Preface. xv 1 NEED FOR QUANTUM MECHANICS AND ITS PHYSICAL BASIS 1 1.1 Inadequacy of Classical Description for Small Systems .1 Planck’s Formula for Energy Distribution in Black-body Radiation 1 1.2 de Broglie Relation and Wave Nature of Material Particles .3 The Photo-electric Effect .4 The Compton Effect .5 Ritz Combination Principle .2 Basis of Quantum Mechanics .1 Principle of Superposition of States .2 Heisenberg Uncertainty Relations .3 Representation of States .4 Dual Vectors: Bra and Ket Vectors .1 Properties of a Linear Operator .6 Adjoint of a Linear Operator .7 Eigenvalues and Eigenvectors of a Linear Operator .1 Physical Interpretation of Eigenstates and Eigenvalues .2 Physical Meaning of the Orthogonality of States .9 Observables and Completeness Criterion .10 Commutativity and Compatibility of Observables .11 Position and Momentum Commutation Relations .12 Commutation Relation and the Uncertainty Product.
26 Appendix 1A1: Basic Concepts in Classical Mechanics .1 Lagrange Equations of Motion .2 Classical Dynamical Variables .1 Meaning of Representation .2 How to Set up a Representation .3 Representatives of a Linear Operator .4 Change of Representation .1 Physical Interpretation of the Wave Function .6 Replacement of Momentum Observable p̂ by −i~ dq̂ d .7 Integral Representation of Dirac Bracket h A2 | F̂ |A1 i .8 The Momentum Representation .1 Physical Interpretation of Φ(p1 , p2 , · · ·pf ) .9 Dirac Delta Function .1 Three-dimensional Delta Function .2 Normalization of a Plane Wave .10 Relation between the Coordinate and Momentum Representations. 56 3 EQUATIONS OF MOTION 67 3.1 Schrödinger Equation of Motion .2 Schrödinger Equation in the Coordinate Representation .3 Equation of Continuity .5 Time-independent Schrödinger Equation in the Coordinate Representation 72 3.6 Time-independent Schrödinger Equation in the Momentum Representation 74 3.1 Two-body Bound State Problem (in Momentum Representation) for Non-local Separable Potential .7 Time-independent Schrödinger Equation in Matrix Form .8 The Heisenberg Picture .9 The Interaction Picture .1 Characteristic Equation of a Matrix .2 Similarity (and Unitary) Transformation of Matrices .3 Diagonalization of a Matrix. 87 4 PROBLEMS OF ONE-DIMENSIONAL POTENTIAL BARRIERS 89 4.1 Motion of a Particle across a Potential Step .2 Passage of a Particle through a Potential Barrier of Finite Extent .3 Tunneling of a Particle through a Potential Barrier .4 Bound States in a One-dimensional Square Potential Well .5 Motion of a Particle in a Periodic Potential. 107 5 BOUND STATES OF SIMPLE SYSTEMS 115 5.2 Motion of a Particle in a Box .1 Density of States .3 Simple Harmonic Oscillator .4 Operator Formulation of the Simple Harmonic Oscillator Problem .1 Physical Meaning of the Operators â and ↠.2 Occupation Number Representation (ONR) .5 Bound State of a Two-particle System with Central Interaction .6 Bound States of Hydrogen (or Hydrogen-like) Atoms .7 The Deuteron Problem .8 Energy Levels in a Three-dimensional Square Well: General Case .9 Energy Levels in an Isotropic Harmonic Potential Well.
147 Appendix 5A1: Special Functions .1 Legendre and Associated Legendre Equations .3 Laguerre and Associated Laguerre Equations. 169 Appendix 5A2: Orthogonal Curvilinear Coordinate Systems .1 Spherical Polar Coordinates .4 General Features of Orthogonal Curvilinear System of Coordinates 178 www.com 6 SYMMETRIES AND CONSERVATION LAWS 181 6.1 Symmetries and Their Group Properties .2 Symmetries in a Quantum Mechanical System .3 Basic Symmetry Groups of the Hamiltonian and Conservation Laws .1 Space Translation Symmetry .2 Time Translation Symmetry .3 Spatial Rotation Symmetry .4 Lie Groups and Their Generators .5 Examples of Lie Group .1 Proper Rotation Group R(3) (or Special Orthogonal Group SO(3)) 191 6.2 The SU(2) Group .3 Isospin and SU(2) Symmetry. 194 Appendix 6A1: Groups and Representations. 199 7 ANGULAR MOMENTUM IN QUANTUM MECHANICS 203 7.2 Raising and Lowering Operators .3 Matrix Representation of Angular Momentum Operators .4 Matrix Representation of Eigenstates of Angular Momentum .5 Coordinate Representation of Angular Momentum Operators and States .6 General Rotation Group and Rotation Matrices .7 Coupling of Two Angular Momenta .8 Properties of Clebsch-Gordan Coefficients .1 The Vector Model of the Atom .2 Projection Theorem for Vector Operators .9 Coupling of Three Angular Momenta .10 Coupling of Four Angular Momenta (L − S and j − j Coupling) .2 Non-degenerate Time-independent Perturbation Theory .3 Time-independent Degenerate Perturbation Theory .4 The Zeeman Effect .6 Particle in a Potential Well .7 Application of WKBJ Approximation to α-decay .8 The Variational Method .9 The Problem of the Hydrogen Molecule .10 System of n Identical Particles: Symmetric and Anti-symmetric States .11 Excited States of the Helium Atom .12 Statistical (Thomas-Fermi) Model of the Atom .13 Hartree’s Self-consistent Field Method for Multi-electron Atoms .14 Hartree-Fock Equations .15 Occupation Number Representation.
290 9 QUANTUM THEORY OF SCATTERING 299 9.2 Laboratory and Center-of-mass (CM) Reference Frames .1 Cross-sections in the CM and Laboratory Frames .3 Scattering Equation and the Scattering Amplitude .4 Partial Waves and Phase Shifts .5 Calculation of Phase Shift .6 Phase Shifts for Some Simple Potential Forms .7 Scattering due to Coulomb Potential .8 The Integral Form of Scattering Equation .9 Lippmann-Schwinger Equation and the Transition Operator .2 Validity of Born Approximation .3 Born Approximation and the Method of Partial Waves. 337 Appendix 9A1: The Calculus of Residues. 342 10 TIME-DEPENDENT PERTURBATION METHODS 351 10.2 Perturbation Constant over an Interval of Time .3 Harmonic Perturbation: Semi-classical Theory of Radiation .6 Electric Dipole Transitions in Atoms and Selection Rules .7 Photo-electric Effect .8 Sudden and Adiabatic Approximations .9 Second Order Effects. 373 11 THE THREE-BODY PROBLEM 377 11.5 Faddeev Equations in Momentum Representation .6 Faddeev Equations for a Three-body Bound System .7 Alt, Grassberger and Sandhas (AGS) Equations.
396 12 RELATIVISTIC QUANTUM MECHANICS 403 12.3 Spin of the Electron .4 Free Particle (Plane Wave) Solutions of Dirac Equation .5 Dirac Equation for a Zero Mass Particle .6 Zitterbewegung and Negative Energy Solutions .7 Dirac Equation for an Electron in an Electromagnetic Field .8 Invariance of Dirac Equation .9 Dirac Bilinear Covariants .10 Dirac Electron in a Spherically Symmetric Potential .11 Charge Conjugation, Parity and Time Reversal Invariance. 436 Appendix 12A1: Theory of Special Relativity .2 Minkowski Space-Time Continuum .3 Four-vectors in Relativistic Mechanics .4 Covariant Form of Maxwell’s Equations .com 13 QUANTIZATION OF RADIATION FIELD 455 13.2 Radiation Field as a Swarm of Oscillators .3 Quantization of Radiation Field .4 Interaction of Matter with Quantized Radiation Field .6 Atomic Level Shift: Lamb-Retherford Shift. 482 Appendix 13A1: Electromagnetic Field in Coulomb Gauge .2 Classical Concept of Field .3 Analogy of Field and Particle Mechanics .4 Field Equations from Lagrangian Density .2 Klein-Gordon Field (Real and Complex) .5 Quantization of a Real Scalar (KG) Field .6 Quantization of Complex Scalar (KG) Field .7 Dirac Field and Its Quantization .8 Positron Operators and Spinors .1 Equations Satisfied by Electron and Positron Spinors .9 Interacting Fields and the Covariant Perturbation Theory .2 S Matrix and Iterative Expansion of S Operator .3 Time-ordered Operator Product in Terms of Normal Constituents 532 14.10 Second Order Processes in Electrodynamics .11 Amplitude for Compton Scattering .1 Compton Scattering Amplitude Using Feynman Rules .2 Electron-positron (e− e+ ) Pair Annihilation .3 Two-photon Annihilation Leading to (e− e+ ) Pair Creation .4 Möller (e− e− ) Scattering .13 Calculation of the Cross-section of Compton Scattering .14 Cross-sections for Other Electromagnetic Processes .1 Electron-Positron Pair Annihilation (Electron at Rest) .2 Möller (e− e− ) and Bhabha (e− e+ ) Scattering. 558 Appendix 14A1: Calculus of Variation and Euler-Lagrange Equations.
564 Appendix 14A2: Functionals and Functional Derivatives. 567 Appendix 14A3: Interaction of the Electron and Radiation Fields. 569 Appendix 14A4: On the Convergence of Iterative Expansion of the S Operator .2 EPR Gedanken Experiment .3 Einstein-Podolsky-Rosen-Bohm Gedanken Experiment .4 Theory of Hidden Variables and Bell’s Inequality .5 Clauser-Horne Form of Bell’s Inequality and Its Violation in Two-photon Correlation Experiments .com Preface This book has grown out of our combined experience of teaching Quantum Mechanics at the graduate level for more than forty years. The emphasis in this book is on logical and consistent development of the subject following Dirac’s classic work Principles of Quan- tum Mechanics.
In this book no mention is made of postulates of quantum mechanics and every concept is developed logically. The alternative ways of representing the state of a physical system are discussed and the mathematical connection between the representa- tives of the same state in different representations is outlined. The equations of motion in Schrödinger and Heisenberg pictures are developed logically. The sequence of other top- ics in this book, namely, motion in the presence of potential steps and wells, bound state problems, symmetries and their consequences, role of angular momentum in quantum me- chanics, approximation methods, time-dependent perturbation methods, etc.
is such that there is continuity and consistency. Special concepts and mathematical techniques needed to understand the topics discussed in a chapter are presented in appendices at the end of the chapter as appropriate. A novel inclusion in this book is a chapter on the Three-body Problem, a subject that has reached some level of maturity.