uan um • anles CI.com QUANTUM MECHANICS www.com THIS PAGE IS BLANK www.com QUANTUM MECHANICS Second Edition V. Thankappan Deparfment oj Physics UnivtrsityojCalicllf, Kerala India PuI!l.lBHIMG FOR OHE WORLD NEW AGE INTERNATIONAL (P) LIl\flTED, PUBLISHERS New Delhi' Bangalore • Cbennai • Cochin • Guwahati • Hyderabad Jalandlmr • Kolkata • Lucknow • Mumbai • Rando Visit us at www.com Copyright © 1993, 1985, New Age International (P) Ltd., Pn blishers Pnblished by New Age International (P) Ltd., Pnblishers All rights reserved. No part of this ebook may be reproduced in any form, by photostat, microfilm, xerography, or any other means, or incorporated into any information retrieval system, electronic or mechanical, without the written permission of the publisher. All inquiries should be emailedtorights@newagepublishers.com ISBN (13) : 978-81-224-2500-0 PuBLISHING FOR ONE WORLD NEW AGE INTERNATIONAL (P) LIMITED, PUBLISHERS 4835/24, Ansari Road, Daryaganj, New Delhi - 110002 Visit us at www.com PREFACE TO THE SECOND EDITION This second edition differs from the first edition mainly in the addition of a chapter on the Interpretational Problem.
Even before the printing of the frrst edition, there was criticism from some quarters that the account of this problem included in the introductory chapter is too sketchy and brief to be of much use to the students. The new chapter, it is hoped, will remove the shortcoming. In addition to a detailed description of the Copenhagen and the Ensemble Interpre- tations, this chapter also contains a brief account of the Hidden-Variable Theories (which are by-products of the interpretational problem) and the associated developments like the Neumann's and Bell's theorems. The important role played by the Einstein-Podolsky-Rosen Paradox in defining and delineating the interpretational problem is emphasized.
Since the proper time to worry over the interpretational aspect is after mastering the mathematical fonnalism, the chapter is placed at the end of the book. Minor additions include the topics of Density Matrix (Chapter 3) and Charge Conjugation (Chapter 10). The new edition thus differs from the old one only in some additions, but no deletions, of material. It is nearly two years since the revision was completed.
an account of certain later developments like the Greenbetger-Home-Zeilinger- Mermin experiment [Mennin N. Physics Today 36 no 4, p. 38 (1985») could not be included in Chapter 12. It would, however, be of interest to note that the arguments against the EPR experiment presented in Section 12.4 could be extended to the case of the GHZ-Mermin thought-experiment also.
For, the quantum mechanically incorrect assumption that a state vector chosen as the eigenvector of a producl of observables is a common eigenvector of the individual (component) observables, is involved in this experiment as well. Several persons have been kind enough to send their critical comments on the book as well as suggestions for improvement. The author is thankful to all of them. The author is also thankful to P.
Gopalakrishna Nambi for permitting to quote. in Chapter 12, from his Ph.D thesis and 10 Ravi K. Menon for the usc of some material from his Ph.D work in this chapter.com THIS PAGE IS BLANK www.com PREFACE TO THE FIRST EDITION This book is intended to serve as a text book for physics students at the M. Phil (Pre-Ph.
It is based, with the exception of Chapter I. on a course on quantum mechanics and quantum field theory that the author taught for many years, starting with 1967, at Kurukshetra University and later at the University of Calicut. At both the Universities the course is covered Over a period of one year (or two semesters) at the final year M. Also at both places, a less formal course, consisting of the developments of the pre-quantum mechanics period (1900-1924) together with some elementary applications of SchrOdinger's wave equation, is offered during the first year.
A fairly good knowledge of classical mechanics. the special theory of relativity, classical elec- trodynamics and mathematical physics (courses on these topics are standard at most universities) is needed at various stages of the book. The mathematics of linear vector spaces and of matrices, which play somewhat an all-pervasive role in this book. are included in the book, the former as part of the text (Chapter 2) and the latter as an Appendix.
Topics covered in this book. with a few exceptions, are the ones usually found in a book on quantum mechanics at this level such as the well known books by L. Schiff and by A. However, the presentation is based on the view that quantum mechanics is a branch of theoretical physics on the same footing as classical mechanics or classical electrodynamics.
As a result, neither accounts of the travails of the pioneers of quantum theory in arriving at the various milestones of the theory nor descriptions of the many experiments that helped them along the way, are induded (though references to the original papers are given). Instead, the empha'iis is on the ba'iic principles, the calculational techniques and the inner consistency and beauty of the theory. Applications to particular problems are taken up only to illustrate a principle or technique under discussion. Also, the Hilbert space fonnalism, which provides a unified view of the different fonnula- tions of nonrelativistic quantum mechanics, is adopted.
In particular, SchrOdin- ger's and Heisenberg's fonnulations appear merely as different representations, analogous respectively to the Hamilton-Jacobi theory and the Hamilton's formalism in classical mechanics. Problems are included with a view to supple- menting the text. From ill) early days, quantum mechanics hm; hccn bedevilled by a controversy among its founders regarding what has come to be known as the Interpretational Prohlem. Judging from the number of papers and books still appearing on this topic.
the controversy is far from settled. While this problem does not affect either the mathematical framework of quantum mechanics or its practical applications, www.com I··· VIII PREFACE a teacher of quantum mechanics cannot afford to be ignorant of it It is with a view to giving an awareness of this problem to the teacher of this book that Chapter 1 is included (students are advised to read this chapter only at the end, or at least after Chapter 4). The chapter is divided into two parts: The first part is a discussion of the two main contestants in the arena of interpretation-the Statis- tical (or, Ensemble) and the Copenhagen. In the second part, the path-integral formalism (which is not considered in any detail in this book) is used to show the connection between the 'If-function of quantum mechanics on the one hand and the Lagrangian function L and the action integral S of classical mechanics on the other.
This too has a bearing on the interpretational problem. For, the interpre- tational problem is, at least partly, due to the proclivity of the Copenhagen school to identify 'If with the particle (as indicated by the notion, held by the advocates of this school, that observing a particle at a point leads to a "collapse" of the 'If-function to that point!). But the relationship between S and 'If suggests that, just as S in classical mechanics, 'If in quantum mechanics is a function that charac- terises the paths of the particle and that its appearance in the dynamical equation of motion need be no more mysterious than the appearance of S or L in the classical equations of motion. The approach adopted in this book as well as its level presumes that the course will be taught by a theoretical physiCist.
The level might be a little beyond that currently followed in some Universities in this country, especially those with few theorists. However, it is well to remember in this connection that, during the last three decades, quantum theory has grown (in the form of quantum field theory) much beyond the developments of the 1920's. As such, a quantum mechanics course at the graduate level can hardly claim to meet the modem needs of the student if it does not take him or her at least to the threshOld of quantum field theory. In a book of this size, it is difficult to reserve one symbol for one quantity.
Care is taken so that the use of the same symbol for different quantities does not lead to any confusion. This book was written under the University Grants Commission's scheme of preparing University level books. Financial assistance under this scheme is gratefully acknowledged. The author is also thankful to the,National Book Trust, India, for subsidising the publication of the book.
Since the book had to be written in the midst of rather heavy teaching assign- ments and since the assistance of a Fellow could be obtained only for a short period of three months, the completion of the book was inordinately delayed. Further delay in the publication of the book was caused in the process of fulfilling certain formalities. The author is indebted to Dr. Ramamurthy and Dr.
Gupta for a thorough reading of the manuscript and for making many valuable suggestions. He is also thankful to the members of the Physics Department, Calicut University, for their help and cooperation in preparing the typescript MarCh 1985 V.com CONTENTS Preface to the Second Edition v Preface to the First Edition vii Chapter 1.1 The Conceptual Aspect 1 1.2 The Mathematical Aspect 9 Chapter 2. LINEAR VECfOR SPACES 19 2.3 Bra and Ket Notation for Vectors 51 2.4 Representation Theory 52 Co-ordinate and Momentum Representation 59 Chapter 3. THE BASIC PRINCIPLES 63 3.1 The Fundamental Postulates 63 3.2 The Uncertainty Principle 75 3.3 Density Matrix 84 Chapter 4.1 The Equations of Motion 87 The SchrOdinger Picture 88 The Heisenberg Picture 94 The Interaction Picture 97 4.2 Illustrative Applications 98 The Linear Hannonic Oscillator 98 The Hydrogen Atom J JJ Chapter 5.
THEORY OF ANGULAR MOMENWM 120 5.2 Eigenvalues and Eigenvectors 122 5.4 Orbital Angular Momentum 129 www.com x QUANTUM MECHANICS 5.5 Addition of Angular Momenta 138 Oebsch-Gordon Coefficients 138 Racah Coefficients 148 The 9j-Symbols 154 5.6 Angular Momentum and Rotations 159 5.8 Consequences of Quantization 179 Chapter 6. INV ARIANCE PRINCIPLES AND CONSERVATION LAWS 181 6.1 Symmetry and Conservation Laws 182 6.2 The Space-Time Symmetries 183 Displacement in Space: ConselVation of Linear Momentum 184 Displacement in Time: ConselVation of Energy 187 Rotations in Space: ConselVation of Angular Momentum 188 Space Inversion: Parity 188 Time Reversal Invariance 191 Chapter 7. THEORY OF SCATIERING 196 7.2 Method of Partial Waves 201 7.3 The Born Approximation 224 Chapter 8.1 The WKB Approximation 237 8.2 The Variational Method 256 Bound States (Ritz Method) 256 Scbwinger's Method for Phase Shifts 263 8.3 Stationary Perturbation Theory 267 Nondegenerate Case 270 Degenerate Case 274 8.4 Time-Dependent Perturbation Theory 284 Constant Perturbation 287 Harmonic Perturbation 293 Coulomb Excitation 300 8.5 Sudden and Adiabatic Approximations 304 Sudden Approximation 304 Adiabatic Approximation 308 Chapter 9.1 The Identity of Particles 319 9.2 Spins and Statistics 324 9.3 Illustrative Examples 325 www.com CONTENTS Chapter 10. RELATIVISTIC WAVE EQUATIONS 322 10.2 The First Order Wave Equations 336 The Dirac Equation 338 The Weyl Equations 374 10.3 The Second Order Wave Equations 377 The Klein-Gordon Equation 378 Wave Equation of the Photon 390 lOA Charge Conjugation 384 Chapter 11.
ELEMENTS OF FIELD QUANTIZATION 390 11.2 Lagrangian Field Theory 390 11.3 Non-Relativistic Fields 398 llA Relativistic Fields 403 The Klein-Gordm Field 405 The Dirac Field 412 The Electromagnetic Field 418 11.5 Interdcting Fields 425 Chapter 12. THE INTERPRET ATIONAL PROBLEM 445 12.1 The EPR Paradox 445 12.2 The Copenhagen Interpretation 448 12.3 The Ensemble Interpretation 454 1204 Explanations of the EPR Paradox 459 12.5 The Hidden-Variable Theories 463 Appendix A. MATRICES 472 Definition 472 Matrix Algebra 473 Important Scalar Numbers Associated with a Square Matrix 476 Special Matrices 479 Matrix Transformations 481 Solution of Linear Algebraic Equations 482 Eigenvalues and Eigenvectors 484 Diagonalizability of a Matrix 488 Bilinear, Quadratic and Hermitian Forms 490 Infinite Matrices 491 www.com xii QUANTUM MECHANICS Appendix B.INEAR OPERATORS 494 Appendix C. FOURIER SERIES AND FOURIER TRANSFORMS 500 Fourier Series 500 Fourier Transforms 504 Appendix D.
DIRAC DELTA FUNCTION 509 Appendix E.