Quantum Mechanics: Theory and Applications I cannot tell how the truth may be I say the tale as it was said to me --- Sir Walter Scott www.com Fundamental Theories of Physics An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application Editor: ALWYN VAN DER MERWE, University ()f Denver, US. Editorial Advisory Board: JAMES T. CUSHING, University of Notre Dame, US. GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P.
HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, UK. CLIVE KILMISTER, University (~f London, UK. LAHTI, University ofTurku, Finland ASHER PERES, Israel Institute ()fTechnology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada TONY SUDBURY, University ()fYork, u.
HANS-JURGEN TREDER, Zentralinstitutfiir Astrophysik der Akademie der Wissenschaften, Germany Volume 137 www.com Quantum Mechanics: Theory and Applications by Ajoy Ghatak Indian Institute of Technology New Delhi, India and S. Lokanathan Jawahar Lal Nehru Planetarium, Bangalore, India SPRINGER SCIENCE+ BUSINESS MEDIA, B. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4020-2129-9 ISBN 978-1-4020-2130-5 (eBook) DOI 10.1007/978-1-4020-2130-5 Printed on acidJree paper All Rights Reserved © 2004 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2004 Softcover reprint of the hardcover 1st edition 2004 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.com To our teachers: Professor H.
Kothari Professor RC. Majumdar Professor R Serber www.com Table of Contents Acknow ledgements ix Preface Xl Prologue XV Chronological Development of Quantum Mechanics: A Very Brief Account xli Physical Constants xlvii Part I: Mathematical Preliminaries Chapter I: The Dirac Delta Function 3 Chapter 2: Fourier Transforms 19 Part II: Basic Quantum Mechanics Chapter 3: Particles and Waves and the Uncertainty Principle 31 Chapter 4: Time Dependent Schrbdinger Equation 71 Chapter 5: Propagation of Wave Packets and Concept of Group Velocity 89 Chapter 6: Bound State Solutions of the SchrOdinger Equation lIS Chapter 7: Linear Harmonic Oscillator: I Solution of the Schrbdinger Equation and Relationship with the Classical Oscillator 159 Chapter 8: One-Dimensional Barrier Transmission Problems 197 Chapter 9: Angular Momentum I-The Spherical Harmonics 221 Chapter 10: Spherically Symmetric Potentials Hydrogen Atom Problem, Rotation Vibration Spectra, Three-Dimensional Oscillator 245 Chapter 11: Dirac's Bra and Ket Algebra 291 Chapter 12: Linear Harmonic Oscillator II Solutions Using Bra and Ket Algebra 308 Vll www.com viii Chapter 13: Angular Momentum II Using Bra and Ket Algebra 339 Chapter 14: Experiments with Spin Half Particles The Stern-Gerlach Experiment, Larmor Precession and Magnetic Resonance 357 Chapter 15: Angular momentum III Eigenfunctions Using Operator Algebra 389 Chapter 16: The Double Well Potential and the Krbnig-Penney Model 401 Chapter 17: The JWKB Approximation 423 Chapter 18: Addition of Angular Momenta: The Clebsch-Gordan Coefficients 478 Chapter 19: Time Independent Perturbation Theory 493 Chapter 20: Effects of Magnetic Field 534 Chapter 21: The Variational Method 560 Chapter 22: The Helium Atom and the Exclusion Principle 579 Chapter 23: Some Select Topics 598 Chapter 24: Elementary Theory of Scattering 625 Chapter 25: Time Dependent Perturbation Theory 676 Chapter 26: The Semi-Classical Theory of Radiation and the Einstein Coefficients 716 Chapter 27: The Quantum Theory of Radiation and Its Interaction with Matter 742 Chapter 28: Relativistic Theory 779 Appendices 809 Index 855 www.com Acknow ledgements We are indebted to our colleagues at lIT Delhi and at Rajasthan University for many enlightening discussions related to many aspects of quantum mechanics. We are indebted to various authors and publishers for their permissions to use material from their publications. We are very grateful to Professor Victor F.
Weis- skopf and W. for allowing us to reprint the essay What is Quan- tum Mechanics-in fact, Professor V. Lakshminarayanan of University of Mis- souri at St. Louis had brought to our attention the essay by Professor Weisskopf.
We express our gratitude to the Niels Bohr Archives in Copenhagen for provid- ing the original photograph on the Stem-Gerlach experiment. We are also grate- ful to Addison-Wesley Publishing Co., Reading; Mc-Graw-Hill Book Co., New York; Oliver and Boyd Ltd., Edinburgh; Oxford University Press, Oxford; Penguin Books Ltd., Middlesex and Prentice Hall Inc., Englewood Cliffs for permission to adapt figures, tables and a few analysis from the following books: G. Baym, Lec- tures on Quantum Mechanics, Addison-Wesley (1969); P. Dirac, Principles of Quantum Mechanics, Oxford University Press (1958); H.
Enge, Introduction to Nuclear Physics, Addison-Wesley (1966); R. Sands, The Feynman Lectures on Physics, Vol. III, Addison-Wesley (1965); LS. Hughes, Elementary Particles, Penguin (1972); v.
Galit- sky, Problems in Quantum Mechanics, Prentice-Hall (1963); R. Cunningham, Quantum Mechanics, Oliver and Boyd (1967); L. Wilson, Introduction to Quantum Mechanics, McGraw-HilI (1935); L. Schiff, Quantum Mechanics, McGraw-Hili (1955); N.
PomiIla, Atomic Theory, McGraw-Hili (1969). We are very grateful to Mr Monish Das, Dr. Rajeev Jindal, Dr. Vipul Rastogi, Dr.
Mukesh Pratap Singh and Mr Prem Bhawnani for their help in creating some of the figures using GNUPLOT and Mathematica. We also thank Mr. Norman Lim of National University of Singapore for carrying out numerical calculations associ- ated with some of the figures in Chapter 8. We are also grateful to Mr Monish Das, Mr Prem Bhawnani and Mrs.
Bhawna Bhardwaj, for their help in the preparation of the manuscript. We end with the quotation (which we found in a book by G. Squires): I have learnt much from my teachers, but more from my pupils. To all our pupils, we owe a very special debt.
Lokanathan IX www.com Preface Quantum mechanics has long been recognized as an essential ingredient in the training of a student in Physics, Chemistry and Electrical Engineering. No matter which specialization the student chooses in his later career, it is necessary for him to develop not merely an understanding of the basic principles of quantum mechanics but an ability to apply these principles in his field. Textbooks on quantum mechanics addressed to students at under-graduate and post-graduate levels generally have few solved problems; moreover, the problems suggested are largely of an academic nature with a greater emphasis on theory than on applications. This book makes an attempt to present the basic concepts in quan- tum mechanics with emphasis on applications in other areas like nuclear physics, astrophysics, solid state physics, quantum optics, etc.
Each chapter is followed by a number of problems. The solutions to most of these problems have been given at the end of the chapter. This will enable the student to refer to the solutions when necessary and at the same time provide him an opportunity, if he so wishes, to solve some or all of these problems by himself before consulting the solutions. The emphasis has not been on rigour but on making results plausible and helping the students to become familiar with methods for solving problems.
The book has evolved from lectures given by the authors (to students of physics and engineering) at the Indian Institute of Technology, New Delhi and at University of Rajasthan, Jaipur. As a prologue to the book we have reprinted an essay, 'What is Quantum Mechanics?', by Professor Victor F. It is so well written that we felt it would motivate the reader to learn more of this fascinating subject. Following the prologue are two short chapters on mathematical preliminaries-one on the Dirac delta function and the other on Fourier transforms.
The two topics are so extensively used in quantum mechanics that even if the students are familiar with them, it would be of great advantage to discuss these topics (again) in about two to three lectures right in the beginning of the course. After the mathematical preliminaries, we have discussed (in Chapter 3) wave- particle duality and the uncertainty principle. In Chapter 4, we have introduced the SchrOdinger equation and in Chapter 5, we have discussed the solutions of the SchrOdinger equation corresponding to a free particle leading to a study of the time evolution of a wave packet. In Chapter 6 bound state solutions of the Schrodinger equation are discussed.com xu We have given a somewhat lengthy account of the Dirac notation of bras and kets; since this is used frequently in advanced work, it was felt that the student would benefit greatly if he became familiar with it right at the beginning.
Imme- diately after solving the SchrOdinger equation for the linear harmonic oscillator problem (Chapter 7), we usually introduce Dirac's bra and ket algebra (Chapter 11) and then solve again the linear harmonic oscillator problem using the bra-ket algebra (Chapter 12). It is then very straightforward to study the time evolution of the coherent state and its relationship with the classical oscillator; such an analysis brings out many salient aspects of quantum mechanics. After this, we usually discuss the angular momentum problem in detail (Chapters 9, 13 and 15), which allows us to obtain very elegantly the expressions for spherical harmonics. We then obtain solutions of the Schrodinger equation for spherically symmetric potentials-in particular, for the hydrogen atom problem and for the three- dimen- sional isotropic oscillator problem (Chapter 10).
Chapter 14 is on the Stern-Gerlach experiment which is indeed one of the most beautiful experiments in quantum mechanics. The principle of magnetic resonance has also been discussed and, in the process, we have been able to give an exact solution of the Schrodinger equation corresponding to a time dependent Hamiltonian. The chapter also has a short essay on the EPR Paradox. Chapter 16 discusses the double well problem and also the Kronig-Penney model.
In Chapter 17 we have also given a detailed account of the JWKB solu- tions of the SchrOdinger equation; the JWKB methodology represents one of the very powerful approximate methods that is extensively used not only in quantum mechanics but also in many other areas. The first seventeen chapters have been developed into a video course which is now available through FITT (Foundation for Innovation & Technology Transfer) at lIT, New Delhi. Chapter 18 discusses addition of angular momenta and introduces the Clebsch- Gordan coefficients. Chapters 19, 20, 2 I and 22 discuss time independent pertur- bation theory, effects of magnetic fields, the variational method and the Helium atom.
In Chapter 23, we have given a detailed discussion of some select topics which range from the concept of quasi-bound states to the Thomas-Fermi model of the atom. Chapters 24, 25, 26 and 27 discuss the elementary theory of scattering, time dependent perturbation theory, the semi-classical theory of radiation and the quan- tum theory of radiation. Finally in Chapter 28, the Dirac equation and some of its solutions are discussed. In order to have a better appreciation of the theory, most of the figures corre- spond to actual numerical calculations; these were generated using GNUPLOT and Mathematica.
We do hope the reader enjoys going through the book; we would greatly appre- ciate receiving suggestions for further improvement.com xiii We have dedicated this book to Professor H. Bethe who taught one of us (AG) at Cornell, to Professor R. Serber who taught the other (SL) at Columbia, and to Professor D. Kothari and Professor R.
Majumdar who taught both of us at Delhi University. To them we would like to extend our gratitude and respect for introducing us to the most beautiful subject in science.com Prologue * What is quantum mechanics? Victor F. Weisskopf There is no doubt that the most outstanding development in modern science was the conception of quantum mechanics. It showed, better than anything else, the human capability to comprehend the fundamental principles that underlie the world in which we live-even when these principles run contrary to our experience in dealing with our everyday environment.