Quantum Mechanics: Theory and Applications

905 Pages·2004·23.62 MB·English

by  Ajoy Ghatak

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
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.A.
Editorial Advisory Board:
JAMES T. CUSHING, University of Notre Dame, US.A.
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.
PEKKA J. LAHTI, University ofTurku, Finland
ASHER PERES, Israel Institute ()fTechnology, Israel
EDUARD PRUGOVECKI, University of Toronto, Canada
TONY SUDBURY, University ()fYork, u.K.
HANS-JURGEN TREDER, Zentralinstitutfiir Astrophysik der Akademie der
Wissenschaften, Germany
Volume 137
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.V.
A c.r.P. 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.
To our teachers:
Professor H.A. Bethe
Professor D.S. Kothari
Professor RC. Majumdar
Professor R Serber
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
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
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.H. Freeman & Co. 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.A.M. Dirac, Principles
of Quantum Mechanics, Oxford University Press (1958); H. Enge, Introduction
to Nuclear Physics, Addison-Wesley (1966); R.P. Feynman, R.B. Leighton and
M. Sands, The Feynman Lectures on Physics, Vol. III, Addison-Wesley (1965);
LS. Hughes, Elementary Particles, Penguin (1972); v.I. Kogan and V.M. Galit
sky, Problems in Quantum Mechanics, Prentice-Hall (1963); R.A. Newing and J.
Cunningham, Quantum Mechanics, Oliver and Boyd (1967); L. Pauling and E.B.
Wilson, Introduction to Quantum Mechanics, McGraw-HilI (1935); L.I. Schiff,
Quantum Mechanics, McGraw-Hili (1955); N. TraIIi and F.R. 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.L. Squires): I have
learnt much from my teachers, but more from my pupils. To all our pupils, we owe
a very special debt.
Ajoy Ghatak
S. Lokanathan
IX
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. Weisskopf. 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.
Xl
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.