Quantum Mechanics I. The Fundamentals

606 Pages·2023·22.148 MB·English

by  S. Rajasekar, R. Velusamy

Quantum Mechanics I
Quantum Mechanics I: The Fundamentals provides a graduate-level account of the behaviour
of matter and energy at the molecular, atomic, nuclear, and sub-nuclear levels. It covers basic
concepts, mathematical formalism, and applications to physically important systems.
This fully updated new edition addresses many topics not typically found in books at this level,
including:
• Bound state solutions of quantum pendulum
• Morse oscillator
• Solutions of classical counterpart of quantum mechanical systems
• A criterion for bound state
• Scattering from a locally periodic potential and reflection-less potential
• Modified Heisenberg relation
• Wave packet revival and its dynamics
• An asymptotic method for slowly varying potentials
• Klein paradox, Einstein-Podolsky-Rosen (EPR) paradox, and Bell’s theorem
• Delayed-choice experiments
• Fractional quantum mechanics
• Numerical methods for quantum systems
A collection of problems at the end of each chapter develops students’ understanding of both
basic concepts and the application of theory to various physically important systems. This book,
along with the authors’ follow-up Quantum Mechanics II: Advanced Topics, provides students
with a broad, up-to-date introduction to quantum mechanics.

Quantum Mechanics I
The Fundamentals
Second Edition
S. Rajasekar and R. Velusamy
Second edition published 2023
by CRC Press
4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
and by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
© 2023 S. Rajasekar and R. Velusamy
First edition published by CRC Press 2015
CRC Press is an imprint of Informa UK Limited
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of the Copyright, Designs and Patents Act 1988.
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British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging‑in‑Publication Data
Names: Rajasekar, S. (Shanmuganathan), 1963- author. | Velusamy, R., 1952-
author.
Title: Quantum mechanics / S. Rajasekar, R. Velusamy.
Description: Second edition. | Boca Raton : CRC Press, 2022. | Includes
bibliographical references and index. | Contents: v. 1. The fundamentals
-- v. 2. Advanced topics. | Summary: “Quantum Mechanics I: The
Fundamentals provides a graduate-level account of the behavior of matter
and energy at the molecular, atomic, nuclear, and sub-nuclear levels. It
covers basic concepts, mathematical formalism, and applications to
physically important systems. This fully updated new edition addresses
many topics not typically found in books at this level, including: Bound
state solutions of quantum pendulum Morse oscillator Solutions of
classical counterpart of quantum mechanical systems A criterion for
bound state Scattering from a locally periodic potential and
reflection-less potential Modified Heisenberg relation Wave packet
revival and its dynamics An asymptotic method for slowly varying
potentials Klein paradox, Einstein-Podolsky-Rosen (EPR) paradox, and
Bell’s theorem Delayed-choice experiments Fractional quantum mechanics
Numerical methods for quantum systems A collection of problems at the
end of each chapter develops students’ understanding of both basic
concepts and the application of theory to various physically important
systems. This book, along with the authors’ follow-up Quantum Mechanics
II: Advanced Topics, provides students with a broad, up-to-date
introduction to quantum mechanics. Print Versions of this book also
include access to the ebook version”-- Provided by publisher.
Identifiers: LCCN 2022021033 | ISBN 9780367769987 (v. 1 ; hardback) | ISBN
9780367776367 (v. 1 ; paperback) | ISBN 9781003172178 (v. 1 ; ebook) |
ISBN 9780367770006 (v. 2 ; hardback) | ISBN 9780367776428 (v. 2 ;
paperback) | ISBN 9781003172192 (v. 2 ; ebook)
Subjects: LCSH: Quantum theory.
Classification: LCC QC174.12 .R348 2022 | DDC 530.12--dc23/eng20220518
LC record available at https://lccn.loc.gov/2022021033
ISBN: 978-0-367-76998-7 (hbk)
ISBN: 978-0-367-77636-7 (pbk)
ISBN: 978-1-003-17217-8 (ebk)
DOI: 10.1201/9781003172178
Typeset in CMR10 font
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
To our wives.

Contents
Preface xv
About the Authors xix
1 Why Was Quantum Mechanics Developed? 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Black Body Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Photoelectric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Hydrogen Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Franck–Hertz Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.6 Stern–Gerlach Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Correspondence Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.8 Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.9 Specific Heat Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.10 de Broglie Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.11 Particle Diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.12 Wave-Particle Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.13 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.14 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2 Schro¨dinger Equation and Wave Function 27
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2 Construction of Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . 27
2.3 Solution of Time-Dependent Equation . . . . . . . . . . . . . . . . . . . . 29
2.4 Physical Interpretation of ψ ψ . . . . . . . . . . . . . . . . . . . . . . . . 30
∗
2.5 Conditions on Allowed Wave Functions . . . . . . . . . . . . . . . . . . . 32
2.6 Box Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 A Special Feature of Occurrence of i in the Schr¨odinger Equation . . . . . 34
2.8 Conservation of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.9 Expectation Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.10 Ehrenfest’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.11 Basic Postulates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.12 Time Evolution of Stationary States . . . . . . . . . . . . . . . . . . . . . 43
2.13 Conditions for Allowed Transitions . . . . . . . . . . . . . . . . . . . . . . 44
2.14 Orthogonality of Two States . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.15 Phase of the Wave Function . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.16 Classical Limit of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 48
2.17 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.18 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.19 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
vii
viii Contents
3 Operators, Eigenvalues and Eigenfunctions 55
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.3 Commuting and Noncommuting Operators . . . . . . . . . . . . . . . . . 58
3.4 Self-Adjoint and Hermitian Operators . . . . . . . . . . . . . . . . . . . . 62
3.5 Discrete and Continuous Eigenvalues . . . . . . . . . . . . . . . . . . . . . 65
3.6 Meaning of Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . 67
3.7 Parity Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.8 Some Other Useful Operators . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.10 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4 Exactly Solvable Systems I: Bound States 79
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Classical Probability Distribution . . . . . . . . . . . . . . . . . . . . . . . 80
4.3 Free Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.4 Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
4.5 Particle in the Potential V(x)=x2k, k =1,2,... . . . . . . . . . . . . . . 92
4.6 Particle in a Box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.7 Morse Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.8 Po¨schl–Teller Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.9 Quantum Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.10 Criteria for the Existence of a Bound State . . . . . . . . . . . . . . . . . 105
4.11 Time-Dependent Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . 107
4.12 Damped and Forced Linear Harmonic Oscillator . . . . . . . . . . . . . . 108
4.13 Two-Dimensional Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.14 Rigid Rotator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.15 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.16 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5 Exactly Solvable Systems II: Scattering States 129
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.2 Potential Barrier: Tunnel Effect . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3 Finite Square-Well Potential . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.4 Potential Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.5 Locally Periodic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.6 Reflectionless Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
5.7 Dynamical Tunnelling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6 Matrix Mechanics 159
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.2 Linear Vector Space and Tensor Products . . . . . . . . . . . . . . . . . . 160
6.3 Matrix Representation of Operators and Wave Function . . . . . . . . . . 162
6.4 Unitary Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5 Schr¨odinger Equation and Other Quantities in Matrix Form . . . . . . . . 165
6.6 Application to Certain Systems . . . . . . . . . . . . . . . . . . . . . . . . 166
Contents ix
6.7 Dirac’s Bra and Ket Notations . . . . . . . . . . . . . . . . . . . . . . . . 171
6.8 Dimensions of Kets and Bras . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.9 Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
6.10 Symmetry Operators in Hilbert Space . . . . . . . . . . . . . . . . . . . . 177
6.11 Projection and Displacement Operators . . . . . . . . . . . . . . . . . . . 179
6.12 Quaternionic Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . 181
6.13 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.14 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
6.15 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7 Various Pictures and Density Matrix 189
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.2 Schr¨odinger Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
7.3 Heisenberg Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.4 Interaction Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
7.5 Comparison of Three Representations . . . . . . . . . . . . . . . . . . . . 197
7.6 Density Matrix for a Single System . . . . . . . . . . . . . . . . . . . . . . 197
7.7 Density Matrix for an Ensemble . . . . . . . . . . . . . . . . . . . . . . . 199
7.8 Time Evolution of Density Operator . . . . . . . . . . . . . . . . . . . . . 201
7.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.10 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
7.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8 Heisenberg Uncertainty Principle 205
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.2 The Classical Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . 206
8.3 Heisenberg Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . . . 206
8.4 Condition for Minimum Uncertainty Product . . . . . . . . . . . . . . . . 211
8.5 Implications of Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . 212
8.6 Illustration of Uncertainty Relation . . . . . . . . . . . . . . . . . . . . . 213
8.7 Some Extensions of Uncertainty Relation . . . . . . . . . . . . . . . . . . 216
8.8 The Modified Heisenberg Relation . . . . . . . . . . . . . . . . . . . . . . 217
8.9 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.10 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
8.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9 Momentum Representation 223
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.2 Momentum Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.3 Schr¨odinger Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.4 Expressions for X and p . . . . . . . . . . . . . . . . . . . . . . . . . 226
h i h i
9.5 Transformation Between Momentum and Coordinate Representations . . 228
9.6 Operators in Momentum Representation . . . . . . . . . . . . . . . . . . . 228
9.7 Momentum Function of Some Systems . . . . . . . . . . . . . . . . . . . . 230
9.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.9 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234