Series in PURE and APPLIED PHYSICS

Concepts in
Quantum
Mechanics

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Concepts in Quantum Mechanics
Vishu Swarup Mathur

Surendra Singh

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A C H A P M A N & H A L L B O O K

CRC Press is an imprint of the
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Series in PURE and APPLIED PHYSICS

Vishnu Swarup Mathur
Surendra Singh

Concepts in
Quantum
Mechanics

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Library of Congress Cataloging-in-Publication Data

Mathur, Vishnu S. (Vishnu Swarup), 1935-
Concepts in quantum mechanics / Vishnu S. Mathur, Surendra Singh.

p. cm. -- (CRC series in pure and applied physics)
Includes bibliographical references and index.
ISBN 978-1-4200-7872-5 (alk. paper)
1. Quantum theory. I. Singh, Surendra, 1953- II. Title. III. Series.

QC174.12.M3687 2008
530.12--dc22 2008044066

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Dedicated to the memory of
Professor P. A. M. Dirac

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Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

1 NEED FOR QUANTUM MECHANICS AND ITS PHYSICAL BASIS 1
1.1 Inadequacy of Classical Description for Small Systems . . . . . . . . . . . 1

1.1.1 Planck’s Formula for Energy Distribution in Black-body Radiation 1
1.1.2 de Broglie Relation and Wave Nature of Material Particles . . . . . 2
1.1.3 The Photo-electric Effect . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.4 The Compton Effect . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.5 Ritz Combination Principle . . . . . . . . . . . . . . . . . . . . . . 6

1.2 Basis of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Principle of Superposition of States . . . . . . . . . . . . . . . . . . 9
1.2.2 Heisenberg Uncertainty Relations . . . . . . . . . . . . . . . . . . . 12

1.3 Representation of States . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Dual Vectors: Bra and Ket Vectors . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5.1 Properties of a Linear Operator . . . . . . . . . . . . . . . . . . . . 16
1.6 Adjoint of a Linear Operator . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.7 Eigenvalues and Eigenvectors of a Linear Operator . . . . . . . . . . . . . 18
1.8 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.8.1 Physical Interpretation of Eigenstates and Eigenvalues . . . . . . . 20
1.8.2 Physical Meaning of the Orthogonality of States . . . . . . . . . . 21

1.9 Observables and Completeness Criterion . . . . . . . . . . . . . . . . . . . 21
1.10 Commutativity and Compatibility of Observables . . . . . . . . . . . . . . 23
1.11 Position and Momentum Commutation Relations . . . . . . . . . . . . . . 24
1.12 Commutation Relation and the Uncertainty Product . . . . . . . . . . . . 26
Appendix 1A1: Basic Concepts in Classical Mechanics . . . . . . . . . . . . . . 31

1A1.1 Lagrange Equations of Motion . . . . . . . . . . . . . . . . . . . . 31
1A1.2 Classical Dynamical Variables . . . . . . . . . . . . . . . . . . . . . 32

2 REPRESENTATION THEORY 35
2.1 Meaning of Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2 How to Set up a Representation . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3 Representatives of a Linear Operator . . . . . . . . . . . . . . . . . . . . . 37
2.4 Change of Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.5 Coordinate Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.5.1 Physical Interpretation of the Wave Function . . . . . . . . . . . . 44
2.6 Replacement of Momentum Observable pˆ by −i~ ddqˆ . . . . . . . . . . . . . 45
2.7 Integral Representation of Dirac Bracket 〈A2| Fˆ |A1 〉 . . . . . . . . . . . 50
2.8 The Momentum Representation . . . . . . . . . . . . . . . . . . . . . . . . 52

2.8.1 Physical Interpretation of Φ(p1 , p2 , · · ·pf ) . . . . . . . . . . . . . . 52
2.9 Dirac Delta Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

2.9.1 Three-dimensional Delta Function . . . . . . . . . . . . . . . . . . 55

2.9.2 Normalization of a Plane Wave . . . . . . . . . . . . . . . . . . . . 56
2.10 Relation between the Coordinate and Momentum Representations . . . . 56

3 EQUATIONS OF MOTION 67
3.1 Schro¨dinger Equation of Motion . . . . . . . . . . . . . . . . . . . . . . . . 67
3.2 Schro¨dinger Equation in the Coordinate Representation . . . . . . . . . . 69
3.3 Equation of Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4 Stationary States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.5 Time-independent Schro¨dinger Equation in the Coordinate Representation 72
3.6 Time-independent Schro¨dinger Equation in the Momentum Representation 74

3.6.1 Two-body Bound State Problem (in Momentum Representation) for
Non-local Separable Potential . . . . . . . . . . . . . . . . . . . . . 76

3.7 Time-independent Schro¨dinger Equation in Matrix Form . . . . . . . . . . 77
3.8 The