Quantum Mechanics: Problems and Solutions [1 ed.] 9814800724, 9789814800723

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Quantum Mechanics

Quantum Mechanics

Problems and Solutions

K. Kong Wan

June 23, 2020 10:49

JSP Book - 9in x 6in

00-Solution˙Manual-Prelims

Published by Jenny Stanford Publishing Pte. Ltd. Level 34, Centennial Tower 3 Temasek Avenue Singapore 039190 Email: [email protected] Web: www.jennystanford.com British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Quantum Mechanics: Problems and Solutions c 2021 Jenny Stanford Publishing Pte. Ltd. Copyright � All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4800-72-3 (Paperback) ISBN 978-0-429-29647-5 (eBook)

June 23, 2020 10:49

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To my beautiful granddaughter Orly Rose, whose new arrival brings infinite joy and jubilation.

00-Solution˙Manual-Prelims

June 23, 2020 10:49

JSP Book - 9in x 6in

00-Solution˙Manual-Prelims

Contents

xi

Preface 1 Structure of Physical Theories

1

2 Classical Systems

3

3 Probability Theory for Discrete Variables

5

4 Probability Theory for Continuous Variables

9

5 Quantum Mechanical Systems

17

6 Three-Dimensional Real Vectors

21

7 Matrices and Their Relations with Vectors

27

 8 Operations on Vectors in IE

3

9 Special Operators on IE 3

35 41

10 Probability, Selfadjoint Operators, Unit Vectors and the Need for Complexness

51

11 Complex Vectors

55

12 N-Dimensional Complex Vectors

59

13 Operators on N-Dimensional Complex Vectors

65

14 Model Theories Based on Complex Vector Spaces

81

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viii Contents

15 Spectral Theory in Terms of Stieltjes Integrals

89

16 Infinite-Dimensional Complex Vectors and Hilbert Spaces

93

 17 Operators in a Hilbert Space H

99

 18 Bounded Operators on H

107

 19 Symmetric and Selfadjoint Operators in H

115

 20 Spectral Theory of Selfadjoint Operators in H

121

 21 Spectral Theory of Unitary Operators on H

127

22 Selfadjoint Operators, Unit Vectors and Probability Distributions

129

23 Physics of Unitary Transformations

133

24 Direct Sums and Tensor Products of Hilbert Spaces and Operators

135

25 Pure States

143

26 Observables and Their Values

145

27 Canonical Quantisation

149

28 States, Observables and Probability Distributions

161

29 Time Evolution

167

30 On States after Measurement

175

31 Pure and Mixed States

177

32 Superselection Rules

181

33 Many-Particle Systems

185

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Contents

34 Conceptual Issues

187

35 Harmonic and Isotropic Oscillators

189

36 Angular Momenta

201

37 Particles in Static Magnetic Fields

225

Bibliography

229

ix

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Preface

This is a solutions manual to accompany the book Quantum Mechanics: A Fundamental Approach by the author published in 2019 by Jenny Stanford Publishing, Singapore. It provides detailed solutions to all the questions listed at the end of each chapter of the book, except for the introductory Chapters 1 and 2. These questions are reproduced here chapter by chapter, followed by their solutions, which are labelled to correspond to the questions. For example, SQ3(1) is the solution to question Q3(1), which is the first question listed in Exercises and Problems at the end of Chapter 3 of the book. The solutions presented make full use of the materials in the book. All the theorems, definitions, examples, comments, properties, postulates and equations in the book are referred to by their chapter or section numbers. For instance, Theorem 13.3.2(2) refers to the second theorem in section 13.3.2, Eq. (4.18) refers to equation (4.18) in Chapter 4, P15.1(5) refers to property (5) in section 15.1, and C28.2(3) refers to comment (3) in section 28.2. Equation labelling in terms of (∗), (∗∗), (∗∗∗) and (∗∗∗∗) is introduced here in some questions when they are needed for reference later in their solutions. K. Kong Wan St Andrews Scotland

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SolutionsManual(C2019)

Chapter 1

Structure of Physical Theories

This introductory chapter sets out a general structure of physical theories which is applicable to both classical and quantum me­ chanics. We start with measurable properties of a given physical system, be it classical or quantum. These properties are called observables. We then introduce a definition of the state of a physical system in terms of measured values of a sufficiently large set of observables. A theory to describe the system should consists of four basic components: 1. Basic mathematical framework This comprises a set of elements endowed with some specific mathematical structure and properties. In mathematics such a set is generally known as a space. 2. Description of states States are described by elements of the space in the chosen mathematical framework. For this reason the space is called the state space of the system. 3. Description of observables Observables are to be described by quantities defined on the state space. The description should yield all possible values of observables. The relationship between observables and states should be explicitly stated. The following two cases are of particular interest:

Quantum Mechanics: Problems and Solutions K. Kong Wan c 2021 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4800-72-3 (Paperback), 978-0-429-29647-5 (eBook) www.jennystanford.com

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2 Structure of Physical Theories

(1) For a deterministic theory like classical mechanics a state should determine the values of all observables. (2) For a probabilistic theory like quantum mechanics a state should determine the probability distribution of the values of all observables. 4. Description of time evolution (dynamics).

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Chapter 2

Classical Systems

This chapter sets out some general physical properties of classical systems which are divided into discrete and continuous: 1. Discrete systems These are systems of discrete point particles. The specific structure of classical mechanics is presented with position, linear and angular momenta serving as basic observables. 2. Continuous systems These systems are illustrated by a vibrat­ ing string. Continuous systems have different kinds of properties and observables, e.g., wave properties. In particular we have discussed: (1) Description of states by solutions of the classical wave equation. (2) The concept of eigenfunctions with orthonormality property and their superposition and interference. (3) The concept of a complete set of states. These discussions are given specifically to provide an intuition to help a better understanding of similar properties of quantum systems.

Quantum Mechanics: Problems and Solutions K. Kong Wan c 2021 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4800-72-3 (Paperback), 978-0-429-29647-5 (eBook) www.jennystanford.com

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Chapter 3

Probability Theory for Discrete Variables

Q3(1)

Prove Theorem 3.4(1).

SQ3(1) Theorem 3.4(1) can be proved using properties PM3.4(1), PM3.4(2) and PM3.4(3) in Definition 3.4(2). (1) To prove Eq. (3.25) let E be any event. Then E ∪ ∅ = E and E ∩ ∅ = ∅. By PM3.4(3) we have,        M p  E ∪ ∅ = M p  E    ⇒ M p ∅ = 0. Mp E ∪ ∅ = Mp E + Mp ∅   (2) To prove Eq. (3.26) we start with M p Sam = 1 by PM3.4(2). Since E ∩ E c = ∅ and E ∪ E c = Sam we have         M p Sam = M p E ∪ E c = M p E + M p E c = 1     ⇒ Mp E c = 1 − Mp E . (3) To that E 1 ⊂ E 2 ⇒ E 2 =  prove Eq. (3.27) we first observe  E 2 − E 1 ∪ E 1 . Since (E 2 − E 1 and E 1 are disjoint, i.e., (E 2 − E 1 ∩ E 1 = ∅, we have      Mp E2 = Mp E2 − E1 ∪ E1       = Mp E2 − E1 + Mp E1 ≥ Mp E1 .

Quantum Mechanics: Problems and Solutions K. Kong Wan c 2021 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4800-72-3 (Paperback), 978-0-429-29647-5 (eBook) www.jennystanford.com

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6 Probability Theory for Discrete Variables

(4) The proof of Eq. (3.28) is based on the decomposition, obvious from the Venn diagram,     E1 = E1 − E2 ∪ E1 ∩ E2 ,     where E 1 − E 2 and E 1 ∩ E 2 are disjoint. Then PM3.4(3) implies       Mp E1 = Mp E1 − E2 + Mp E1 ∩ E2 , which is Eq. (3.28).

Two cases are noteworthy:

(a) E 1 and E 2 are disjoint, i.e., E 1 ∩ E 2 = ∅. Then E 1 − E 2 = E 1 , we have M p (E 1 ∩ E 2 ) = 0 and

M p (E 1 − E 2 ) = M p (E 1 ).

Equation (3.28) is again satisfied. (b) E 1 is a subset of E 2 . Then E1 ∩ E2 = E1

and

E 1 − E 2 = ∅.

Equation (3.28) is again satisfied in these two cases. (5) To prove Eq. (3.29) we first note that if E 1 and E 2 are disjoint the equation is obviously true, i.e.,       Mp E1 ∪ E2 = Mp E1 + Mp E2 by PM3.4(3). If E 1 and E 2 are not disjoint, then the Venn diagram tells us that   E 1 ∪ E 2 = E 1 − E 2 ∪ E 2,   where E 1 − E 2 and E 2 are disjoint. We have, by PM3.4(3),       Mp E1 ∪ E2 = Mp E1 − E2 + Mp E2 . Using Eq. (3.28) we immediately get         Mp E1 ∪ E2 = Mp E1 + Mp E2 − Mp E1 ∩ E2 , which is Eq. (3.29). We have assumed E 2 is a subset of E 1 in the above proof. When E 1 is a subset of E 2 we can similarly establish the result.

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Probability Theory for Discrete Variables

Q3(2)

Prove Theorem 3.5(1).

SQ3(2)

Equation (3.36) is proved as follows:

Var (℘ A ) =

n  2  a − E(℘ A ) ℘ A (a )

=1

n    = a2 − 2a E(℘ A ) + E(℘ A )2 ℘ A (a ) =1 n 

 =

a2 ℘ A (a )

− E(℘ A )2 ,

=1

using the fact that n  n    − 2a E(℘ A ) ℘ A (a ) = −2E(℘ A )2 and ℘ A (a ) = 1. =1

=1

Q3(3) What is the value F(a4 ) − F(a3 ) of the probability distribution function in Eq. (3.38)? SQ3(3)

The value F(a4 ) − F(a3 ) is equal to ℘(a4 ).

Q3(4) In an experiment of tossing a fair die a number from 1 to 6 will be obtained with equal probabilities. (a) Write down the probability mass function ℘ and evaluate the expectation value and the uncertainty. (b) Write down the corresponding probability distribution function F(τ ) and sketch a plot of F(τ ) versus τ . What are the values F(τ ) at τ = 0.9, 1, 2.5, 6 and 6.1? SQ3(4)(a) For a fair die every number is equally likely to appear in a toss. The probability mass function is a function ℘ on the sample space Sam := {a1 = 1, a2 = 2, a3 = 3, a4 = 4, a5 = 5, a6 = 6} defined by ℘(a ) = 1/6 for all a ∈ Sam . The expectation value is 1 1 1 1 1 1 1 × + 2 × + 3 × + 4 × + 5 × + 6 × = 3.5. 6 6 6 6 6 6 The variance is given by Theorem 3.5(1) to be

 1 1 1 1 1 1 1 × + 4 × + 9 × + 16 × + 25 × + 36 × − 3.52 6 6 6 6 6 6

≈ 2.9.

7

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8 Probability Theory for Discrete Variables

The uncertainty =

√ √ variance with an approximate value 2.9.

SQ3(4)(b) The probability distribution function F(τ ) is piecewise-constant with discontinuous steps occurring at τ = 1, 2, 3, 4, 5, 6. Explicitly F(τ ) is related to ℘(a ) by ⎧ 0, τ