Problems and Solutions in Nonrelativistic Quantum Mechanics

507 Pages·2002·41.234 MB·English

by  Anton Z. Capri

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NONRELATIVISTIC
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MECHANICS
Anton Z. Capri
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NONRELATIVISTIC
QUANTUM
MECHANICS
Anton Z. Capri
Department of Physics
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University of Alberta, Canada
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Published by
World Scientific Publishing Co. Pte. Ltd.
P O Box 128, Farrer Road, Singapore 912805
USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661
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Library of Congress Cataloging-in-Publication Data
Capri, Anton Z.
Problems & solutions in nonrelativistic quantum mechanics / Anton Z. Capri.
p. cm.
Includes bibliographical references.
ISBN 9810246331 (alk. paper) — ISBN 9810246501 (pbk.: alk. paper)
1. Nonrelativistic quantum mechanics - Problems, exercises, etc. I. Title: Problems and
solutions in nonrelativistic quantum mechanics. II. Title: Nonrelativistic quantum
mechanics. III. Title.
QC174.24.N64 C374 2002
530.12'076--dc21 2002029614
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2002 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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This book is printed on acid-free paper.
To bkaidrite, who knows that physics is simple because "everything equals
zero".

Preface
Soon after the first edition of Nonrelativistic Quantum Mechanics appeared,
I received numerous requests for solutions to the problems in that book. To
remedy this situation I started by writing out solutions to the more difficult
problems, but as I proceeded with the third edition of Nonrelativistic Quantum
Mechanics I also revised some of the problems and added quite a few others.
Since in constructing these new problems I had to solve them, in the first place,
to be sure that they were indeed problems that students could solve, I finally
went on to write out solutions to all the problems. However, I did not simply
want a compendium of solutions of the Schrodinger equation since with pro-
grams such as Maple or Mathematica these solutions are accessible to every
student. Instead I wanted to concentrate on problems that teach quantum me-
chanics. It is with this in mind that I began to collect and solve problems. My
idea was to provide a means for students to learn quantum mechanics by "doing
it". This is why the book begins with extremely simple problems and progresses
to more difficult ones.
Some of the problems extend results that are usually taught in a course on
quantum mechanics. But, by having the students obtain the results themselves
they are more likely to retain the ideas and at the same time gain confidence in
their own abilities.
As usual, I tested most of these problems on my students. Sometimes they
came up with very original ways of looking at old problems. I have learned a lot
from my students. It is this learning process that led me to occasionally intro-
duce more than one way of solving a problem since the solutions are intended
to help students to obtain a better understanding of the techniques involved in
tackling problems in quantum mechanics.
The notation and methods used are those explained in Nonrelativistic Quan-
tum Mechanics and I frequently refer to chapters from that book. The chapter
headings are also the same as in Nonrelativistic Quantum Mechanics. Never-
theless, the present book is independent and should serve as a companion to
any of the numerous excellent books on quantum mechanics. Throughout the
book I have used Gaussian units since these are the units most commonly used
in atomic physics. I also tried to arrange the problems according to increasing
degree of difficulty. This, was not always possible since it would have meant
losing the possibility of arranging them according to topic.
It is a pleasure to thank Professor M. Razavy for his generous help in, not
only providing me with some wonderful problems and supplying me with nu-
merous references, but also for his constant moral support.
Of course the students who suffered through the courses in which I subjected
them to all sorts of quantum problems also deserve my heartfelt thanks. To their
credit, the undergraduates seldom complained. On the other hand, there was
many an evening, after I had assigned some more than usually difficult problems
in the graduate course on quantum mechanics, that walking down the hall of
the fourth floor of the physics building I heard my name muttered with less than
flattering epithets. Nevertheless, the graduate students survived and many, after
they completed their degree, even thanked me for what they had learned.
It is my hope that these problems and solutions will be of use to future
generations of physics students. At any rate they should provide more enter-
tainment than solving cross word puzzles.
A.Z.Capri
Edmonton, Alberta
July, 2002.
Contents
1 The Breakdown of Classical Mechanics 1
1.1 Quantum Number of the Earth 1
1.2 Thermal Wavelength 2
1.3 Photons in a Beam 2
1.4 Hydrogen Atom and de Broglie 3
1.5 Vibrations in NaCl 4
1.6 Crystal Powder 4
1.7 Einstein Coefficients 5
2 Review of Classical Mechanics 8
2.1 Lagrangian and Hamiltonian for SHO 8
2.2 Lagrangian and Hamiltonian: Simple Pendulum 9
2.3 Bohr-Sommerfeld Quantization: SHO 9
2.4 Bohr-Sommerfeld: Particle in a Box 10
2.5 Larmor Frequency 11
2.6 Applicability of Bohr-Sommerfeld Quantization 12
2.7 Schrodinger and Hamilton-Jacobi 12
2.8 WKB Approximation 13
2.9 Dumbbell Molecule: Bohr-Sommerfeld 14
3 Elementary Systems 15
3.1 Commutator Identities 15
3.2 Complex Potential 16
3.3 Group and Phase Velocity 17
3.4 Linear Operators 18
3.5 Probability Density 19
3.6 Angular Momentum Operators 20
3.7 Beam of Particles 21
3.8 Time Evolution of Wave Function 22
3.9 Operator Hamiltonian 23
3.10 Zero of Energy 24
3.11 Some Commutators 25
3.12 Eigenfunction for a Simple Hamiltonian 26
4 One-Dimensional Problems 29
4.1 Potential Step 29
4.2 Deep Square Well 30
4.3 Hydrogenic Wavefunction 32
4.4 Bound State Wavefunction, Current, Momentum 33
4.5 Time Evolution for Particle in a Box 34
4.6 Particle in Box: Energy and Eigenfunctions 35
4.7 Particle in a Box 36
4.8 Particles Incident on a Potential 36
4.9 Two Beams Incident on a Potential 38
4.10 Ramsauer-Townsend Effect 39
4.11 Wronskian and Non-degeneracy in 1 Dimension 40
4.12 Symmetry of Reflection 41
4.13 Parity and Electric Dipole Moment 43
4.14 Bound State Degeneracy and Current 44
4.15 Car Reflected from a Cliff 45
5 More One-Dimensional Problems 48
5.1 Motion of a Wavepacket 48
5.2 Lowest Energy States 50
5.3 Particle at Rest 51
5.4 Scattering from Two Delta Functions 51
5.5 Reflection and Transmission Amplitudes: Phase Shifts 53
5.6 Oscillator Against a Solid Wall 55
5.7 Periodic Potential 56
5.8 Reflection and Transmission Through a Barrier 57
5.9 Hermite Polynomials: Integral Representation 58
5.10 Matrix Element Between Degenerate States 60
5.11 Hellmann-Feynman Theorem 61
6 Mathematical Foundations 63
6.1 Cauchy Sequence in a Finite Vector Space 63
6.2 Nonuniqueness of Schrodinger Representation 64
6.3 Degeneracy and Commutator 64
6.4 von Neumann's Example 65
6.5 Projection Operator 67
6.6 Spectral Resolution 68
6.7 Resolvent Operator 68
6.8 Deficiency Indices 69
6.9 Adjoint Operator 71
6.10 Projection Operator 72
6.11 Commutator of L and <p 74
z
6.12 Uncertainty Relation: L and cos tp, siny? 74
z
6.13 Domain of Kinetic Energy: Polar Coordinates 75
6.14 Self-Adjoint Extensions of p4 76
7 Physical Interpretation 79
7.1 Tetrahedral Die 79
7.2 Probabilities, Expectation Values, Evolution 79
7.3 (L) and (L) in an Eigenstate of L 80
x y z
7.4 Free Particle Propagator 81
7.5 Minimum Uncertainty Wavefunction 82
7.6 Spreading of a Wave Packet 84
7.7 Time-dependent Expectation Values 85
7.8 Ehrenfest Theorem 87
7.9 Compatibility Theorem 89
7.10 Constant of the Motion 90
7.11 Spreading of a Gaussian Wavepacket 90
7.12 Incorrect Time Operator 92
7.13 Probability to Find a Particle 92
7.14 Sphere Bouncing on Sphere 93
7.15 Cloud Chamber Tracks 95
7.16 Spin 1 Measurement in Two Directions 95
7.17 Particle in a Box: Probabilities and Evolution 97
7.18 Free Wave Equation: Translation Invariance 99
7.19 Free Wave Equation: Accelerated Frame 101
7.20 The Wigner Function 103
7.21 Properties of the Wigner Function 105
7.22 Uncertainty Relation and Wigner Function 106
8 Distributions and Fourier Transforms 108
8.1 Properties of the Delta Function 108
8.2 Representation of Delta Function 110
8.3 Normalization of Scattering Solution 112
8.4 Tempered Distribution 114
8.5 Fourier Transform of V 114
8.6 Tempered Distribution of Fast Decrease 115
8.7 A Useful Identity 116
8.8 A Representation of ¿(a:) 118
8.9 Fourier Transform of ¿'"'(x) 118
8.10 Value of xmS<-n\x) 119
8.11 Distribution Occurring in Fermi's Golden Rule 119
9 Algebraic Methods 122
9.1 An Operator Identity 122
9.2 Expectation Values: Simple Harmonic Oscillator 123
9.3 Angular Momentum Matrices 124
9.4 Displaced Oscillator 127
9.5 Dipole Matrix Elements 128
9.6 Scalar Operator 129
9.7 Probability to Obtain I, m 129
9.8 Probability to Obtain I, m Along Different Axis 130