Elementary quantum mechanics

438 Pages·1968·55.845 MB·English

by  Saxon, David S

ELEMENTARY
QUANTUM
MECHANICS
ELEMENTARY
QUANTUM
MECHANICS
David S. Saxon
University o/California, LosAngeles
no
HOLDEN-DAY
San Francisco, Cambridge. London,Amsterdam
© Copyright 1968 by Holden-Day, Inc., 500 Sansome
Street, San Francisco, California. All rights reserved. No
part of this book may be reproduced in any form, by
mimeograph or any other means, without permission in
writing from the publisher.
Library ofCongress Catalog Card Number 68-16996.
Printed in the United States ofAmerica.
Preface
This book is based on lectures given by the author in an intensive
undergraduate course in quantum mechanics which occupies a central
role in the physics curriculum at UCLA. It is a required course for all
third-year physics and astrophysics students, but it is taken by some
seniors and manygraduate students,both in physics and in related fields.
Students enrolling in the course are expected to have hadan introduc
tion to elementary Hamiltonian mechanics, to the extent of knowing,
for simple systems, what the Hamiltonian function is and what the
Hamiltonian equations are. Students are also expected to have had
training in mathematics through differential equations and Fouriersenes
and to have at least seen many ofthe special functions ofmathematical
physics. In an effort to keep the mathematics as simple as possible,
however, the first two-thirds of the book is largely confined to the con
sideration ofone-dimensional systems.
The stress throughout is on the formulation of quantum mechanics
and not on its applications. At UCLA the applications follow in immedi
ately subsequent courses selected from atomic, nuclear, solid state and
elementary particle physics. The lastchapteris intended to pave the way
for these applications; in it a number of relatively advanced topics are
somewhat briefly presented.
The coverage is rather broad and not everything is treated in depth.
Wherever the text is frankly introductory, however, references to a
complete treatment are given. In all other respects the book is self
contained. Experience with a preliminary edition has shown that it is
accessible to students and that they can learn from it largely by them
selves. To aconsiderabledegree the teacher is thus leftfree to illuminate
the subject in his own way.
One hundred and fifty problems are presented, and these play an im-
vi PREFACE
portant pedagogical role. The problems are not exclusively illustrative
of material presented in the text; they also amplify it. A significant
number are intended to broaden the scope ofthe course by pointing the
way to new topics and new points ofview. Many problems are too diffi
cult for the student to master in his first attempt. He is encouraged to
return to them again and again as his understanding grows. Eventually
he should be able to handle any and all of them. Answers or complete
solutions to some fifty representativeproblemsaregivenin Appendix Ill.
Aboutforty exercises are scattered throughoutthetext. Theseare mostly
concerned with the working out ofdetails, but not all ofthem are trivial.
At UCLA the materialinthetextis presentedinasequencecovering
two quarters. However, the text isalsointendedforuseinaone-semester
course; any, or all, of the starred sections in the table of contentS can
be omitted without harm to the logical development. Ifit is desired, on
the other hand, to use the textfor aone-yearcourse, some supplementa
tion would be desirable. The Heisenbergand interaction representations,
and transformation theory in general, are topics which at once come to
mind. At the applied level, the Zeeman and Stark effects, Bloch waves,
the Hartree-Fock and Fermi-Thomas methods, simple molecules and
isotopic spin are a suitable list from which to choose.
The author has benefited from numerous criticisms and suggestions
from a host of colleagues and students. To each of them, he expresses
his deep gratitude and especially to Dr. Ronald Blum for his meticulous
reading of both the preliminary edition and the final manuscript. The
author wi11 be equally grateful for additional comments and for the
correction ofmisprints and errors.
DavidS. Saxon
November, 1967
Contents
I. THE DUAL NATURE OF MATTER AND RADIATION
I. The breakdown ofclassical physics . I
2. Quantum mechanical concepts. . . . . . . 3
3. The wave aspects of particles. . . . . . . 5
4. Numerical magnitudes and the quantum domain 12
5. The particle aspects ofwaves . 13
6. Complementarity . . . . . 16
7. The correspondence principle. 16
II. STATE FUNCTIONS AND THEIR INTERPRETATION
I. The idea ofa state function; superposition ofstates 18
2. Expectation values. . . . . . . . . . . 23
3. Comparison between the classical and quantum
descriptions ofa state; wave packets . . . . 25
Ill. LINEAR MOMENTUM
I. State functions corresponding to a definite momentum 29
2. Construction ofwave packets by superposition 31
3. Fouriertransforms: the Dirac deltafunction. 34
4. Momentum and configuration space. . 38
5. The momentum and position operators. 39
6. Commutation relations . 45
7. The uncertainty principle . . . . . 47
IV. MOTION OF A FREE PARTICLE
I. Motion ofa wave packet; group velocity 56
viii CONTENTS
2. The correspondence principle requirement 59
3. Propagation ofafree particle wave packet in
configuration space. 60
4. Propagation ofa free particle wave packet in momentum
space; the energy operator . 62
5. Time development ofa Gaussian wave packet 64
6. The free particle Schrodingerequation . 66
7. Conservation ofprobability. 68
8. Dirac bracket notation 72
9. Stationary states 73
10. A particle in a box. 75
II. Summary 81
V. SCHRODINGER'S EQUATION
I. The requirement ofconservation ofprobability. 84
2. Hermitian operators 85
3. The correspondence principle requirement 9\
4. Schrodinger's equation in configuration and
momentum space 95
5. Stationary states 97
6. Eigenfunctions and eigenvalues ofHermitian operators 101
7. Simultaneous observables and complete sets ofoperators 104
8. The uncertainty principle 106
*9. Wave packets and their motion 110
10. Summary: The postulates ofquantum mechanics III
VI. STATES OF A PARTICLE IN ONE DIMENSION
I. General features 117
2. Classification by symmetry; the parity operator. 119
3. Bound states in a square well 121
4. The harmonic oscillator . 127
*5. The creation operator representation 139
*6. Motion ofa wave packet in the harmonic oscillator
potential 145
7. Continuum states in a square well potential 147
8. Continuum states in general; the probability flux 153
*9. Passage ofa wave packet through a potential 155
* 10. Numerical solution ofSchrodinger's equation 159
VII. APPROXIMATION METHODS
I. The WKB approximation 175
2. The Rayleigh-Ritz approximation 185
3. Stationary state perturbation theory. 189
*For a one-semester course, any or all of the starred sections can be omitted without
harm to the logical development(see Preface).
CONTENTS ix
4. Matrices . . . . . . . . 201
5. Degenerate or close-lying states 205
6. Time dependent perturbation theory 209
VIII. SYSTEMS OF PARTICLES IN ONE DIMENSION
I. Formulation. . . . . . . . . . . . . 227
2. Two particles: Center-of-mass coordinates 229
3. Interacting particles in the presence ofuniform
external forces . . . . . . . . . . . . 233
*4. Coupled harmonic oscillators . . . . . . 237
5. Weakly interacting particles in the presence of
general external forces . . . . . . . . 239
6. Identical particles and exchange degeneracy. 241
7. Systems oftwo identical particles. . . . . 243
8. Many-particle systems; symmetrization and the
Pauli exclusion principle. . . . . . . 245
*9. Systems ofthree identical particles. . . 249
10. Weakly interacting identical particles in the
presence ofgeneral external forces 255
IX. MOTION IN THREE DIMENSIONS
I. Formulation: Motion ofa free particle. 263
*2. Potentials separable in rectangularcoordinates 265
3. Central potentials; angular momentum states. 269
4. Some examples. . . 279
5. The hydrogenic atom. . . . . . . . . 287
X. ANGULAR MOMENTUM AND SPIN
I. Orbital angular momentum operators and
commutation relations . . . . . . . 299
2. Angular momentum eigenfunctions and eigenvalues 303
*3. Rotation and translation operators 313
4. Spin: The Pauli operators . . 317
*5. Addition ofangular momentum . 327
XI. SOME APPLICATIONS AND FURTHER GENERALIZATIONS
*I. The helium atom; the periodic table. . . . . . .. 345
*2. Theory ofscattering . . . . . . . . . . . .. 351
*3. Green's function for scattering; the Born approximation. 361
*4. Motion in an electromagnetic field . 373
*5. Dirac theory ofthe electron 377
*6. Mixed states and the density matrix. 387
x CONTENTS
APPENDICES
I. Evaluation ofintegrals containing Gaussian functions. 397
II. Selected references. . . . . . . . . 401
III. Answers and solutions to selected problems . . . . 403
"And now reader,- bestir thyself-for
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HENRY FIELDING