Introduction to Quantum Mechanics: A Comprehensive Textbook, Transcriptions of Quantum Mechanics

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Introduction to

Quantum Mechanics

David J. Griffiths

Reed College

Prentice Hall Upper Saddle River, New Jersey 07458

Fundamental Equations

Schrodinger equation:

ili a\I1 = H\I

at

Time independent Schrodinger equation:

Hf/I = Et/J,

Standard Hamiltonian:

li 2 H= --V2+V

2m

Time dependence of an expectation value:

d(Q) =.i. ([H QJ) + (aQ)

dt li' at

Generalized uncertainty principle:

U AUB ~ 2i {[A, BJ)

Heisenberg uncertainty principle:

Caaonical commutator:

[x,p] = ili

Aaau!ar momentum:

Pauli mairicM:

Library of Congress Cataloging-in-Publication Data

Griffiths, David J. (David Jeffrey) Introduction to quantum mechanics / David J. Griffiths. p. em. Includes bibliographical references and index. ISBN 0-13-124405-

1. Quantum theory. I. Title. QC174.12.G75 1994 530.1'2-dc

Acquisitions Editor: Ray Henderson Assistant Acquisition Editor: Wendy Rivers Editorial Assistant: Pam Holland-Moritz Production Editors: Rose Kernan and Fred Dahl Copy Editor: Rose Kernan Production Coordinator: Trudy Pisciotti

© 1995 by Prentice Hall, Inc. Upper Saddle River, NJ 07458

All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher.

Printed in the United States of America

10

ISBN 0-13-124405-

94- CIP

Prentice-Hall International (UK) Limited, London Prentice-Hall of Australia Pty. Limited, Sydney Prentice-Hall of Canada, Inc., Toronto Prentice-Hall Hispancamericana, S. A., Mexico Prentice-Hall of India Private Limited, New Delhi Prentice-Hall of Japan, Inc., Tokyo Pearson Education Asia Pte. Ltd., Singapore Editora Prentice-Hall do Brasil, Ltda., Rio de Janeiro

ISBN 0-13-124405-

90000

CONTENTS

PREFACE, vii

PART I

• CHAPTER THEORY
• THE WAVE FUNCTION, - 1.1 The Schrodinger Equation, - 1.2 The Statistical Interpretation, - 1.3 Probability, - 1.4 Normalization, - 1.5 Momentum, - 1.6 The Uncertainty Principle,
• CHAPTER
• THE TIME-INDEPENDENT SCHRODINGER EQUATION, - 2.1 Stationary States, - 2.2 The Infinite Square Well, - 2.3 The Harmonic Oscillator,
  • 2.4 The Free Particle,
    • 2.5 The Delta-Function Potential,
    • 2.6 The Finite Square Well, - 2.7 The Scattering Matrix, iv Contents - Further Problems for Chapter 2,
    • CHAPTER
  • FORMAliSM, - 3.1 Linear Algebra, - 3.2 Function Spaces, - 3.3 The Generalized Statistical Interpretation, - 3.4 The Uncertainty Principle, - Further Problems for Chapter 3,
  • CHAPTER
  • QUANTUM MECHANICS IN THREE DIMENSIONS, - 4.1 Schrodinger Equations in Spherical Coordinates, - 4.2 The Hydrogen Atom, - 4.3 Angular Momentum, - 4.4 Spin, - Further Problems for Chapter 4,
  • CHAPTER
  • IDENTICAL PARTICLES, - 5.1 Two-Particle Systems, - 5.2 Atoms, - 5.3 Solids, - 5.4 Quantum Statistical Mechanics, - Further Problems for Chapter 5,
  • CHAPTER APPLICATIONS
• TIME-INDEPENDENT PERTURBATION THEORY, - 6.1 Nondegenerate Perturbation Theory, - 6.2 Degenerate Perturbation Theory, - 6.3 The Fine Structure of Hydrogen, - 6.4 The Zeeman Effect, - 6.5 Hyperfined Splitting, - Further Problems for Chapter 6, - CHAPTER - THE VARIATIONAL PRINCIPLE, - 7.1 Theory, - 7.2 The Ground State of Helium, - 7.3 The Hydrogen Molecule Ion, - Further Problems for Chapter 7, - CHAPTER - THE WKB APPROXIMATION, - 8.1 The "Classical" Region, - 8.2 Tunneling, - 8.3 The Connection Formulas, - Further Problems for Chapter 8, - CHAPTER
  • TIME-DEPENDENT PERTURBATION THEORY, - 9.1 Two-Level Systems, - 9.2 Emission and Absorption of Radiation, - 9.3 Spontaneous Emission, - Further Problems for Chapter 9,
  • CHAPTER
  • THE ADIABATIC APPROXIMATION, - 10.1 The Adiabatic Theorem, - 10.2 Berry's Phase, - Further Problems for Chapter 10,
  • CHAPTER
  • SCATTERING, - 11.1 Introduction, - 11.2 Partial Wave Analysis, - 11.3 The Born Approximation, - Further Problems for Chapter 11,
• AFTERWORD,
  • INDEX,

viii Preface

This book is intended for a one-semester or one-year course at the junior or senior level. A one-semester course will have to concentrate mainly on Part I; a full-year course should have room for supplementary material beyond Part II. The reader must be familiar with the rudiments of linear algebra, complex numbers, and calculus up through partial derivatives; some acquaintance with Fourier analysis and the Dirac delta function would help. Elementary classical mechanics is essential, of course, and a little electrodynamics would be useful in places. As always, the more physics and math you know the easier it will be, and the more you will get out of your study. But I would like to emphasize that quantum mechanics is not, in my view, something that flows smoothly and naturally from earlier theories. On the contrary, it represents an abrupt and revolutionary departure from classical ideas, calling forth a wholly new and radically counterintuitive way of thinking about the world. That, indeed, is what makes it such a fascinating subject. At first glance, this book may strike you as forbiddingly mathematical. We en- counter Legendre, Hermite, and Laguerre polynomials, spherical harmonics, Bessel, Neumann, and Hankel functions, Airy functions, and even the Riemann Zeta function -not to mention Fourier transforms, Hilbert spaces, Hermitian operators, Clebsch- Gordan coefficients, and Lagrange multipliers. Is all this baggage really necessary? Perhaps not, but physics is like carpentry: Using the right tool makes the job easier, not more difficult, and teaching quantum mechanics without the appropriate mathe- matical equipment is like asking the student to dig a foundation with a screwdriver. (On the other hand, it can be tedious and diverting if the instructor feels obliged to give elaborate lessons on the proper use of each tool. My own instinct is to hand the students shovels and tell them to start digging. They may develop blisters at first, but I still think this is the most efficient and exciting way to learn.) At any rate, I can assure you that there is no deep mathematics in this book, and if you run into something unfamiliar, and you don't find my explanation adequate, by all means ask someone about it, or look it up. There are many good books on mathematical methods-I par- ticularly recommend Mary Boas, Mathematical Methods in the Physical Sciences, 2nd ed., Wiley, New York (1983), and George Arfken, Mathematical Methods for Physicists, 3rd ed., Academic Press, Orlando (1985). But whatever you do, don't let the mathematics-which, for us, is only a tool-interfere with the physics. Several readers have noted that there are fewer worked examples in this book than is customary, and that some important material is relegated to the problems. This is no accident. I don't believe you can learn quantum mechanics without doing many exercises for yourself. Instructors should, of course, go over as many problems in class as time allows, but students should be warned that this is not a subject about which anyone has natural intuitions-you're developing a whole new set of muscles here, and there is simply no substitute for calisthenics. Mark Semon suggested that I offer a "Michelin Guide" to the problems, with varying numbers of stars to indicate the level of difficulty and importance. This seemed like a good idea (though, like the quality of a restaurant, the significance of a problem is partly a matter of taste); I have adopted the following rating scheme:

Preface ix

• an essential problem that every reader should study; ** a somewhat more difficult or more peripheral problem;
• ** an unusually challenging problem that may take over an hour.

(No stars at all means fast food: OK if you're hungry, but not very nourishing.) Most of the one-star problems appear at the end of the relevant section; most of the three-star problems are at the end of the chapter. A solution manual is available (to instructors only) from the publisher. I have benefited from the comments and advice of many colleagues, who sug- gested problems, read early drafts, or used a preliminary version in their courses. I would like to thank in particular Burt Brody (Bard College), Ash Carter (Drew Uni- versity), Peter Collings (Swarthmore College), Jeff Dunham (Middlebury College), Greg Elliott (University of Puget Sound), Larry Hunter (Amherst College), Mark Semon (Bates College), Stavros Theodorakis (University of Cyprus), Dan Velleman (Amherst College), and all my colleagues at Reed College. Finally, I wish to thank David Park and John Rasmussen (and their publishers) for permission to reproduce Figure 8.6, which is taken from Park's Introduction to the Quantum Theory (footnote 1), adapted from I. Perlman and J. O. Rasmussen, "Alpha Radioactivity," in Encyclopedia of Physics, vol. 42, Springer-Verlag, 1957.

PART I

THEORY

2 Chap. 1 The Wave Function

m

'rr-rl -----J~~ F (x,t)

x(t)

Figure 1.1: A "particle" constrained to move in one dimension under the influ- ence of a specified force.

x

[1.2]

Here i is the square root of -1, and h is Planck's constant-or rather, his original constant (h) divided by 'In :

1i = ~ = 1.054573 x 10- 34 J s.

2:rr The Schrodinger equation plays a role logically analogous to Newton's second law: Given suitable initial conditions [typically, \II(x, 0)], the Schrodinger equation de- termines \II (x, t) for all future time, just as, in classical mechanics, Newton's law determines x(t) for all future time.

1.2 THE STATISTICAL INTERPRETATION

But what exactly is this "wave function", and what does it do for you once you've got it? After all, a particle, by its nature, is localized at a point, whereas the wave function

(as its name suggests) is spread out in space (it's a function of x, for any given time

t). How can such an object be said to describe the state of a particle? The answer is provided by Born's statistical interpretation of the wave function, which says that I\II (x, 1)1 2 gives the probability of finding the particle at point x, at time t-or, more precisely,"

1\II(x,t)1 2dx^ = Iprobabilityoffindingtheparticle I [1.3] between x and (x + dx), at time t.

For the wave function in Figure 1.2, you would be quite likely to find the particle in the vicinity of point A, and relatively unlikely to find it near point B. The statistical interpretation introduces a kind of indeterminacy into quantum mechanics, for even if you know everything the theory has to tell you about the

2The wave function itself is complex, but 11JI1 2 = IJIIJI (where IJI is the complex conjugate of IJI) is real and nonnegative-as a probability, of course, must be.

Sec. 1.2: The Statistical Interpretation 3

dx (^) A B C x

Figure 1.2: A typical wave function. The particle would be relatively likely to be found near A, and unlikely to be found near B. The shaded area represents the probability of finding the particle in the range dx.

particle (to wit: its wave function), you cannot predict with certainty the outcome of a simple experiment to measure its position-all quantum mechanics has to offer is statistical information about the possible results. This indeterminacy has been profoundly disturbing to physicists and philosophers alike. Is it a peculiarity of nature, a deficiency in the theory, a fault in the measuring apparatus, or what? Suppose I do measure the position of the particle, and I find it to be at the point C. Question: Where was the particle just before I made the measurement? There are three plausible answers to this question, and they serve to characterize the main schools of thought regarding quantum indeterminacy:

1. The realist position: The particle was at C. This certainly seems like a

sensible response, and it is the one Einstein advocated. Note, however, that if this is

true then quantum mechanics is an incomplete theory, since the particle really was at

C, and yet quantum mechanics was unable to tell us so. To the realist, indeterminacy is not a fact of nature, but a reflection of our ignorance. As d'Espagnat put it, "the position of the particle was never indeterminate, but was merely unknown to the experimenter,'? Evidently \II is not the whole story-some additional information

(known as a hidden variable) is needed to provide a complete description of the

particle.

2. The orthodox position: The particle wasn 't really anywhere. It was the act

of measurement that forced the particle to "take a stand" (though how and why it decided on the point C we dare not ask). Jordan said it most starkly: "Observations not only disturb what is to be measured, they produce it. ... We compel [the particle]

to assume a definite position?" This view (the so-called Copenhagen interpretation)

is associated with Bohr and his followers. Among physicists it has always been the

3Bemard d'Espagnat, The Quantum Theory and Reality, Scientific American, Nov. 1979 (Vol. 241), p. 165. 4Quoted in a lovely article by N. David Mermin, Is the moon there when nobody looks?, Physics Today, April 1985, p. 38.

c

Sec. 1.3: Probability 5

x

Figure 1.3: Collapse of the wave function: graph of 11111 2 immediately after a measurement has found the particle at point C.

are, then, two entirely distinct kinds of physical processes: "ordinary" ones, in which the wave function evolves in a leisurely fashion under the Schrodinger equation, and "measurements", in which \11 suddenly and discontinuously collapses.'

1.3 PROBABILITY

Because of the statistical interpretation, probability plays a central role in quantum mechanics, so I digress now for a brief discussion of the theory of probability. It is mainly a question of introducing some notation and terminology, and I shall do it in the context of a simple example. Imagine a room containing 14 people, whose ages are as follows:

one person aged 14

one person aged 15

three people aged 16 two people aged 22 two people aged 24 five people aged 25.

If we let N (j) represent the number of people of age j, then

7The role of measurement in quantum mechanics is so critical and so bizarre that you may well be wondering what precisely constitutes a measurement. Does it have to do with the interaction between a microscopic (quantum) system and a macroscopic (classical) measuring apparatus (as Bohr insisted), or is it characterized by the leaving of a permanent "record" (as Heisenberg claimed), or does it involve the intervention of a conscious "observer" (as Wigner proposed)? I'll return to this thorny issue in the Afterword; for the moment let's take the naive view: A measurement is the kind of thing that a scientist does in the laboratory, with rulers, stopwatches, Geiger counters, and so on.

6 Chap. 1 The Wave Function

N(j)

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 j

Figure 1.4: Histogram showing the number of people, N(j), with age j, for the example in Section 1.3.

N(14) = 1 N(15) = 1 N(16) = 3 N(22) = 2 N(24) = 2

N(25) = 5

while N (17), for instance, is zero. The total number of people in the room is 00 N=LN(j). }=o

[1.4]

[1.5]

(In this instance, of course, N = 14.) Figure 1.4 is a histogram of the data. The following are some questions one might ask about this distribution. »>: Question 1. If you selected one individual at random from this group, what is the probability that this person's age would be 15? Answer: One chance in 14, since there are 14 possible choices, all equally likely, of whom only one has this particular

age. If P(j) is the probability of getting age i. then P(14) = 1/14, P(15) =

1/14, P(16) = 3/14, and so on. In general,

P(j) _ N(j)

- N.

Notice that the probability of getting either 14 or 15 is the sum of the individual probabilities (in this case, 1/7). In particular, the sum of all the probabilities is 1- you're certain to get some age: 00 L P(j) = 1. }=

[1.6]