Operator Methods in Quantum Mechanics

By Martin Schechter

This advanced undergraduate and graduate-level text introduces the power of operator theory as a tool in the study of quantum mechanics, assuming only a working knowledge of advanced calculus and no background in physics. The author presents a few simple postulates describing quantum theory, gradually introducing the mathematical techniques that help answer questions important to the physical theory; in this way, readers see clearly the purpose of the method and understand the accomplishment. The entire book is devoted to the study of a single particle moving along a straight line. By posing questions about the particle, the text gradually leads readers into the development of mathematical techniques that provide the answers. Lebesgue integration theory and complex variables are sometimes involved, but most of the book can be understood without them. Exercises at the end of each chapter provide helpful reinforcement.

• Language: English
• Publisher: Dover Publications
• Release date: Jun 10, 2014
• ISBN: 9780486150048

Book preview

Mechanics

Operator Methods

in

Quantum Mechanics

Martin Schechter

University of California, Irvine

DOVER PUBLICATIONS, INC.

Mineola, New York

Copyright

Copyright © 1981 by Elsevier North Holland, Inc.

All rights reserved under Pan American and International Copyright Conventions.

Bibliographical Note

This Dover edition, first published in 2002, is an unabridged republication of the work originally published in 1981 by Elsevier North Holland, Inc., New York.

Library of Congress Cataloging-in-Publication Data

Schechter, Martin.

Operator methods in quantum mechanics / Martin Schechter.

      p. cm.

Originally published: New York: North Holland, c1981.

Includes bibliographical references and index.

eISBN 13: 978-0-486-15004-8

  1. Operator theory. 2. Quantum theory. I. Title.

QC174.17.O63S33 2002

515’.724—dc21

2002031300

Manufactured in the United States of America

Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

BS″D

To the memory of my parents, Z″L,

Joshua and Rose

who left this world before their nachas was complete.

Contents

Preface

Acknowledgments

A Message to the Reader

List of Symbols

Chapter 1.  One-Dimensional Motion

1.1.  Position

1.2.  Mathematical Expectation

1.3.  Momentum

1.4.  Energy

1.5.  Observables

1.6.  Operators

1.7.  Functions of Observables

1.8.  Self-Adjoint Operators

1.9.  Hilbert Space

1.10.  The Spectral Theorem

Exercises

Chapter 2.  The Spectrum

2.1.  The Resolvent

2.2.  Finding the Spectrum

2.3.  The Position Operator

2.4.  The Momentum Operator

2.5.  The Energy Operator

2.6.  The Potential

2.7.  A Class of Functions

2.8.  The Spectrum of H

Exercises

Chapter 3.  The Essential Spectrum

3.1.  An Example

3.2.  A Calculation

3.3.  Finding the Eigenvalues

3.4.  The Domain of H

3.5.  Back to Hilbert Space

3.6.  Compact Operators

3.7.  Relative Compactness

3.8.  Proof of Theorem 3.7.5

Exercises

Chapter 4.  The Negative Eigenvalues

4.1.  The Possibilities

4.2.  Forms Extensions

4.3.  The Remaining Proofs

4.4.  Negative Eigenvalues

4.5.  Existence of Bound States

4.6.  Existence of Infinitely Many Bound States

4.7.  Existence of Only a Finite Number of Bound States

4.8.  Another Criterion

Exercises

Chapter 5.  Estimating the Spectrum

5.1.  Introduction

5.2.  Some Crucial Lemmas

5.3.  A Lower Bound for the Spectrum

5.4.  Lower Bounds for the Essential Spectrum

5.5.  An Inequality

5.6.  Bilinear Forms

5.7.  Intervals Containing the Essential Spectrum

5.8.  Coincidence of the Essential Spectrum with an Interval

5.9.  The Harmonic Oscillator

5.10.  The Morse Potential

Exercises

Chapter 6.  Scattering Theory

6.1.  Time Dependence

6.2.  Scattering States

6.3.  Properties of the Wave Operators

6.4.  The Domains of the Wave Operators

6.5.  Local Singularities

Exercises

Chapter 7.  Long-Range Potentials

7.1.  The Coulomb Potential

7.2.  Some Examples

7.3.  The Estimates

7.4.  The Derivatives of V(x)

7.5.  The Relationship Between Xt and V(x)

7.6.  An Identity

7.7.  The Reduction

7.8.  Mollifiers

Exercises

Chapter 8.  Time-Independent Theory

8.1.  The Resolvent Method

8.2.  The Theory

8.3.  A Simple Criterion

8.4.  The Application

Exercises

Chapter 9.  Completeness

9.1.  Definition

9.2.  The Abstract Theory

9.3.  Some Identities

9.4.  Another Form

9.5.  The Unperturbed Resolvent Operator

9.6.  The Perturbed Operator

9.7.  Compact Operators

9.8.  Analytic Dependence

9.9.  Projections

9.10.  An Analytic Function Theorem

9.11.  The Combined Results

9.12.  Absolute Continuity

9.13.  The Intertwining Relations

9.14.  The Application

Exercises

Chapter 10.  Strong Completeness

10.1.  The More Difficult Problem

10.2.  The Abstract Theory

10.3.  The Technique

10.4.  Verification for the Hamiltonian

10.5.  An Extension

10.6.  The Principle of Limiting Absorption

Exercises

Chapter 11.  Oscillating Potentials

11.1.  A Surprise

11.2.  The Hamiltonian

11.3.  The Estimates

11.4.  A Variation

11.5.  Examples

Exercises

Chapter 12.  Eigenfunction Expansions

12.1.  The Usefulness

12.2.  The Problem

12.3.  Operators on Lp

12.4.  Weighted Lp-Spaces

12.5.  Extended Resolvents

12.6.  The Formulas

12.7.  Some Consequences

12.8.  Summary

Exercises

Chapter 13.  Restricted Particles

13.1.  A Particle Between Walls

13.2.  The Energy Levels

13.3.  Compact Resolvents

13.4.  One Opaque Wall

13.5.  Scattering on a Half-Line

13.6.  The Spectral Resolution for the Free Particle on a Half-Line

Exercises

Chapter 14.  Hard-Core Potentials

14.1.  Local Absorption

14.2.  The Modified Hamiltonian

14.5.  Propagation

14.6.  Proof of Theorem 14.5.1

14.9.  A Regularity Theorem

14.10.  A Family of Spaces

Exercises

Chapter 15.  The Invariance Principle

15.1.  Introduction

15.2.  A Simple Result

15.3.  The Estimates

15.4.  An Extension

15.5.  Another Form

Exercises

Chapter 16.  Trace Class Operators

16.1.  The Abstract Theorem

16.2.  Some Consequences

16.3.  Hilbert–Schmidt Operators

16.4.  Verification for the Hamiltonian

Exercises

Appendix A.  The Fourier Transform

Exercises A

Appendix B.  Hilbert Space

Exercises B

Appendix C.  Hölder’s Inequality and Banach Space

Bibliography

Index

Preface

The interaction between physics and mathematics has always played an important role in the development of both sciences. The physicist who does not have the latest mathematical knowledge available to him is at a distinct disadvantage. The mathematician who shies away from physical applications will most likely miss important insights and motivations. For these reasons it has become a practice to expose science students to a course in mathematical physics and mathematics students to a course in applied mathematics. Such courses usually present various mathematical techniques that are useful in science and engineering. The student is told that the mathematics is important, but he is rarely shown why it is needed. If applications are discussed, the student may find them artificially manufactured.

For several years I have been giving a course in applications of operator theory. Rather than presenting the usual mixture of theorems, I decided to pick a branch of physics and to develop the theory from the very beginning. I did it in a mathematical way without describing the experimental evidence that gave rise to the theory. I gave a few simple postulates assuming no background in physics and as little background in mathematics as possible. The students were asked to accept these few postulates and not to try to understand the reasoning that led to them. Then I gradually introduced the powerful mathematical techniques that help answer questions that are important to the physical theory. Only those mathematical methods that are needed for the physics are introduced, and they are introduced when they are required. From the very beginning it is possible to motivate the entire exposition. The student sees clearly the purpose of the method and understands the accomplishment. This book is an outgrowth of this course.

I found quantum theory to be very fruitful from a mathematical point of view. The theory gives rise to the many questions in operator theory and in the study of differential equations. I found the theory so rich that I was able to fill this entire volume analyzing only a single particle in one dimension. In fact, I was not able to cover all of the one-dimensional topics that I would have liked. It is true that most of the methods developed can be applied to systems of particles in higher dimensions, but I found that there was enough to do in the simplest case. In this way the student is introduced to the methods without being overwhelmed by details. The consideration of one situation throughout the book adds unity to the exposition.

My main thrust has been towards the understanding of the physical and mathematical principles and not towards the achieving of the strongest results. I have tried to minimize technical details wherever possible and to keep the discussion elementary. Each mathematical method is introduced as needed in the development. It is then shown how the new technique solves the problem at hand.

The book may be used at an advanced undergraduate or a beginning graduate level. The main prerequisite is advanced calculus. Elementary theory of Lebesgue integration and analytic functions of a complex variable are used unavoidably in a few sections, but the reader will have no difficulty skipping these sections if he is willing to believe certain statements. Most theorems in analysis and functional analysis are proved either on the spot or in an Appendix. There are a few theorems that I do not prove. However, the student will have no difficulty understanding their meaning, purpose, and applicability. These few theorems are standard in mathematics courses at the beginning graduate level (references are given in the Bibliography).

Theorems are designated by three numbers. The first refers to the chapter, the second to the section and the third to the order in which it appears in the section. Lemmas and propositions are similarly designated.

To my mind, the application of operator theory to quantum mechanics forms one of the most beautiful areas of knowledge. I hope this book has captured some of the beauty.

Martin Schechter

New York

TVSLB″ 0

Acknowledgements

The material presented in the book is based upon results contained in various research articles. I have tried to include all of them in the Bibliography.

I wish to thank my students Merl Altabet and Alexander Gelman for reading parts of the manuscript and making important suggestions and corrections. My daughter, Sara, helped me greatly by typing parts of the manuscript, and my wife, Deborah, made invaluable contributions to the effort.

A Message to the Reader

The purpose of this volume is to show how mathematics is used to answer questions in science. We begin by assuming that the reader has a working knowledge of advanced calculus, and we present a few simple postulates describing quantum theory. The entire book is devoted to the study of a single particle moving along a straight line. We ask questions about this particle and gradually develop mathematical techniques that give the answers. Most of these techniques can be used for more complicated systems of particles, but I did not want to get involved in too many details. There are places where I need Lebesgue integration theory and complex variables, but most of the book can be understood without them. There are a few places where I ask you to believe me (of course I give references), but there should be no difficulty in understanding the statements. I hope this book can help you see how operator theory can be used as a powerful tool in the study of quantum mechanics.

List of Symbols

The page numbers indicate where the symbols are defined or are explained.

C, 142

, 61

(J), 235

D (A), 9

EI, 13

E (I), 24, 140

Ē(I), 140

G0 (z), 177

G (z), 177

G0± (s), 192

(H), 182

(H), 139

(H), 205

(H), 182

Hs, 276

ħ, 5

iff, if and only if

, 153

Jn, 131

L², 9

Lp, 200

Lp, β, 220

M±, 106

R (A), 20

R (Z), 26

R±, 107

R± (H, H0), 152

S, 107

W±, 106

W± (H, H0), 152

δa (λ), 140

Φ, 228

χI (λ), 13

ψ, 1

ρ (A), 26

ρ(x), 113

σ (A), 26

σe (A), 48

ωI, 164

( , ), 7

Ā, 12

ā, 2

|| ||, 9

I → R, 16

N⊥, 109

|I|, 79

M± (H, H0), 152

P (a I), 13

p, 5

Q0 (z), 177

|u|s, 276

A*, 18

Z*, complex conjugate

Operator Methods

in

Quantum Mechanics

1

One-Dimensional Motion

1.1. Position

We begin by studying a single particle restricted to motion along a line. The first postulate we shall use is

Postulate 1. There is a function ψ(x, t) of position x and time t such that the probability that the particle is in an interval I at the time t is given by

Note that this postulate does not tell us how to determine the position of the particle, but only how to determine the probability that it is located in an interval. The postulate’s probabilistic nature represents a basic philosophical tenet of quantum mechanics. Later we shall see that it is seldom that we can determine the position (or any other quantity) precisely. For those who are unfamiliar with probability theory, it suffices at the moment to note that the probability of an event occurring is a number P, 0 ≤ P ≤ 1, which, in a sense, represents the chances that the event will take place. We shall have more to say about this later.

The function ψ is called the state function for the particle. Usually it is complex valued, but only in the sense that it takes on complex values for each x and t. There need not be any relationship between its real and imaginary parts (i.e., it need not be analytic in the sense of complex variables). We shall not be able to justify the appearance of the absolute value and the square power in (1.1.1); although we shall see that they are highly desirable from a mathematical point of view.

Since it is certain that the particle must be somewhere along the line, we have

at each time t.

1.2. Mathematical Expectation

Suppose w is a measurable quantity which can take on the values w1,…, wN. Assume that one can perform an indefinite number of independent experiments (i.e., the outcome of one does not affect the outcome of another) in which w is measured. Suppose the probability that w takes on the value wk is Pk, 1 ≤ k ≤ N. Since w must take on one of the given values, we have

The quantity

is called the mathematical expectation or average value of w. The reason for this terminology is the following theorem from the theory of probability (cf. Feller, 1950).

Theorem 1.2.1. Suppose a sequence of identical experiments is performed and the values S1, S2, . . ., Sn, . . ., are observed (the numbers Sn are among the values w1, . . ., wN). Then the average value

converges to "in the sense of probability."

In order for this theorem to make sense, we must know what it means to converge in the sense of probability. By this we mean that for each ε > 0 the probability that

tends to 0 as n → ∞. Life would be a lot simpler if are small.

If w can take on any value in some interval, we must assign to each subinterval I the probability PI that the value of w lies in I. To compute the mathematical expectation of w, assume first that it is restricted to a bounded interval [a, b]. We divide this interval up into small subintervals Ik and form the sum

where xk is an arbitrary point of Ik. If these sums converge to a limit as the maximum length of the intervals Ik tends to 0 independently of the manner in which the intervals Ik were chosen and independently of the choice of the points xk ∈ Ik the mathematical expectation of w. If w is not restricted to a bounded interval, we compute first the limit as above for a bounded interval [a, bof w.

To illustrate this concept, consider the probability (1.1.1) that a particle is in an interval I. Let I be a bounded interval, and subdivide it into smaller intervals Ik. If xk is an arbitrary point in Ik, (1.2.5) becomes

The following is true:

Lemma 1.2.1. If ψ(x, t) is continuous with respect to x in I, then (1.2.6) converges to

as the maximum length of the intervals Ik tends to 0 independently of the choice of the Ik and xk.

PROOF. We have

Let ε > 0 be given, and take the length of each Ik < ε. Then |x–xk |< ε in Ik. Thus the right-hand side of (1.2.8) is less than

The definition and Lemma 1.2.1 lead immediately to

Theorem 1.2.2. If ψ(x, t) is continuous with respect to x and

then the mathematical expectation of position is given by

Now we give a proof of Theorem 1.2.1 for the case of the position of a particle. Suppose the positions of n identical particles are measured at a time t and that they all have the same state function. Suppose the observed values are x1, . . ., xn. If x0 and a > 0 are given, then the probability that |x0 – Σxk| > a is

This is less than

Set

and

where the asterisk denotes complex conjugation. Since

(1.2.11) is equal to

is bounded by

By , while the second term tends to 0 as n → ∞.

Theorem 1.2.3. If f(x) is a continuous function satisfying

then the mathematical expectation of f(x) is given by

The proof of Theorem 1.2.3 is simple. In fact, let I be a bounded interval, and let I1, . . ., IN be a partition of I into smaller intervals with maximum length δ. If xk is an arbitrary point in Ik, we have

Since f(x) is uniformly continuous in I, we can make |f(x) – f(xk)| < ε in Ik by taking δ sufficiently small. For such a δ the left-hand side of (1.2.14) is less than

Once we have established this, we merely note that

1.3. Momentum

In classical physics the momentum of a particle is defined as

The second postulate we shall make concerns momentum. We state it as follows:

Postulate 2. The probability that the momentum p of the particle is contained in the interval I is given by

where ħ is the Fourier transform of ψ with respect to x defined by

(We apologize to the mathematicians for using k as a variable.) As in the case of position, the average value of p turns out to be

As in Section 1.2, we have

Theorem 1.3.1. If g(p) is continuous and satisfies

then

The proof of Theorem 1.3.1 is the same as that of Theorem 1.2.3.

In order to deal with momentum, we shall need to know certain properties of the Fourier transform. We state them here without hypotheses. Precise statements and proofs will be given in Appendix A.

a.The inverse Fourier transform

b.Parseval’s identity

(The asterisk denotes the complex conjugate.)

denotes the inverse Fourier transform.)

without using the Fourier transform. For by (1.3.2)

Thus

where

This is called the momentum operator. Note that (1.3.7) is free of Fourier transforms, but the penalty paid is the introduction of the partial differential operator L.

Repeated applications of (e) give

Consequently, we have by (1.3.4)

where we have used the notation

1.4. Energy

In classical physics the kinetic energy T of the particle is given by

By (1.3.10) the expectation of T is

The potential energy is given by a real-valued function of position V(x). The total energy is given by

If V(x) is continuous and satisfies

then Theorem 1.2.3 gives as the expectation value of the potential energy

How can we compute the average of the total energy? It turns out to be a simple matter if we use

Theorem 1.4.1. The mathematical expectation of a sum is equal to the sum of the mathematical expectations.

Thus, if we add (1.4.2) and (1.4.5), we get

where

is the energy operator or Hamiltonian.

It should be noted that Theorem 1.4.1 holds whether or not the quantities are independent. We shall give the proof only for the case of discrete variables.

Let u be a measurable quantity that can only take on the values u1, . . ., uM with probabilities P1, . . ., PM and let υ be a measurable quantity that can take on only the values υ1, . . ., υN with probabilities Q1, . . ., QN. Let Rij be the probability that u = ui and that υ = υj. (If the quantities are independent, then Rij = PiQj. Otherwise, this need not be so.) Thus

Now the mathematical expectation of u + υ is

This proves the theorem for the discrete case.

Although we have proved (1.4.5) only for continuous functions V(x), there are many potentials of physical interest which possess discontinuities or even singularities. For such functions we shall take (1.4.5) as the definition of the expectation value.

1.5. Observables

Any quantity that can be measured is called an observable. We have discussed three, namely, position, momentum, and energy. In each of these cases, we have noticed that there corresponds an operator in the sense that if a denotes the variable and ā is its average value, then there is an operator A such that

We can make a table:

In each of the cases mentioned the observable took on real values only. In such cases the average value must be real as well. This requires that (Aψ, ψ) be real for any state function ψ. This leads to the question, what operators A have the property that (Aψ, ψ) is real for every state function ψ? Fortunately the answer is simple. An operator A is called Hermitian if

holds for all ψ, φ. The answer to our question is given by

Lemma 1.5.1. An operator A is Hermitian if and only if (Aψ, ψ) is real for all ψ.

Before we give the proof of Lemma 1.5.1, we had better clarify certain concepts. An operator . The set of elements on which it acts is called its domain. An operator A is called linear if (1) αx + βy is in its domain D(A) whenever x, y are in its domain and α, β are scalars, and (2) A(αx + βy) = αAx + βAy. We shall take our scalars to be complex, and