A Mathematical Companion to Quantum Mechanics

by Shlomo Sternberg

Overview

This original 2019 work, based on the author's many years of teaching at Harvard University, examines mathematical methods of value and importance to advanced undergraduates and graduate students studying quantum mechanics. Its intended audience is students of mathematics at the senor university level and beginning graduate students in mathematics and physics.
Early chapters address such topics as the Fourier transform, the spectral theorem for bounded self-joint operators, and unbounded operators and semigroups. Subsequent topics include a discussion of Weyl's theorem on the essential spectrum and some of its applications, the Rayleigh-Ritz method, one-dimensional quantum mechanics, Ruelle's theorem, scattering theory, Huygens' principle, and many other subjects.

Product Details

ISBN-13:	9780486839820
Publisher:	Dover Publications
Publication date:	03/20/2019
Series:	Dover Books on Physics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	336
File size:	27 MB
Note:	This product may take a few minutes to download.

About the Author

Shlomo Sternberg received his Ph.D. from Johns Hopkins University in 1957 and taught in the Mathematics Department at Harvard University from 1959 until his retirement as Professor Emeritus of Mathematics in 2017. He is the author of many frequently cited and highly regarded books in mathematics and related fields, including two published by Dover: Dynamical Systems and Curvature in Mathematics and Physics. Among many other honors and awards, Dr. Sternberg was elected to the American Academy of Arts and Sciences in 1984 and to the National Academy of Sciences in 1986.

Read an Excerpt

CHAPTER 1

Introduction

Over the years, I taught "Theory of Functions of a Real Variable" at Harvard many times. In addition to standard material such as functional analysis, measure, and integration theory, I included elementary mathematics for quantum mechanics. I thought it would be useful to extract this material and gather it together.

There are many books on this subject. The closest competitor is the excellent book Mathematical Methods in Quantum Mechanics by Teschl, but my approach to many topics is sufficiently different to warrant the effort of organizing this book.

Of course, the multivolume book by Reed and Simon is the classic text on the subject, which goes far beyond anything I cover, and is absolutely necessary for any serious student.

I don't deal with the philosophical and mathematical foundations of quantum mechanics for which I refer to Mackey's classic. This book presents mathematical methods.

As this book is assembled from many lectures, it is not in linear order. It should be regarded as a potpourri of various topics. Nevertheless, here is a rough outline:

A key ingredient is the spectral theorem for self-adjoint operators. For example, the first proof for possibly unbounded self-adjoint operators occurs in Wintner's book of 1929, and the subtitle of the book is "Introduction to the Analytical Apparatus of Quantum Theory." I reproduce the frontispiece to his book at the end of Chapter 12. I know at least eight different proofs of this theorem. I derive the spectral theorem from the Fourier inversion formula, which says that (for nice functions) f,

[MATHEMATICAL EXPRESSION OMITTED]

where f denotes the Fourier transform of f. If we replace x by A and write U(t) instead of eitA, this suggests that we define

[MATHEMATICAL EXPRESSION OMITTED]

This defines a "functional calculus" f -> f(A) which has nice properties, as we check in Chapter 3. The problem with this approach is how to make sense of eitA. For bounded operators A, we can use the power series for the exponential. But quantum mechanics requires unbounded operators for which the power series makes no sense. In Chapter 4, I explain unbounded operators, and define their resolvents and spectra. In Chapter 6, I give the rather subtle definition of a (possibly unbounded) self-adjoint operator. Chapters 5, 7, and 8 are devoted to the study of semi-groups and their generators, culminating with the celebrated Hille-Yosida theorem. We derive Stone's theorem on the nature of unitary one parameter groups on a Hilbert space as a special case of the Hille-Yosida theorem. From Stone's theorem we get the spectral theorem in functional calculus form, at least for continuous functions f vanishing at infinity. In this chapter I also present the important Dynkin-Helffer-Sjöstrand formula for the functional calculus. The main points of these first eight chapters are summarized at the end of Chapter 8.

The next big ticket item is Weyl's 1909 theorem on the stability of the "essential spectrum." This general theorem is proved in Chapter 9 and worked out in detail for the square well Hamiltonian of elementary quantum mechanics. Chapter 10 involves more discussion on the details of Weyl's theorem. One important consequence is to a Schrödinger operator of the form H = H0 + V , where H0 is the "free Hamiltonian" and V is a potential. Weyl's theorem implies that if V(x) -> ∞ as x -> ∞ then there is no essential spectrum so the spectrum of H consists entirely of eigenvalues of finite multiplicity, while if V(x) -> 0 as x -> ∞ then the essential spectrum of H is the same as that of H0.

In Chapters 11 and 12, I return to the spectral theorem. In the version given in the first eight chapters, the functional calculus f -> f(A) is limited to continuous functions vanishing at infinity. We need to extend it to bounded Borel functions. (For those who don't know what these are, see section 26.1 below.) In Chapter 11, I use the Riesz representation theorem to get the needed extension.

In Chapter 12, I present Wintner's proof, which gives the functional calculus directly for bounded Borel functions. I did not do the Wintner approach from the start because it depends on the Borel transform and the Stieltjes inversion formula, which are not as well known (or as standard) as the Fourier transform and its inversion formula.

Chapter 13 is devoted to the L2 version of the spectral theorem, which says that any self-adjoint operator is unitarily equivalent to multiplication by a real function on a measure space. In contrast to the functional calculus versions, this isomorphism is highly non-canonical, but it allows us to discuss fractional powers of a non-negative self-adjoint operator.

Chapter 14 is devoted to the celebrated Rayleigh-Ritz approximation method for eigenvalues, with applications to "chemical theories."

Chapters 15 and 16 are devoted to quantum mechanics in one dimension, with some results going back to the work of Sturm circa 1850.

Chapter 17 does some three-dimensional computations.

Chapter 18 is devoted to Ruelle's theorem: It is a truism in atomic physics or quantum chemistry courses that the eigenstates of the Schrödinger operator for atomic electrons are the bound states — the ones that remain bound to the nucleus — and that the "scattering states" which fly off in large positive or negative times correspond to the continuous spectrum. Ruelle's theorem gives a mathematical justification for this truism.

At the other extreme, Agmon's theorem says that under appropriate conditions the eigenvectors (which correspond to "bound states") die off exponentially at infinity. Chapter 19 gives a watered down version of Agmon's theorem.

In Chapter 20 I return once more to the spectral theorem, this time giving Lorch's proof which makes use of a beautifully "complex variable style" calculus due Riesz and Dunford.

Chapters 21 to 23 give a smattering of quantum mechanical scattering theory, including a chapter on Hughens' principle of wave mechanics.

Chapter 24 gives the Groenewold-van Hove Theorem, which shows that Dirac's proposal that "quantization" gives a correspondence between the Poisson bracket of classical mechanics and the commutator bracket of quantum mechanics does not work beyond quadratic polynomials.

Chapter 25 returns to the subject of semigroups and presents an important theorem of Chernoff and some of its consequences.

Chapter 26 gives some background material.

The first two chapters of provide enough background in real variable theory for this book. In fact, I assume much less and provide a lot of the background, including a "primer" on Hilbert spaces in Chapter 26. I suggest as a good source for background material. Both are available gratis on the web. I use the "uniform boundedness principle" (also known as the "Banach-Steinhaus Theorem") several times. A good treatment can be found on page 39 of and on page 98 of. I use the Stone-Weierstrass theorem (see pp. 59–61) several times.

I use the Cauchy integral formula and Cauchy's theorem (which can be found near the opening of any text on complex variable theory) throughout the book.

I thank Chen He and Rob Lowry for proofreading the text.

CHAPTER 2

The Fourier Transform

I will start with standard facts about the Fourier transform from which I will derive the spectral theorem for bounded self-adjoint operators in Chapter 3.

2.0.1 Conventions

The space S consists of all complex valued functions on R that are infinitely differentiable and vanish at infinity rapidly with all their derivatives in the sense that

[MATHEMATICAL EXPRESSION OMITTED]

The ||·||m,n give a family of semi-norms on S making S into a Fréchet space — that is, a vector space whose topology is determined by a countable family of semi-norms.

We use the measure

[MATHEMATICAL EXPRESSION OMITTED]

on R and so define the Fourier transform of an element of S by

[MATHEMATICAL EXPRESSION OMITTED]

and the convolution of two elements of S by

[MATHEMATICAL EXPRESSION OMITTED]

2.1 Basic Facts about the Fourier Transform Acting on S

We are allowed to differentiate [MATHEMATICAL EXPRESSION OMITTED] with respect to [??] under the

[MATHEMATICAL EXPRESSION OMITTED]

integral sign since f(x) vanishes so rapidly at ∞. We get

[MATHEMATICAL EXPRESSION OMITTED]

So the Fourier transform of (-ix)f(x) is [MATHEMATICAL EXPRESSION OMITTED]

Integration by parts (with vanishing values at the end points) gives

[MATHEMATICAL EXPRESSION OMITTED]

So the Fourier transform of f' is [MATHEMATICAL EXPRESSION OMITTED]. Putting these two facts together gives

Theorem 2.1.1.The Fourier transform is well defined on S and

[MATHEMATICAL EXPRESSION OMITTED]

This follows by differentiation under the integral sign and by integration by parts, showing that the Fourier transform mapsS to S.

Convolution goes to multiplication.

[MATHEMATICAL EXPRESSION OMITTED]

so

[MATHEMATICAL EXPRESSION OMITTED]

Scaling.

For any f [member of] S and a > 0 define Saf by (Sa) f(x) := f (ax). Then setting u = ax so dx = (1/a)du we have

[MATHEMATICAL EXPRESSION OMITTED]

so

[MATHEMATICAL EXPRESSION OMITTED]

The Fourier transform of a Gaussian is a Gaussian. The polar coordinate trick evaluates

[MATHEMATICAL EXPRESSION OMITTED]

The integral

[MATHEMATICAL EXPRESSION OMITTED]

converges for all complex values of η, uniformly in any compact region. Hence it defines an analytic function of η that can be evaluated by taking η to be real and then using analytic continuation.

For real η we complete the square and make a change of variables:

[MATHEMATICAL EXPRESSION OMITTED]

We will make much use of this equation in what follows.

Scaling the unit Gaussian.

If we set a = ε in our scaling equation and define ρε := Sεn so

[MATHEMATICAL EXPRESSION OMITTED]

then

[MATHEMATICAL EXPRESSION OMITTED]

Notice that for any g [member of] S we have (by a change of variables)

[MATHEMATICAL EXPRESSION OMITTED]

so setting a = ε we conclude that

[MATHEMATICAL EXPRESSION OMITTED]

for all ε.

Let

[MATHEMATICAL EXPRESSION OMITTED]

and

[MATHEMATICAL EXPRESSION OMITTED]

Then

[MATHEMATICAL EXPRESSION OMITTED]

so

[MATHEMATICAL EXPRESSION OMITTED]

An approximation. Since g [member of] S it is uniformly continuous on R, for any δ > 0 we can find ε0 so that the above integral is less than ? in absolute value for all 0 < ε < ε0. In short,

[MATHEMATICAL EXPRESSION OMITTED]

The multiplication formula. This says that

[MATHEMATICAL EXPRESSION OMITTED]

for any f, g ε S. Indeed the left-hand side equals

[MATHEMATICAL EXPRESSION OMITTED]

We can write this integral as a double integral and then interchange the order of integration, which gives the right-hand side.

The inversion formula. This says that for any f [member of] S

[MATHEMATICAL EXPRESSION OMITTED]

Proof. First observe that for any h [member of] S the Fourier transform of [MATHEMATICAL EXPRESSION OMITTED] is just [MATHEMATICAL EXPRESSION OMITTED] as follows directly from the definition. Taking [MATHEMATICAL EXPRESSION OMITTED] in the multiplication formula gives

[MATHEMATICAL EXPRESSION OMITTED]

We know that the right-hand side approaches f(x) as ε -> 0. Also, [MATHEMATICAL EXPRESSION OMITTED] for each fixed t, and in fact uniformly on any bounded t interval. Furthermore, [MATHEMATICAL EXPRESSION OMITTED] for all t. So choosing the interval of integration large enough, we can take the left-hand side as close as we like to [MATHEMATICAL EXPRESSION OMITTED] by then choosing ε sufficiently small.

Plancherel's theorem. This says that

[MATHEMATICAL EXPRESSION OMITTED]

Proof. Let

[MATHEMATICAL EXPRESSION OMITTED]

Then the Fourier transform of [??] is given by

[MATHEMATICAL EXPRESSION OMITTED]

so

[MATHEMATICAL EXPRESSION OMITTED]

Thus

[MATHEMATICAL EXPRESSION OMITTED]

The inversion formula applied to f * [??] and evaluated at 0 gives

[MATHEMATICAL EXPRESSION OMITTED]

The left-hand side of this equation is

[MATHEMATICAL EXPRESSION OMITTED]

Thus we have proved Plancherel's formula

[MATHEMATICAL EXPRESSION OMITTED]

The Poisson summation formula. This says that for any g [member of] S we have

[MATHEMATICAL EXPRESSION OMITTED]

Proof. Let [MATHEMATICAL EXPRESSION OMITTED] so h is a smooth function, periodic of period 2π and

[MATHEMATICAL EXPRESSION OMITTED]

Expand h into a Fourier series [MATHEMATICAL EXPRESSION OMITTED] where

[MATHEMATICAL EXPRESSION OMITTED]

Setting x = 0 in the Fourier expansion

[MATHEMATICAL EXPRESSION OMITTED]

gives

[MATHEMATICAL EXPRESSION OMITTED]

The Shannon sampling theorem.

Let f [member of] S be such that its Fourier transform is supported in the interval [-π, π]. Then a knowledge off(n)for alln [member of] Z determines f . This theorem is the basis for all digital sampling used in information technology. More explicitly,

[MATHEMATICAL EXPRESSION OMITTED]

Proof. Let g be the periodic function (of period 2]pi]) which extends [??], the Fourier transform of f. So

[MATHEMATICAL EXPRESSION OMITTED]

and is periodic.

Expand g into a Fourier series:

[MATHEMATICAL EXPRESSION OMITTED]

where

[MATHEMATICAL EXPRESSION OMITTED]

or, by the Fourier inversion formula,

[MATHEMATICAL EXPRESSION OMITTED]

But

[MATHEMATICAL EXPRESSION OMITTED]

Replacing n by -n in the sum, and interchanging summation and integration, which is legitimate since the f(n) decrease very fast, this becomes

[MATHEMATICAL EXPRESSION OMITTED]

But

[MATHEMATICAL EXPRESSION OMITTED]

Rescaling the Shannon sampling theorem. It is useful to reformulate this via rescaling so that the interval [-π, π] is replaced by an arbitrary interval symmetric about the origin: In the engineering literature, the frequency λ is defined by

[MATHEMATICAL EXPRESSION OMITTED]

Suppose we want to apply (2.1) to g = Sa f . We know that the Fourier transform of g is (1/a)S1/a [??] and

[MATHEMATICAL EXPRESSION OMITTED]

So if

[MATHEMATICAL EXPRESSION OMITTED]

we want to choose a so that [MATHEMATICAL EXPRESSION OMITTED] or

[MATHEMATICAL EXPRESSION OMITTED] (2.2)

For a in this range (2.1) says that

[MATHEMATICAL EXPRESSION OMITTED]

or setting t = ax,

[MATHEMATICAL EXPRESSION OMITTED] (2.3)

(Continues…)

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Excerpted from "A Mathematical Companion to Quantum Mechanics"
by .
Copyright © 2019 Shlomo Sternberg.
Excerpted by permission of Dover Publications, Inc..
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.

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Table of Contents

Introduction
The Fourier transform
The spectral theorem, I
Unbounded operators
Semi-groups, I
Self-adjoint operators
Semi-groups, II
Semi-groups, III
Weyl's theorem on the essential spectrum
More from Weyl's theorem
Extending the functional analysis via Riesz
Wintner's proof of the spectral theorem
The L2 version of a spectral theorem
Rayleigh-Ritz
Some one-dimensional quantum mechanics
More one-dimensional quantum mechanics
Some three-dimensional computations
Bound states and scattering states
The exponential decay of eigenstates
Lorch's proof of the spectral theorem
Scattering theory via Lax and Phillips
Huygens' principle
Some quantum mechanical scattering theory
The Groenwald-van Hove theorem
Chernoff's theorem
Some background material