Quantum theory for mathematicians (Graduate Texts in Mathematics 267)

Graduate Texts in Mathematics

Brian C. Hall

Quantum

Theory for

Mathematicians

Graduate Texts in Mathematics 267

Graduate Texts in Mathematics

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Brian C. Hall

Quantum Theory for

Mathematicians

123

Brian C. Hall

Department of Mathematics

University of Notre Dame

Notre Dame, IN, USA

ISSN 0072-5285

ISBN 978-1-4614-7115-8 ISBN 978-1-4614-7116-5 (eBook)

DOI 10.1007/978-1-4614-7116-5

Springer New York Heidelberg Dordrecht London

Library of Congress Control Number: 2013937175

Mathematics Subject Classification: 81-01, 81S05, 81R05, 46N50, 81Q20, 81Q10, 81S40, 53D50

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For as the heavens are higher than the earth, so are my ways higher than

your ways, and my thoughts than your thoughts, says the Lord.

Isaiah 55:9

Preface

Ideas from quantum physics play important roles in many parts of modern

mathematics. Many parts of representation theory, for example, are moti￾vated by quantum mechanics, including the Wigner–Mackey theory of in￾duced representations, the Kirillov–Kostant orbit method, and, of course,

quantum groups. The Jones polynomial in knot theory, the Gromov–Witten

invariants in topology, and mirror symmetry in algebraic topology are other

notable examples. The awarding of the 1990 Fields Medal to Ed Witten, a

physicist, gives an idea of the scope of the influence of quantum theory in

mathematics.

Despite the importance of quantum mechanics to mathematics, there is

no easy way for mathematicians to learn the subject. Quantum mechan￾ics books in the physics literature are generally not easily understood by

most mathematicians. There is, of course, a lower level of mathematical

precision in such books than mathematicians are accustomed to. In addi￾tion, physics books on quantum mechanics assume knowledge of classical

mechanics that mathematicians often do not have. And, finally, there is a

subtle difference in “culture”—differences in terminology and notation—

that can make reading the physics literature like reading a foreign language

for the mathematician. There are few books that attempt to translate quan￾tum theory into terms that mathematicians can understand.

This book is intended as an introduction to quantum mechanics for math￾ematicians with little prior exposure to physics. The twin goals of the book

are (1) to explain the physical ideas of quantum mechanics in language

mathematicians will be comfortable with, and (2) to develop the neces￾sary mathematical tools to treat those ideas in a rigorous fashion. I have

vii

viii Preface

attempted to give a reasonably comprehensive treatment of nonrelativistic

quantum mechanics, including topics found in typical physics texts (e.g.,

the harmonic oscillator, the hydrogen atom, and the WKB approximation)

as well as more mathematical topics (e.g., quantization schemes, the Stone–

von Neumann theorem, and geometric quantization). I have also attempted

to minimize the mathematical prerequisites. I do not assume, for example,

any prior knowledge of spectral theory or unbounded operators, but pro￾vide a full treatment of those topics in Chaps. 6 through 10 of the text.

Similarly, I do not assume familiarity with the theory of Lie groups and

Lie algebras, but provide a detailed account of those topics in Chap. 16.

Whenever possible, I provide full proofs of the stated results.

Most of the text will be accessible to graduate students in mathematics

who have had a first course in real analysis, covering the basics of L2 spaces

and Hilbert spaces. Appendix A reviews some of the results that are used in

the main body of the text. In Chaps. 21 and 23, however, I assume knowl￾edge of the theory of manifolds. I have attempted to provide motivation for

many of the definitions and proofs in the text, with the result that there

is a fair amount of discussion interspersed with the standard definition￾theorem-proof style of mathematical exposition. There are exercises at the

end of each chapter, making the book suitable for graduate courses as well

as for independent study.

In comparison to the present work, classics such as Reed and Simon [34]

and Glimm and Jaffe [14], along with the recent book of Schm¨udgen [35],

are more focused on the mathematical underpinnings of the theory than

on the physical ideas. Hannabuss’s text [22] is fairly accessible to math￾ematicians, but—despite the word “graduate” in the title of the series—

uses an undergraduate level of mathematics. The recent book of Takhtajan

[39], meanwhile, has an expository bent to it, but provides less physical

motivation and is less self-contained than the present book. Whereas, for

example, Takhtajan begins with Lagrangian and Hamiltonian mechanics

on manifolds, I begin with “low-tech” classical mechanics on the real line.

Similarly, Takhtajan assumes knowledge of unbounded operators and Lie

groups, while I provide substantial expositions of both of those subjects.

Finally, there is the work of Folland [13], which I highly recommend, but

which deals with quantum field theory, whereas the present book treats

only nonrelativistic quantum mechanics, except for a very brief discussion

of quantum field theory in Sect. 20.6.

The book begins with a quick introduction to the main ideas of classical

and quantum mechanics. After a brief account in Chap. 1 of the historical

origins of quantum theory, I turn in Chap. 2 to a discussion of the neces￾sary background from classical mechanics. This includes Newton’s equa￾tion in varying degrees of generality, along with a discussion of important

physical quantities such as energy, momentum, and angular momentum,

and conditions under which these quantities are “conserved” (i.e., constant

along each solution of Newton’s equation). I give a short treatment here

Preface ix

of Poisson brackets and Hamilton’s form of Newton’s equation, deferring a

full discussion of “fancy” classical mechanics to Chap. 21.

In Chap. 3, I attempt to motivate the structures of quantum mechanics in

the simplest setting. Although I discuss the “axioms” (in standard physics

terminology) of quantum mechanics, I resolutely avoid a strictly axiomatic

approach to the subject (using, say, C∗-algebras). Rather, I try to provide

some motivation for the position and momentum operators and the Hilbert

space approach to quantum theory, as they connect to the probabilistic as￾pect of the theory. I do not attempt to explain the strange probabilistic

nature of quantum theory, if, indeed, there is any explanation of it. Rather,

I try to elucidate how the wave function, along with the position and mo￾mentum operators, encodes the relevant probabilities.

In Chaps. 4 and 5, we look into two illustrative cases of the Schr¨odinger

equation in one space dimension: a free particle and a particle in a square

well. In these chapters, we encounter such important concepts as the dis￾tinction between phase velocity and group velocity and the distinction be￾tween a discrete and a continuous spectrum.

In Chaps. 6 through 10, we look into some of the technical mathematical

issues that are swept under the carpet in earlier chapters. I have tried to

design this section of the book in such a way that a reader can take in as

much or as little of the mathematical details as desired. For a reader who

simply wants the big picture, I outline the main ideas and results of spec￾tral theory in Chap. 6, including a discussion of the prototypical example

of an operator with a continuous spectrum: the momentum operator. For

a reader who wants more information, I provide statements of the spec￾tral theorem (in two different forms) for bounded self-adjoint operators in

Chap. 7, and an introduction to the notion of unbounded self-adjoint op￾erators in Chap. 9. Finally, for the reader who wants all the details, I give

proofs of the spectral theorem for bounded and unbounded self-adjoint

operators, in Chaps. 8 and 10, respectively.

In Chaps. 11 through 14, we turn to the vitally important canonical com￾mutation relations. These are used in Chap. 11 to derive algebraically the

spectrum of the quantum harmonic oscillator. In Chap. 12, we discuss the

uncertainty principle, both in its general form (for arbitrary pairs of non￾commuting operators) and in its specific form (for the position and momen￾tum operators). We pay careful attention to subtle domain issues that are

usually glossed over in the physics literature. In Chap. 13, we look at differ￾ent “quantization schemes” (i.e., different ways of ordering products of the

noncommuting position and momentum operators). In Chap. 14, we turn to

the celebrated Stone–von Neumann theorem, which provides a uniqueness

result for representations of the canonical commutation relations. As in the

case of the uncertainty principle, there are some subtle domain issues here

that require attention.

In Chaps. 15 through 18, we examine some less elementary issues in quan￾tum theory. Chapter 15 addresses the WKB (Wentzel–Kramers–Brillouin)

x Preface

approximation, which gives simple but approximate formulas for the eigen￾vectors and eigenvalues for the Hamiltonian operator in one dimension.

After this, we introduce (Chap. 16) the notion of Lie groups, Lie alge￾bras, and their representations, all of which play an important role in

many parts of quantum mechanics. In Chap. 17, we consider the example

of angular momentum and spin, which can be understood in terms of the

representations of the rotation group SO(3). Here a more mathematical

approach—especially the relationship between Lie group representations

and Lie algebra representations—can substantially clarify a topic that is

rather mysterious in the physics literature. In particular, the concept of

“fractional spin” can be understood as describing a representation of the

Lie algebra of the rotation group for which there is no associated represen￾tation of the rotation group itself. In Chap. 18, we illustrate these ideas by

describing the energy levels of the hydrogen atom, including a discussion

of the hidden symmetries of hydrogen, which account for the “accidental

degeneracy” in the levels. In Chap. 19, we look more closely at the concept

of the “state” of a system in quantum mechanics. We look at the notion

of subsystems of a quantum system in terms of tensor products of Hilbert

spaces, and we see in this setting that the notion of “pure state” (a unit

vector in the relevant Hilbert space) is not adequate. We are led, then, to

the notion of a mixed state (or density matrix). We also examine the idea

that, in quantum mechanics, “identical particles are indistinguishable.”

Finally, in Chaps. 21 through 23, we examine some advanced topics in

classical and quantum mechanics. We begin, in Chap. 20, by considering the

path integral formulation of quantum mechanics, both from the heuristic

perspective of the Feynman path integral, and from the rigorous perspective

of the Feynman–Kac formula. Then, in Chap. 21, we give a brief treatment

of Hamiltonian mechanics on manifolds. Finally, we consider the machinery

of geometric quantization, beginning with the Euclidean case in Chap. 22

and continuing with the general case in Chap. 23.

I am grateful to all who have offered suggestions or made corrections

to the manuscript, including Renato Bettiol, Edward Burkard, Matt Cecil,

Tiancong Chen, Bo Jacoby, Will Kirwin, Nicole Kroeger, Wicharn Lewkeer￾atiyutkul, Jeff Mitchell, Eleanor Pettus, Ambar Sengupta, and Augusto

Stoffel. I am particularly grateful to Michel Talagrand who read almost

the entire manuscript and made numerous corrections and suggestions. Fi￾nally, I offer a special word of thanks to my advisor and friend, Leonard

Gross, who started me on the path toward understanding the mathemati￾cal foundations of quantum mechanics. Readers are encouraged to send me

comments or corrections at [email protected].

Notre Dame, IN, USA Brian C. Hall

Contents

1 The Experimental Origins of Quantum Mechanics 1

1.1 Is Light a Wave or a Particle? ................ 1

1.2 Is an Electron a Wave or a Particle? ............ 7

1.3 Schr¨odinger and Heisenberg . . . . . . . . . . . . . . . . . 13

1.4 A Matter of Interpretation . . . . . . . . . . . . . . . . . . 14

1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 A First Approach to Classical Mechanics 19

2.1 Motion in R1 . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Motion in Rn . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 Systems of Particles . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Angular Momentum . . . . . . . . . . . . . . . . . . . . . . 31

2.5 Poisson Brackets and Hamiltonian Mechanics . . . . . . . 33

2.6 The Kepler Problem and the Runge–Lenz Vector . . . . . 41

2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 A First Approach to Quantum Mechanics 53

3.1 Waves, Particles, and Probabilities . . . . . . . . . . . . . 53

3.2 A Few Words About Operators and Their Adjoints . . . . 55

3.3 Position and the Position Operator . . . . . . . . . . . . . 58

3.4 Momentum and the Momentum Operator . . . . . . . . . 59

3.5 The Position and Momentum Operators . . . . . . . . . . 62

3.6 Axioms of Quantum Mechanics: Operators

and Measurements . . . . . . . . . . . . . . . . . . . . . . 64

xi

xii Contents

3.7 Time-Evolution in Quantum Theory . . . . . . . . . . . . 70

3.8 The Heisenberg Picture . . . . . . . . . . . . . . . . . . . . 78

3.9 Example: A Particle in a Box . . . . . . . . . . . . . . . . 80

3.10 Quantum Mechanics for a Particle in Rn . . . . . . . . . . 82

3.11 Systems of Multiple Particles . . . . . . . . . . . . . . . . 84

3.12 Physics Notation . . . . . . . . . . . . . . . . . . . . . . . 85

3.13 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

4 The Free Schr¨odinger Equation 91

4.1 Solution by Means of the Fourier Transform . . . . . . . . 92

4.2 Solution as a Convolution . . . . . . . . . . . . . . . . . . 94

4.3 Propagation of the Wave Packet: First Approach . . . . . 97

4.4 Propagation of the Wave Packet: Second Approach . . . . 100

4.5 Spread of the Wave Packet . . . . . . . . . . . . . . . . . 104

4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5 A Particle in a Square Well 109

5.1 The Time-Independent Schr¨odinger Equation . . . . . . . 109

5.2 Domain Questions and the Matching Conditions . . . . . . 111

5.3 Finding Square-Integrable Solutions . . . . . . . . . . . . . 112

5.4 Tunneling and the Classically Forbidden Region . . . . . 118

5.5 Discrete and Continuous Spectrum . . . . . . . . . . . . . 119

5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6 Perspectives on the Spectral Theorem 123

6.1 The Difficulties with the Infinite-Dimensional Case . . . . 123

6.2 The Goals of Spectral Theory . . . . . . . . . . . . . . . . 125

6.3 A Guide to Reading . . . . . . . . . . . . . . . . . . . . . . 126

6.4 The Position Operator . . . . . . . . . . . . . . . . . . . . 126

6.5 Multiplication Operators . . . . . . . . . . . . . . . . . . . 127

6.6 The Momentum Operator . . . . . . . . . . . . . . . . . . 127

7 The Spectral Theorem for Bounded Self-Adjoint

Operators: Statements 131

7.1 Elementary Properties of Bounded Operators . . . . . . . 131

7.2 Spectral Theorem for Bounded Self-Adjoint

Operators, I . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.3 Spectral Theorem for Bounded Self-Adjoint

Operators, II . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8 The Spectral Theorem for Bounded Self-Adjoint

Operators: Proofs 153

8.1 Proof of the Spectral Theorem, First Version . . . . . . . . 153

Contents xiii

8.2 Proof of the Spectral Theorem, Second Version . . . . . . 162

8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

9 Unbounded Self-Adjoint Operators 169

9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 169

9.2 Adjoint and Closure of an Unbounded Operator . . . . . . 170

9.3 Elementary Properties of Adjoints and Closed

Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

9.4 The Spectrum of an Unbounded Operator . . . . . . . . . 177

9.5 Conditions for Self-Adjointness and Essential

Self-Adjointness . . . . . . . . . . . . . . . . . . . . . . . . 179

9.6 A Counterexample . . . . . . . . . . . . . . . . . . . . . . 182

9.7 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . 184

9.8 The Basic Operators of Quantum Mechanics . . . . . . . . 185

9.9 Sums of Self-Adjoint Operators . . . . . . . . . . . . . . . 190

9.10 Another Counterexample . . . . . . . . . . . . . . . . . . . 193

9.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

10 The Spectral Theorem for Unbounded Self-Adjoint

Operators 201

10.1 Statements of the Spectral Theorem . . . . . . . . . . . . . 202

10.2 Stone’s Theorem and One-Parameter Unitary Groups . . . 207

10.3 The Spectral Theorem for Bounded Normal

Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

10.4 Proof of the Spectral Theorem for Unbounded

Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . 220

10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

11 The Harmonic Oscillator 227

11.1 The Role of the Harmonic Oscillator . . . . . . . . . . . . 227

11.2 The Algebraic Approach . . . . . . . . . . . . . . . . . . . 228

11.3 The Analytic Approach . . . . . . . . . . . . . . . . . . . . 232

11.4 Domain Conditions and Completeness . . . . . . . . . . . 233

11.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

12 The Uncertainty Principle 239

12.1 Uncertainty Principle, First Version . . . . . . . . . . . . . 241

12.2 A Counterexample . . . . . . . . . . . . . . . . . . . . . . 245

12.3 Uncertainty Principle, Second Version . . . . . . . . . . . . 246

12.4 Minimum Uncertainty States . . . . . . . . . . . . . . . . . 249

12.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

13 Quantization Schemes for Euclidean Space 255

13.1 Ordering Ambiguities . . . . . . . . . . . . . . . . . . . . . 255

13.2 Some Common Quantization Schemes . . . . . . . . . . . . 256

xiv Contents

13.3 The Weyl Quantization for R2n . . . . . . . . . . . . . . . 261

13.4 The “No Go” Theorem of Groenewold . . . . . . . . . . . 271

13.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

14 The Stone–von Neumann Theorem 279

14.1 A Heuristic Argument . . . . . . . . . . . . . . . . . . . . 279

14.2 The Exponentiated Commutation Relations . . . . . . . . 281

14.3 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 286

14.4 The Segal–Bargmann Space . . . . . . . . . . . . . . . . . 292

14.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

15 The WKB Approximation 305

15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 305

15.2 The Old Quantum Theory and the Bohr–Sommerfeld

Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

15.3 Classical and Semiclassical Approximations . . . . . . . . . 308

15.4 The WKB Approximation Away from the Turning

Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

15.5 The Airy Function and the Connection Formulas . . . . . 315

15.6 A Rigorous Error Estimate . . . . . . . . . . . . . . . . . . 320

15.7 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . 328

15.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

16 Lie Groups, Lie Algebras, and Representations 333

16.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

16.2 Matrix Lie Groups . . . . . . . . . . . . . . . . . . . . . . 335

16.3 Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . 338

16.4 The Matrix Exponential . . . . . . . . . . . . . . . . . . . 339

16.5 The Lie Algebra of a Matrix Lie Group . . . . . . . . . . . 342

16.6 Relationships Between Lie Groups and Lie Algebras . . . . 344

16.7 Finite-Dimensional Representations of Lie Groups

and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 350

16.8 New Representations from Old . . . . . . . . . . . . . . . . 358

16.9 Infinite-Dimensional Unitary Representations . . . . . . . 360

16.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363

17 Angular Momentum and Spin 367

17.1 The Role of Angular Momentum

in Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 367

17.2 The Angular Momentum Operators in R3 . . . . . . . . . 368

17.3 Angular Momentum from the Lie Algebra Point

of View . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369

17.4 The Irreducible Representations of so(3) . . . . . . . . . . 370

17.5 The Irreducible Representations of SO(3) . . . . . . . . . . 375

17.6 Realizing the Representations Inside L2(S2) . . . . . . . . 376