Graduate Texts in Mathematics Brian C. Hall Quantum Theory for Mathematicians www.com Graduate Texts in Mathematics 267 www.com Graduate Texts in Mathematics Series Editors: Sheldon Axler San Francisco State University, San Francisco, CA, USA Kenneth Ribet University of California, Berkeley, CA, USA Advisory Board: Colin Adams, Williams College, Williamstown, MA, USA Alejandro Adem, University of British Columbia, Vancouver, BC, Canada Ruth Charney, Brandeis University, Waltham, MA, USA Irene M. Gamba, The University of Texas at Austin, Austin, TX, USA Roger E. Howe, Yale University, New Haven, CT, USA David Jerison, Massachusetts Institute of Technology, Cambridge, MA, USA Jeffrey C.
Lagarias, University of Michigan, Ann Arbor, MI, USA Jill Pipher, Brown University, Providence, RI, USA Fadil Santosa, University of Minnesota, Minneapolis, MN, USA Amie Wilkinson, University of Chicago, Chicago, IL, USA Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study. For further volumes: http://www.com/series/136 www.
Hall Quantum Theory for Mathematicians 123 www. 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 © Springer Science+Business Media New York 2013 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissim- ilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the pur- pose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
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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 www.com viii Preface attempted to give a reasonably comprehensive treatment of nonrelativistic quantum mechanics, including topics found in typical physics texts (e., the harmonic oscillator, the hydrogen atom, and the WKB approximation) as well as more mathematical topics (e., 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.
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. 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 Schmüdgen [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.
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., constant along each solution of Newton’s equation). I give a short treatment here www.com Preface ix of Poisson brackets and Hamilton’s form of Newton’s equation, deferring a full discussion of “fancy” classical mechanics to 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. 4 and 5, we look into two illustrative cases of the Schrödinger 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. 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. 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.
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. 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. 13, we look at differ- ent “quantization schemes” (i., different ways of ordering products of the noncommuting position and momentum operators). 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.
15 through 18, we examine some less elementary issues in quan- tum theory. Chapter 15 addresses the WKB (Wentzel–Kramers–Brillouin) www.com 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.
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.