《量子力学》课程教学资源（书籍文献）GTM267 Quantum Theory for Mathematicians，Brian C. Hall，Sheldon Axler、Kenneth Ribet（电子书）

GraduateTextsinMathematicsGTMBrian C. HallQuantumTheory forMathematiciansSpringer

Graduate Texts in Mathematics Brian C. Hall Quantum Theory for Mathematicians

GraduateTextsinMathematicsSeries Editors:SheldonAxlerSan Francisco State University, SanFrancisco, CA, USAKennethRibetUniversityofCalifomia,Berkeley,CA,USAAdvisory Board:ColinAdams,WilliamsCollege,Williamstown,MA,USAAlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,CanadaRuthCharney,BrandeisUniversityWaltham,MA,USAIreneM.Gamba,TheUniversityofTexas atAustin,Austin,TX,USARoger E.Howe,YaleUniversity,NewHaven,CT,USADavid Jerison,MassachusettsInstituteofTechnologyCambridge,MA,USAJeffreyC.Lagarias,UniversityofMichigan,AnnArbor,Ml,USAJillPipher,BrownUniversity,Providence,RI,USAFadil Santosa,University ofMinnesota,Minneapolis,MN,USAAmieWilkinson,UniversityofChicago,Chicago,IL,USAGraduate Texts in Mathematics bridge the gap between passive study andcreative understanding, offering graduate-level introductions to advanced topicsin mathematics.The volumes are carefully written as teaching aids and highlightcharacteristic features of the theory.Although these books arefrequently used astextbooks in graduate courses,they are also suitablefor individual study.Forfurthervolumes:http://www.springer.com/series/136

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.springer.com/series/136

Brian C.HallDepartment of MathematicsUniversityofNotreDameNotre Dame, IN, USAISSN0072-5285ISBN 978-1-4614-7115-8ISBN978-1-4614-7116-5 (eBook)DOI10.1007/978-1-4614-7116-5Springer New York Heidelberg Dordrecht LondonLibrary of Congress Control Number: 2013937175Mathematics Subject Classification: 81-01,81S05,81R05,46N50,81Q20, 81Q10,81S40,53D50SpringerScience+Business Media NewYork2013This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof 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 orinfomation 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 briefexcerpts 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 thework. Duplication of this publication or parts thereof is permitted only under the provisions of theCopyright Law of the Publisher's location, in its current version, and permission for use must alwaysbe obtained from Springer. Pemissions for use may be obtained through RightsLink at the CopyrightClearance Center. Violations are liable to prosecution under the respective Copyright Law.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publi-cation does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.While the advice and information in this book are believed to be true and accurate at the date ofpublication, neither the authors nor the editors nor the publisher can accept any legal responsibility forany errors or omissions that may be made. The publisher makes no warranty, express or implied, withrespect to the material contained herein.Printed on acid-free paperSpringer is part of Springer Science+Business Media (www.springer.com)

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 © 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. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher’s location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publi￾cation does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

PrefaceIdeas from quantumphysics playimportant roles in many parts of modernmathematics. 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,quantumgroups.TheJones polynomial inknot theory,the Gromov-Witteninvariants in topology,and mirror symmetry in algebraic topology are othernotable examples.The awarding of the 1990 Fields Medal to Ed Witten, aphysicist, gives an idea of the scope of the influence of quantum theory inmathematics.Despite the importance of quantum mechanics to mathematics, there isno easy way for mathematicians to learn the subject. Quantum mechan-ics books in the physics literature are generally not easily understood bymost mathematicians. There is, of course, a lower level of mathematicalprecision in such books than mathematicians are accustomed to. In addi-tion, physics books on quantum mechanics assume knowledge of classicalmechanics that mathematicians often do not have. And, finally, there is asubtle difference in “"culture"differences in terminology and notation-that can make reading the physics literature like reading a foreign languagefor themathematician.Thereare 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 bookare (1) to explain the physical ideas of quantum mechanics in languagemathematicians will be comfortable with, and (2) to develop the neces-sary mathematical tools to treat those ideas in a rigorous fashion. I havevii

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

viiPrefaceattempted to give a reasonably comprehensive treatment of nonrelativisticquantum mechanics, including topics found in typical physics texts (e.g.,theharmonic 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).Ihave also attemptedto 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 andLie 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 mathematicswho have had a first course in real analysis, covering the basics of L? spacesand Hilbert spaces. Appendix A reviews some of the results that are used inthe 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 formany of the definitions and proofs in the text, with the result that thereis a fair amount of discussion interspersed with the standard definition-theorem-proof style of mathematical exposition. There are exercises at theend of each chapter,making the book suitable for graduate courses as wellas forindependentstudy.In comparison to thepresent work, classics such as Reed and Simon [34]and Glimm and Jaffe[14],along with therecentbook of Schmidgen [35],are more focused on the mathematical underpinnings of the theory thanon 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 physicalmotivation and is less self-contained than the present book. Whereas, forexample, Takhtajan begins with Lagrangian and Hamiltonian mechanicson manifolds, I begin with"low-tech"classical mechanics on the real line.Similarly, Takhtajan assumes knowledge of unbounded operators and Liegroups, while I provide substantial expositions of both of those subjects.Finally, there is the work of Folland [13], which I highly recommend, butwhich deals with quantum field theory, whereas the present book treatsonly nonrelativistic quantum mechanics, except for a very brief discussionofquantumfieldtheoryin Sect.20.6.The book begins with a quick introduction to the main ideas of classicaland quantum mechanics. After a brief account in Chap.1 of the historicalorigins 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 importantphysical quantities such as energy, momentum, and angular momentum,and conditions under which these quantities are“conserved"(i.e., constantalong each solution of Newton's equation).I give a short treatment here

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

Prefaceixof Poisson brackets and Hamilton's form of Newton's equation, deferring afull discussion of "fancy"classical mechanicsto Chap.2l.In Chap.3,Iattempt to motivate the structures of quantum mechanics inthe simplest setting.Although I discuss the"axioms"(in standard physicsterminology) of quantum mechanics,I resolutely avoid a strictly axiomaticapproach to the subject (using, say, C*-algebras).Rather,I try to providesomemotivationforthepositionandmomentumoperatorsandtheHilbertspaceapproachtoquantumtheory,astheyconnecttotheprobabilisticas-pect of the theory.I do not attempt to erplain the strange probabilisticnature of quantum theory,if, indeed, there is any explanation of it. Rather,I try to elucidatehow the wave function, along with the position and mo-mentum operators, encodes therelevant probabilities.In Chaps.4 and 5, we look into two illustrative cases of the Schrodingerequation in one space dimension: a free particle and a particle in a squarewell. In these chapters, we encounter such important concepts as the dis-tinction between phase velocity and group velocity and the distinction be-tween adiscrete and a continuous spectrum.In Chaps. 6 through 10, we look into some of the technical mathematicalissues that are swept under the carpet in earlier chapters. I have tried todesign this section of the book in such a way that a reader can take in asmuch or as little of the mathematical details as desired. For a reader whosimply wants the big picture, I outline the main ideas and results of spectral theory in Chap.6, including a discussion of the prototypical exampleof an operator with a continuous spectrum: the momentum operator.Fora reader who wants more information, I provide statements of the spec-tral theorem (in two different forms) for bounded self-adjoint operators inChap.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 giveproofs of the spectral theorem for bounded and unbounded self-adjointoperators,in Chaps.8and 10,respectively.In Chaps. 11 through 14, we turn to the vitally important canonical com-mutation relations. These are used in Chap.l1 to derive algebraically thespectrum ofthe quantum harmonic oscillator.In Chap.12, we discusstheuncertaintyprinciple,both inits general form (for arbitrarypairsof non-commuting operators)and in its specificform (for theposition and momen-tum operators). We pay careful attention to subtle domain issues that areusually glossed over in the physics literature. In Chap.13, we look at differ-ent“quantization schemes"(i.e.,different ways of ordering products of thenoncommuting position and momentum operators).In Chap.14, we turn tothe celebrated Stone-von Neumann theorem, which provides a uniquenessresult for representations of the canonical commutation relations. As in thecase of the uncertainty principle, there are some subtle domain issues herethatrequireattention.In Chaps.15 through 18, we examine someless elementary issues in quantumtheory.Chapter15addressestheWKB(Wentzel-Kramers-Brillouin)

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)

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 inmany parts of quantum mechanics. In Chap.17, we consider the exampleof angular momentum and spin, which can be understood in terms of therepresentations of the rotation group SO(3).Here a more mathematicalapproach-especially the relationship between Lie group representationsand Lie algebra representations--can substantiallyclarifyatopicthatisrather mysterious in the physics literature. In particular, the concept of"fractional spin" can be understood as describing a representation of theLie 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 bydescribing the energy levels of the hydrogen atom, including a discussionof the hidden symmetries of hydrogen, which account for the"accidentaldegeneracy" in the levels. In Chap.19, we look more closely at the conceptof the"state"of a system in quantum mechanics. We look at the notionof subsystems of a quantum system in terms of tensor products of Hilbertspaces, and we see in this setting that the notion of"pure state"(a unitvectorin the relevantHilbert space)is not adequate.We areled, then,tothe notion of a mixed state (or density matrix). We also examine the ideathat, in quantum mechanics,"identical particles are indistinguishable."Finally, in Chaps.21 through 23, we examine some advanced topics inclassical and quantum mechanics. We begin, in Chap.20, by considering thepath integral formulation of quantum mechanics, both from the heuristicperspective of the Feynman path integral, and from the rigorous perspectiveof theFeynman-Kac formula.Then, in Chap.21, we give a brief treatmentof Hamiltonian mechanics onmanifolds.Finally,we consider themachineryof geometric quantization, beginning with theEuclidean casein Chap.22and continuing with the general case in Chap.23.I am grateful to all who have offered suggestions or made correctionsto 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 AugustoStoffel.I am particularly grateful to Michel Talagrand who read almostthe entire manuscript and made numerous corrections and suggestions. Fi-nally, I offer a special word of thanks to my advisor and friend, LeonardGross, who started me on the path toward understanding the mathemati-cal foundations of quantum mechanics. Readers are encouraged to send mecomments or corrections at bhall@nd.edu.NotreDame, IN, USABrian C.Hall

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 bhall@nd.edu. Notre Dame, IN, USA Brian C. Hall

Contents1The Experimental Origins of Quantum Mechanics11.1Is Light a Wave or a Particle?71.2Is an Electron a Wave or a Particle?131.3Schrodinger and Heisenberg141.4AMatterof Interpretation1.516Exercises192A First Approach to Classical Mechanics2.1Motion in Ri192.2Motion in Rn232.326Systems of Particles2.431Angular Momentum2.533PoissonBrackets andHamiltonian Mechanics2.641The Kepler Problem and the Runge-Lenz Vector2.746Exercises53AFirst Approach to Quantum Mechanics3533.1Waves.Particles,and Probabilities553.2A Few Words About Operators and Their Adjoints3.358Position and the Position Operator593.4Momentum and the Momentum Operator623.5ThePosition and Momentum Operators3.6Axioms of Quantum Mechanics: Operators64and Measurementsxi

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

xiiContents3.770Time-Evolution in QuantumTheory783.8The Heisenberg Picture3.980Example: A Particle in a Box823.10Quantum Mechanics fora Particle in R843.11Systems of Multiple Particles853.12Physics Notation883.13Exercises91The Free Schrodinger Equation924.1Solution by Means of the Fourier Transform4.294Solution as a Convolution974.3Propagation of the Wave Packet: First Approach4.4100Propagation of the Wave Packet: SecondApproach4.5104Spread of the Wave Packet4.6106ExercisesA Particle in a Square Well10955.1109The Time-Independent Schrodinger Equation5.2111Domain Questions and the Matching Conditions5.3112Finding Square-Integrable Solutions5.4118Tunneling and the Classically Forbidden Region5.5119Discrete and Continuous Spectrum5.6120Exercises123Perspectives on the Spectral Theorem61236.1TheDifficulties with theInfinite-Dimensional Case6.2125The Goals of Spectral Theory1266.3A Guide to Reading6.4126The Position Operator1276.5Multiplication Operators6.6127The Momentum OperatorThe Spectral TheoremforBounded Self-Adjoint131Operators:Statements1317.1Elementary Properties of Bounded Operators7.2Spectral Theorem for Bounded Self-Adjoint137Operators,I7.3Spectral Theorem for Bounded Self-Adjoint144Operators, II7.4150Exercises8The Spectral Theorem for BoundedSelf-Adjoint153Operators:Proofs1538.1Proof of the Spectral Theorem, First Version

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

xiliContents8.2162Proof of the Spectral Theorem, Second Version8.3166Exercises169Unbounded Self-Adjoint Operators91699.1Introduction9.2170Adjoint and Closure of an Unbounded Operator9.3Elementary Properties of Adjoints and Closed173Operators9.4177The Spectrum of an Unbounded Operator9.5Conditions for Self-Adjointness and Essential179Self-Adjointness9.6182A Counterexample9.7184An Example9.8185TheBasicOperatorsof QuantumMechanics9.9190Sums of Self-Adjoint Operators1939.10Another Counterexample1969.11Exercises10 TheSpectral Theoremfor Unbounded Self-Adjoint201Operators20210.1Statements of the Spectral Theorem20710.2Stone's Theorem and One-ParameterUnitary Groups10.3TheSpectral TheoremforBoundedNormal213Operators10.4Proof of the Spectral Theorem for Unbounded220Self-Adjoint Operators22410.5Exercises22711 The Harmonic Oscillator22711.1The Role of the Harmonic Oscillator22811.2The Algebraic Approach11.3TheAnalytic Approach23223311.4Domain Conditions and Completeness23611.5Exercises23912 The Uncertainty Principle12.1241Uncertainty Principle, First Version12.2245A Counterexample12.3246Uncertainty Principle, Second Version12.4249Minimum Uncertainty States25112.5Exercises25513Quantization Schemes for Euclidean Space25513.1Ordering Ambiguities25613.2SomeCommon Quantization Schemes

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