Mathematical Methods in Quantum Mechanics

1 Mathematical Methodsin Quantum MechanicsWith Applicationsto Schr odinger OperatorsGerald TeschlNote: The AMS has granted the permission to post this online edition!This version is for personal online use only! If you like this book and wantto support the idea of online versions, please consider buying this book: Studiesin MathematicsVolume 99 American Mathematical SocietyProvidence, Rhode IslandEditorial BoardDavid Cox (Chair)Steven G. KrantsRafe MazzeoMartin Scharlemann2000 Mathematics subject , 81 Qxx, 46-01, 34 Bxx, book provides a self-contained introduction to Mathematical Methods in quan-tum Mechanics (spectral theory) with applications to Schr odinger operators. The first part cov-ers Mathematical foundations of Quantum Mechanics from self-adjointness, the spectral theorem, Quantum dynamics (including Stone s and the RAGE theorem) to perturbation theory for self-adjoint second part starts with a detailed study of the free Schr odinger operator respectivelyposition, momentum and angular momentum operators.

2 Then we develop Weyl Titchmarsh the-ory for Sturm Liouville operators and apply it to spherically symmetric problems, in particularto the hydrogen atom. Next we investigate self-adjointness of atomic Schr odinger operators andtheir essential spectrum, in particular the HVZ theorem. Finally we have a look at scatteringtheory and prove asymptotic completeness in the short range additional information and updates on this book, visit: by LATEXand Makeindex. Version: February 12, of Congress Cataloging-in-Publication DataTeschl, Gerald, 1970 Mathematical Methods in Quantum Mechanics : with applications to Schr odinger operators/ Gerald cm. (Graduate Studies in Mathematics ; v. 99)Includes bibliographical references and 978-0-8218-4660-5 (alk. paper)1. Schr odinger operators. 2. Quantum theory Mathematics. I. 20092008045437515 .724 dc22 Copying and readers of this publication, and nonprofit libraries actingfor them, are permitted to make fair use of the material, such as to copy a chapter for usein teaching or research.

3 Permission is granted to quote brief passages from this publication inreviews, provided the customary acknowledgement of the source is , systematic copying, or multiple reproduction of any material in this pub-lication (including abstracts) is permitted only under license from the American MathematicalSociety. Requests for such permissions should be addressed to the Assistant to the Publisher,American Mathematical Society, Box 6248, Providence, Rhode Island 02940-6248. Requestscan also be made by e-mail 2009 by the American Mathematical Society. All rights American Mathematical Society retains all rightsexcept those granted too the United States Susanne, Simon, and JakobContentsPrefacexiPart 0. PreliminariesChapter 0. A first look at Banach and Hilbert spaces3 Warm up: Metric and topological spaces3 The Banach space of continuous functions12 The geometry of Hilbert spaces16 Completeness22 Bounded operators22 LebesgueLpspaces25 Appendix: The uniform boundedness principle32 Part 1.

4 Mathematical Foundations of Quantum MechanicsChapter 1. Hilbert spaces37 Hilbert spaces37 Orthonormal bases39 The projection theorem and the Riesz lemma43 Orthogonal sums and tensor products45 TheC algebra of bounded linear operators47 Weak and strong convergence49 Appendix: The Stone Weierstra theorem51 Chapter 2. Self-adjointness and spectrum55viiviiiContents Some Quantum mechanics55 Self-adjoint operators58 Quadratic forms and the Friedrichs extension67 Resolvents and spectra73 Orthogonal sums of operators79 Self-adjoint extensions81 Appendix: Absolutely continuous functions84 Chapter 3. The spectral theorem87 The spectral theorem87 More on Borel measures99 Spectral types104 Appendix: The Herglotz theorem106 Chapter 4. Applications of the spectral theorem111 Integral formulas111 Commuting operators115 The min-max theorem117 Estimating eigenspaces119 Tensor products of operators120 Chapter 5.

5 Quantum dynamics123 The time evolution and Stone s theorem123 The RAGE theorem126 The Trotter product formula131 Chapter 6. Perturbation theory for self-adjoint operators133 Relatively bounded operators and the Kato Rellich theorem 133 More on compact operators136 Hilbert Schmidt and trace class operators139 Relatively compact operators and Weyl s theorem145 Relatively form bounded operators and the KLMN theorem 149 Strong and norm resolvent convergence153 Part 2. Schr odinger OperatorsChapter 7. The free Schr odinger operator161 The Fourier transform161 The free Schr odinger operator167 Contentsix The time evolution in the free case169 The resolvent and Green s function171 Chapter 8. Algebraic methods173 Position and momentum173 Angular momentum175 The harmonic oscillator178 Abstract commutation179 Chapter 9. One-dimensional Schr odinger operators181 Sturm Liouville operators181 Weyl s limit circle, limit point alternative187 Spectral transformations I195 Inverse spectral theory202 Absolutely continuous spectrum206 Spectral transformations II209 The spectra of one-dimensional Schr odinger operators214 Chapter 10.

6 One-particle Schr odinger operators221 Self-adjointness and spectrum221 The hydrogen atom222 Angular momentum225 The eigenvalues of the hydrogen atom229 Nondegeneracy of the ground state235 Chapter 11. Atomic Schr odinger operators239 Self-adjointness239 The HVZ theorem242 Chapter 12. Scattering theory247 Abstract theory247 Incoming and outgoing states250 Schr odinger operators with short range potentials253 Part 3. AppendixAppendix A. Almost everything about Lebesgue integration259 Borel measures in a nut shell259 Extending a premeasure to a measure263 Measurable functions268xContents The Lebesgue integral270 Product measures275 Vague convergence of measures278 Decomposition of measures280 Derivatives of measures282 Bibliographical notes289 Bibliography293 Glossary of notation297 Index301 PrefaceOverviewThe present text was written for my courseSchr odinger Operatorsheldat the University of Vienna in winter 1999, summer 2002, summer 2005,and winter 2007.

7 It gives a brief but rather self-contained introductionto the Mathematical Methods of Quantum Mechanics with a view towardsapplications to Schr odinger operators. The applications presented are highlyselective and many important and interesting items are not touched 1 is a stripped down introduction to spectral theory of unboundedoperators where I try to introduce only those topics which are needed forthe applications later on. This has the advantage that you will (hopefully)not get drowned in results which are never used again before you get tothe applications. In particular, I am not trying to present an encyclopedicreference. Nevertheless I still feel that the first part should provide a solidbackground covering many important results which are usually taken forgranted in more advanced books and research approach is built around the spectral theorem as the central I try to get to it as quickly as possible.

8 Moreover, I do not take thedetour over bounded operators but I go straight for the unbounded addition, existence of spectral measures is established via the Herglotztheorem rather than the Riesz representation theorem since this approachpaves the way for an investigation of spectral types via boundary values ofthe resolvent as the spectral parameter approaches the real 2 starts with the free Schr odinger equation and computes thefree resolvent and time evolution. In addition, I discuss position, momen-tum, and angular momentum operators via algebraic Methods . This isusually found in any physics textbook on Quantum Mechanics , with theonly difference that I include some technical details which are typicallynot found there. Then there is an introduction to one-dimensional mod-els (Sturm Liouville operators) including generalized eigenfunction expan-sions (Weyl Titchmarsh theory) and subordinacy theory from Gilbert andPearson.

9 These results are applied to compute the spectrum of the hy-drogen atom, where again I try to provide some Mathematical details notfound in physics textbooks. Further topics are nondegeneracy of the groundstate, spectra of atoms (the HVZ theorem), and scattering theory (the En method).PrerequisitesI assume some previous experience with Hilbert spaces and boundedlinear operators which should be covered in any basic course on functionalanalysis. However, while this assumption is reasonable for mathematicsstudents, it might not always be for physics students. For this reason thereis a preliminary chapter reviewing all necessary results (including proofs).In addition, there is an appendix (again with proofs) providing all necessaryresults from measure present book is highly influenced by the four volumes of Reed andSimon [40] [43] (see also [14]) and by the book by Weidmann [60] (anextended version of which has recently appeared in two volumes [62], [63],however, only in German).

10 Other books with a similar scope are for example[14], [15], [21], [23], [39], [48], and [55]. For those who want to know moreabout the physical aspects, I can recommend the classical book by Thirring[58] and the visual guides by Thaller [56], [57]. Further information can befound in the bibliographical notes at the s guideThere is some intentional overlap between Chapter 0, Chapter 1, andChapter 2. Hence, provided you have the necessary background, you canstart reading in Chapter 1 or even Chapter 2. Chapters 2 and 3 are keyPrefacexiiichapters and you should study them in detail (except for Section whichcan be skipped on first reading). Chapter 4 should give you an idea of howthe spectral theorem is used. You should have a look at ( ) the firstsection and you can come back to the remaining ones as needed. Chapter 5contains two key results from Quantum dynamics: Stone s theorem and theRAGE theorem.