Mathematical concepts of quantum mechanics

397 Pages·2011·2.125 MB·English

by  Stephen J. Gustafson, Israel Michael Sigal (auth.)

Universitext
Universitext
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity
VincenzoCapasso
Universita`degliStudidiMilano
CarlesCasacuberta
UniversitatdeBarcelona
AngusJ.MacIntyre
QueenMary,UniversityofLondon
KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
CNRS,E´colePolytechnique
EndreSu¨li
UniversityofOxford
WojborA.Woyczynski
CaseWesternReserveUniversity
Universitextisaseriesoftextbooksthatpresentsmaterialfromawidevarietyof
mathematicaldisciplinesatmaster’slevelandbeyond.Thebooks,oftenwell
class-testedbytheirauthor,mayhaveaninformal,personalevenexperimental
approachtotheirsubjectmatter.Someofthemostsuccessfulandestablishedbooks
intheserieshaveevolvedthroughseveraleditions,alwaysfollowingtheevolution
ofteachingcurricula,toverypolishedtexts.
Thus as research topics trickle down into graduate-level teaching, first textbooks
writtenfornew,cutting-edgecoursesmaymaketheirwayintoUniversitext.
Forfurthervolumes:
www.springer.com/series/223
Stephen J. Gustafson
Israel Michael Sigal
Mathematical Concepts
of Quantum Mechanics
Second Edition
123
StephenJ.Gustafson IsraelMichaelSigal
UniversityofBritishColumbia UniversityofToronto
Dept.Mathematics Dept.Mathematics
Vancouver,BCV6T1Z2 40St.GeorgeStreet
Canada Toronto,ONM5S2E4
[email protected] Canada
[email protected]
ISBN978-3-642-21865-1 e-ISBN978-3-642-21866-8
DOI10.1007/978-3-642-21866-8
SpringerHeidelbergDordrechtLondonNewYork
LibraryofCongressControlNumber:2011935880
MathematicsSubjectClassification(2010):81S,47A,46N50
(cid:2)c Springer-VerlagBerlinHeidelberg2011
Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermitted onlyundertheprovisions oftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsare
liabletoprosecutionundertheGermanCopyrightLaw.
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Coverdesign:deblik,Berlin
Printedonacid-freepaper
SpringerispartofSpringerScience+BusinessMedia(www.springer.com)
Preface
Preface to the second edition
Oneof the main goals motivating this new edition was to enhance the
elementarymaterial.To this end,in additionto somerewriting andreorgani-
zation,severalnewsectionshavebeenadded(covering,forexample,spin,and
conservationlaws),resultinginafairlycompletecoverageofelementarytopics.
A second main goal was to address the key physical issues of stability of
atomsandmolecules,andmean-fieldapproximationsoflargeparticlesystems.
Thisisreflectedinnewchapterscoveringtheexistenceofatomsandmolecules,
mean-field theory, and second quantization.
Our final goal was to update the advanced material with a view toward
reflecting current developments,and this led to a complete revisionand reor-
ganizationofthematerialonthetheoryofradiation(non-relativisticquantum
electrodynamics), as well as the addition of a new chapter.
Inthiseditionwehavealsoaddedanumberofproofs,whichwereomitted
in the previous editions. As a result, this book could be used for senior level
undergraduate,as wellas graduate,coursesin both mathematics andphysics
departments.
Prerequisites for this book are introductory real analysis (notions of vec-
tor space, scalar product, norm, convergence, Fourier transform) and com-
plex analysis, the theory of Lebesgue integration, and elementary differential
equations.Thesetopicsaretypicallycoveredbythethirdyearinmathematics
departments.Thefirstandthirdtopicsarealsofamiliartophysicsundergrad-
uates. However, even in dealing with mathematics students we have found it
useful, if not necessary, to review these notions, as needed for the course.
Hence, to make the book relatively self-contained,we briefly coverthese sub-
jects, with the exception of Lebesgue integration. Those unfamiliar with the
latter can think about Lebesgue integrals as if they were Riemann integrals.
This said,the pace ofthe book is nota leisurely oneandrequires,atleastfor
beginners, some amount of work.
Though, as in the previous two issues of the book, we tried to increase
the complexity of the material gradually, we were not always successful, and
VI Preface
first in Chapter 12, and then in Chapter 18, and especially in Chapter 19,
there is a leap in the level of sophistication required from the reader. One
maysaythebookproceedsatthreelevels.Thefirstone,coveringChapters1-
11,is elementary and could be used for senior levelundergraduate,as well as
graduate, courses in both physics and mathematics departments; the second
one, covering Chapters 12 - 17, is intermediate; and the last one, covering
Chapters 18 - 22, advanced.
During the last few years since the enlarged second printing of this book,
there have appeared four books on Quantum Mechanics directed at mathe-
maticians:
F. Strocchi, An Introduction to the Mathematical Structure of Quantum Me-
chanics: a Short Course for Mathematicians. World Scientific, 2005.
L. Takhtajan, Quantum Mechanics for Mathematicians. AMS, 2008.
L.D. Faddeev, O.A. Yakubovskii, Lectures on Quantum Mechanics for Math-
ematics Students. With an appendix by Leon Takhtajan. AMS, 2009.
J.Dimock,QuantumMechanicsandQuantumFieldTheory.CambridgeUniv.
Press, 2011.
These elegant and valuable texts have considerably different aims and rather
limited overlap with the present book. In fact, they complement it nicely.
Acknowledgment:TheauthorsaregratefultoI.Anapolitanos,Th.Chen,J.
Faupin,Z.Gang,G.-M.Graf,M.Griesemer,L.Jonsson,M.Merkli,M.Mu¨ck,
Yu.Ovchinnikov,A.Soffer,F.Ting,T.Tzaneteas,andespeciallyJ.Fr¨ohlich,
W. Hunziker andV. Buslaevfor useful discussions,andto J. Feldman,G.-M.
Graf, I. Herbst, L. Jonsson, E. Lieb, B. Simon and F. Ting for reading parts
of the manuscript and making useful remarks.
Vancouver/Toronto, Stephen Gustafson
May 2011 Israel Michael Sigal
Preface to the enlarged second printing
Forthesecondprinting,wecorrectedafewmisprintsandinaccuracies;for
some help with this, we are indebted to B. Nachtergaele.We have also added
a small amount of new material. In particular, Chapter 11, on perturbation
theory via the Feshbach method, is new, as are the short sub-sections 13.1
and 13.2 concerning the Hartree approximationand Bose-Einstein condensa-
tion. We also note a change in terminology, from “point” and “continuous”
spectrum, to the mathematically more standard “discrete” and “essential”
spectrum, starting in Chapter 6.
Vancouver/Toronto, Stephen Gustafson
July 2005 Israel Michael Sigal
Preface VII
From the preface to the first edition
The first fifteen chapters of these lectures (omitting four to six chapters
each year) cover a one term course taken by a mixed group of senior under-
graduate and junior graduate students specializing either in mathematics or
physics. Typically, the mathematics students have some background in ad-
vanced analysis, while the physics students have had introductory quantum
mechanics.Tosatisfysuchadisparateaudience,wedecidedtoselectmaterial
which is interesting from the viewpoint of modern theoretical physics, and
whichillustratesaninterplayofideasfromvariousfieldsofmathematicssuch
as operator theory, probability, differential equations, and differential geome-
try. Given our time constraint, we have often pursued mathematical content
at the expense of rigor. However, wherever we have sacrificed the latter, we
have tried to explain whether the result is an established fact, or, mathemat-
ically speaking, a conjecture, and in the former case, how a given argument
can be made rigorous. The present book retains these features.
Vancouver/Toronto, Stephen Gustafson
Sept. 2002 Israel Michael Sigal
•
Contents
1 Physical Background ...................................... 1
1.1 The Double-Slit Experiment ............................. 2
1.2 Wave Functions ........................................ 4
1.3 State Space............................................ 5
1.4 The Schr¨odinger Equation............................... 5
2 Dynamics ................................................. 9
2.1 Conservation of Probability.............................. 9
2.2 Self-adjointness ........................................ 10
2.3 Existence of Dynamics .................................. 14
2.4 The Free Propagator.................................... 16
3 Observables ............................................... 19
3.1 Mean Values and the Momentum Operator ................ 19
3.2 Observables ........................................... 20
3.3 The Heisenberg Representation .......................... 21
3.4 Spin .................................................. 22
3.5 Conservation Laws ..................................... 23
3.5.1 Probability current ............................. 24
4 Quantization .............................................. 27
4.1 Quantization .......................................... 27
4.2 Quantization and Correspondence Principle................ 29
4.3 Complex Quantum Systems ............................. 32
4.4 Supplement: Hamiltonian Formulation of Classical
Mechanics............................................. 36
5 Uncertainty Principle and Stability of Atoms and
Molecules ................................................. 41
5.1 The Heisenberg Uncertainty Principle..................... 41
5.2 A Refined Uncertainty Principle.......................... 42
5.3 Application: Stability of Atoms and Molecules ............. 43