Table Of ContentUndergraduate Texts in Mathematics
Editors
S. Axler
K.A. Ribet
UndergraduateTextsinMathematics
Abbott:UnderstandingAnalysis. Chambert-Loir:AFieldGuidetoAlgebra.
Anglin:Mathematics:AConciseHistory Childs:AConcreteIntroductionto
andPhilosophy. HigherAlgebra.Secondedition.
ReadingsinMathematics. Chung/AitSahlia:ElementaryProbability
Anglin/Lambek:TheHeritageof Theory:WithStochasticProcessesand
Thales. anIntroductiontoMathematical
ReadingsinMathematics. Finance.Fourthedition.
Apostol:IntroductiontoAnalytic Cox/Little/O’Shea:Ideals,Varieties,
NumberTheory.Secondedition. andAlgorithms.Secondedition.
Armstrong:BasicTopology. Croom:BasicConceptsofAlgebraic
Armstrong:GroupsandSymmetry. Topology.
Axler:LinearAlgebraDoneRight. Curtis:LinearAlgebra:AnIntroductory
Secondedition. Approach.Fourthedition.
Beardon:Limits:ANewApproachto Daepp/Gorkin:Reading,Writing,and
RealAnalysis. Proving:ACloserLookat
Bak/Newman:ComplexAnalysis. Mathematics.
Secondedition. Devlin:TheJoyofSets:Fundamentals
Banchoff/Wermer:LinearAlgebra ofContemporarySetTheory.Second
ThroughGeometry.Secondedition. edition.
Berberian:AFirstCourseinReal Dixmier:GeneralTopology.
Analysis. Driver:WhyMath?
Bix:ConicsandCubics:A Ebbinghaus/Flum/Thomas:
ConcreteIntroductiontoAlgebraic MathematicalLogic.Secondedition.
Curves. Edgar:Measure,Topology,andFractal
Bre´maud:AnIntroductionto Geometry.
ProbabilisticModeling. Elaydi:AnIntroductiontoDifference
Bressoud:FactorizationandPrimality Equations.Thirdedition.
Testing. Erdo˜s/Sura´nyi:TopicsintheTheoryof
Bressoud:SecondYearCalculus. Numbers.
ReadingsinMathematics. Estep:PracticalAnalysisinOneVariable.
Brickman:MathematicalIntroduction Exner:AnAccompanimenttoHigher
toLinearProgrammingandGame Mathematics.
Theory. Exner:InsideCalculus.
Browder:MathematicalAnalysis: Fine/Rosenberger:TheFundamental
AnIntroduction. TheoryofAlgebra.
Buchmann:Introductionto Fischer:IntermediateRealAnalysis.
Cryptography. Flanigan/Kazdan:CalculusTwo:Linear
Buskes/vanRooij:TopologicalSpaces: andNonlinearFunctions.Second
FromDistancetoNeighborhood. edition.
Callahan:TheGeometryofSpacetime: Fleming:FunctionsofSeveralVariables.
AnIntroductiontoSpecialandGeneral Secondedition.
Relavitity. Foulds:CombinatorialOptimizationfor
Carter/vanBrunt:TheLebesgue– Undergraduates.
StieltjesIntegral:APractical Foulds:OptimizationTechniques:An
Introduction. Introduction.
Cederberg:ACourseinModern Franklin:MethodsofMathematical
Geometries.Secondedition. Economics.
(continuedafterindex)
Stephanie Frank Singer
Linearity, Symmetry,
and Prediction in
the Hydrogen Atom
StephanieFrankSinger
Philadelphia,PA19103
U.S.A.
[email protected]
EditorialBoard
S.Axler K.A.Ribet
CollegeofScienceandEngineering DepartmentofMathematics
SanFranciscoStateUniversity UniversityofCaliforniaatBerkeley
SanFrancisco,CA94132 Berkeley,CA94720-3840
U.S.A. U.S.A.
MathematicsSubjectClassification(2000):Primary–81-01,81R05,20-01,20C35,
22-01,22E70,22C05,81Q99;Secondary–15A90,20G05,20G45
LibraryofCongressCataloging-in-PublicationData
Singer,StephanieFrank,1964–
Linearity,symmetry,andpredictioninthehydrogenatom/StephanieFrank
Singer.
p.cm.—(Undergraduatetextsinmathematics)
Includesbibliographicalreferencesandindex.
ISBN0-387-24637-1(alk.paper)
1.Grouptheory.2.Hydrogen.3.Atoms.4.Linearalgebraicgroups.5.Symmetry
(Physics)6.Representationsofgroups.7.Quantumtheory.I.Title.II.Series.
QC20.7.G76S56 2005
530.15′22—dc22 2005042679
ISBN-100-387-24637-1 e-ISBN0-387-26369-1 Printedonacid-freepaper.
ISBN-13978-0387-24637-6
©2005StephanieFrankSinger
All rights reserved. This work may not be translated or copied in whole or in part
without the written permission of the publisher (Springer Science+Business Media,
Inc.,233SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnec-
tionwithreviewsorscholarlyanalysis.Useinconnectionwithanyformofinforma-
tionstorageandretrieval,electronicadaptation,computersoftware,orbysimilaror
dissimilarmethodologynowknownorhereafterdevelopedisforbidden.
The use in this publication of trade names, trademarks, service marks, and similar
terms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionof
opinionastowhetherornottheyaresubjecttoproprietaryrights.
PrintedintheUnitedStatesofAmerica. (TXQ/EB)
9 8 7 6 5 4 3 2 1 SPIN10940815
springeronline.com
Tomymother,MaxineFrankSinger,
whoalwaysencouragedmetofollowmyowninstincts:
IthinkImaybereadytolearnsomechemistrynow.
Contents
Preface xi
1 SettingtheStage 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 FundamentalAssumptionsofQuantumMechanics . . . . . 2
1.3 TheHydrogenAtom . . . . . . . . . . . . . . . . . . . . . 8
1.4 ThePeriodicTable . . . . . . . . . . . . . . . . . . . . . . 13
1.5 PreliminaryMathematics . . . . . . . . . . . . . . . . . . . 17
1.6 SphericalHarmonics . . . . . . . . . . . . . . . . . . . . . 27
1.7 EquivalenceClasses . . . . . . . . . . . . . . . . . . . . . . 33
1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2 LinearAlgebraovertheComplexNumbers 41
2.1 ComplexVectorSpaces . . . . . . . . . . . . . . . . . . . . 42
2.2 Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3 LinearTransformations . . . . . . . . . . . . . . . . . . . . 48
2.4 KernelsandImagesofLinearTransformations. . . . . . . . 51
2.5 LinearOperators . . . . . . . . . . . . . . . . . . . . . . . 55
2.6 CartesianSumsandTensorProducts . . . . . . . . . . . . . 62
2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
viii Contents
3 ComplexScalarProductSpaces(a.k.a.HilbertSpaces) 77
3.1 LebesgueEquivalenceand L2(R3) . . . . . . . . . . . . . . 78
3.2 ComplexScalarProducts . . . . . . . . . . . . . . . . . . . 81
3.3 Euclidean-styleGeometryinComplexScalar
ProductSpaces . . . . . . . . . . . . . . . . . . . . . . . . 85
3.4 NormsandApproximations . . . . . . . . . . . . . . . . . . 94
3.5 UsefulSpanningSubspaces . . . . . . . . . . . . . . . . . . 99
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4 LieGroupsandLieGroupRepresentations 111
4.1 GroupsandLieGroups . . . . . . . . . . . . . . . . . . . . 112
4.2 TheKeyPlayers:SO(3),SU(2)andSO(4) . . . . . . . . . . 117
4.3 TheSpectralTheoremforSU(2)andthe
DoubleCoverofSO(3) . . . . . . . . . . . . . . . . . . . . 120
4.4 Representations:DefinitionandExamples . . . . . . . . . . 127
4.5 RepresentationsinQuantumMechanics . . . . . . . . . . . 133
4.6 HomogeneousPolynomialsinTwoVariables . . . . . . . . 137
4.7 CharactersofRepresentations . . . . . . . . . . . . . . . . 141
4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5 NewRepresentationsfromOld 153
5.1 Subrepresentations . . . . . . . . . . . . . . . . . . . . . . 153
5.2 CartesianSumsofRepresentations . . . . . . . . . . . . . . 158
5.3 TensorProductsofRepresentations. . . . . . . . . . . . . . 160
5.4 DualRepresentations . . . . . . . . . . . . . . . . . . . . . 164
5.5 TheRepresentationHom . . . . . . . . . . . . . . . . . . . 168
5.6 PullbackandPushforwardRepresentations. . . . . . . . . . 172
5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6 IrreducibleRepresentationsandInvariantIntegration 179
6.1 DefinitionsandSchur’sLemma. . . . . . . . . . . . . . . . 180
6.2 ElementaryStatesofQuantumMechanicalSystems . . . . . 185
6.3 InvariantIntegrationandCharacters
ofIrreducibleRepresentations . . . . . . . . . . . . . . . . 187
6.4 IsotypicDecompositions(Optional) . . . . . . . . . . . . . 193
6.5 ClassificationoftheIrreducibleRepresentationsof SU(2) . 199
6.6 ClassificationoftheIrreducibleRepresentationsofSO(3) . . 202
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Contents ix
7 RepresentationsandtheHydrogenAtom 209
7.1 HomogeneousHarmonicPolynomialsofThreeVariables . . 209
7.2 SphericalHarmonics . . . . . . . . . . . . . . . . . . . . . 213
7.3 TheHydrogenAtom . . . . . . . . . . . . . . . . . . . . . 219
7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
8 TheAlgebraso(4)SymmetryoftheHydrogenAtom 229
8.1 LieAlgebras. . . . . . . . . . . . . . . . . . . . . . . . . . 230
8.2 RepresentationsofLieAlgebras . . . . . . . . . . . . . . . 241
8.3 RaisingOperators,LoweringOperatorsand
IrreducibleRepresentationsofsu(2) . . . . . . . . . . . . . 246
8.4 TheCasimirOperatorand
IrreducibleRepresentationsofso(4) . . . . . . . . . . . . . 255
8.5 BoundStatesoftheHydrogenAtom . . . . . . . . . . . . . 262
8.6 TheHydrogenRepresentationsofso(4) . . . . . . . . . . . 267
8.7 TheHeinousDetails . . . . . . . . . . . . . . . . . . . . . 271
8.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
9 TheGroupSO(4)SymmetryoftheHydrogenAtom 283
9.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 284
9.2 Fock’sOriginalArticle . . . . . . . . . . . . . . . . . . . . 286
9.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
10 ProjectiveRepresentationsandSpin 299
10.1 ComplexProjectiveSpace . . . . . . . . . . . . . . . . . . 299
10.2 TheQubit . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
10.3 ProjectiveHilbertSpaces . . . . . . . . . . . . . . . . . . . 311
10.4 ProjectiveUnitaryIrreducibleRepresentationsandSpin . . . 318
10.5 PhysicalSymmetries . . . . . . . . . . . . . . . . . . . . . 323
10.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 335
11 IndependentEventsandTensorProducts 339
11.1 IndependentMeasurements . . . . . . . . . . . . . . . . . . 340
11.2 PartialMeasurement . . . . . . . . . . . . . . . . . . . . . 342
11.3 EntanglementandQuantumComputing . . . . . . . . . . . 346
11.4 TheStateSpaceofaMobileSpin-1/2Particle . . . . . . . . 354
11.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 356
11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
x Contents
A SphericalHarmonics 359
B ProofoftheCorrespondencebetweenIrreducible
LinearRepresentationsofSU(2)and
IrreducibleProjectiveRepresentationsofSO(3) 369
C SuggestedPaperTopics 377
Bibliography 379
GlossaryofSymbolsandNotation 385
Index 391
Preface
Itjustmeanssomuchmoretosomuchmorepeoplewhenyou’rerappin’and
youknowwhatfor.
—Eminem,“Business”[Mat]
This is a textbook for a senior-level undergraduate course for math, physics
and chemistry majors. This one course can play two different but comple-
mentary roles: it can serve as a capstone course for students finishing their
education, and it can serve as motivating story for future study of mathe-
matics.
Sometextbooksarelikeavigorousregularphysicaltrainingprogram,pre-
paringpeopleforawiderangeofchallengesbyhoningtheirbasicskillsthor-
oughly. Some are like a series of day hikes. This book is more like an ex-
tendedtrektoaparticularlybeautifulgoal.We’lltaketheeasiestroutetothe
top,andwe’llstoptoappreciatelocalfloraaswellasdistantpeaksworthyof
thevigoroustrainingonewouldneedtoscalethem.
Advice to the Student
This book was written with many different readers in mind. Some will be
mathematicsstudentsinterestedtoseeabeautifulandpowerfulapplicationof
a “pure” mathematical subject. Some will be students of physics and chem-
istry curious about the mathematics behind some tools they use, such as
