Quantum Mechanics: Genesis and Achievements

293 Pages·2013·2.299 MB·English

by  Alexander Komech (auth.)

Quantum Mechanics: Genesis and Achievements
Alexander Komech
Quantum Mechanics:
Genesis and
Achievements
AlexanderKomech
FacultyofMathematics
UniversityofVienna
Vienna,Austria
Resume Theintentofthisbookisaconciseintroductiontothefieldofnonrelativisticand
relativisticquantummechanics.AhistoricalaccountoftheHeisenbergandtheSchrödinger
theoriesisgiven,andtraditionalapplicationstothehydrogenatomareoutlined.Alldetails
ofcalculationsaregivenandsupplementedwithrelatedmaterialsfromtheMaxwellElectro-
dynamicsandSpecialRelativity.
Thebookcanbeusedasasourceforatwo-semesterlecturecourseonNonrelativisticQuan-
tumMechanics,aswellasforaone-semestercourseoneithertheRelativisticQuantumMe-
chanics,ClassicalElectrodynamics,ortheClassicalFieldTheory.
ISBN978-94-007-5541-3 ISBN978-94-007-5542-0(eBook)
DOI10.1007/978-94-007-5542-0
SpringerDordrechtHeidelbergNewYorkLondon
LibraryofCongressControlNumber:2012951292
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To thememoryof
Vera andAlexeyKomech,
EstherBraginskaya, andMarkVishik
Preface
This book is based on a three-semester course of lectures delivered by the author
attheFacultyofMathematicsoftheViennaUniversity.Forpreliminaryversionsof
themanuscript,see[113,114].
The main emphasis of this book is on nonrelativistic and relativistic quantum
mechanics with standard applications to the hydrogen atom. Our main intention
is to present the quantum mechanics in a comprehensive manner, accessible for
amathematician.Theexpositionisformalized(wheneverpossible)onthebasisof
coupledMaxwell–SchrödingerandMaxwell–Diracequations.Thisintentionagrees
with Hilbert’s 6th Problem (“Axiomatize Theoretical Physics”), and Heisenberg’s
nonlinearprogramme[83,84].
Our exposition starts with a chronological analysis of crucial empirical obser-
vations and their theoretical systematization, explaining inter alia the motivation
behindtheHeisenbergandSchrödingerequations.Theintroductionofquantumob-
servables stems from the agreement with corresponding classical observables for
short wavelength solutions, Hamilton–Jacobi’s theorem being taken into account.
Also, the relation between the quantum observables and the Noether symmetry
theory is discussed. The Lagrangian formalism is used as a fundamental unifying
principletolaythebasisforintroductionofthecoupledMaxwell–Schrödingerand
Maxwell–Diracequations.
Ofcourse,theequationsandobservablescouldbe(andformallyshouldbe)ac-
cepted as axioms. On the other hand, it is crucially important to know the experi-
mentalandmathematicalfactsbehindquantumformalismtoembeditinthewhole
ofphysics.
Moreover, the modern form of quantum theory seems to be far from its com-
pleteness,likegeometryinpre-Euclideanera.Itisthereforeparticularlyimportant
to understand the degree of confidence to individual constituents of the quantum
formalism.Thisiswhywepaysomuchattentiontotheoriginandmotivationofthe
formalism.
The hydrogen spectrum and the atom radiation are calculated with all detail.
Parallelsbetweenquantumandclassicaldescriptionaretracedeverywheretomoti-
vate the introductionof quantum differential cross section,magneticmoment,etc.
vii
viii Preface
The scattering problems are solved by application of the perturbation procedure
to the coupled Maxwell–Schrödinger equations. We point out some deficiency in
theperturbationprocedure,whichshouldbefixedwithanonperturbativeapproach,
howeverthiscorrectionisstillanopenproblem.
The introduction of the electron spin is discussed in detail from experimental
and theoretical point of view. We calculate the Landé formula for the gyromag-
neticratioviathespin-orbitalinteractionofRussell–Saunders,whichexplainsthe
Einstein–deHaasexperimentandtheanomalousZeemaneffect.Furtherweprove
therelativisticcovarianceoftheDiracequation,obtainthecorrespondingintrinsic
spinmomentum,andthecorrespondingnonrelativisticapproximations.Finally,we
calculatethehydrogenspectrumviatheDiracequation.
Wemakeexplicitinvokedassumptionsandapproximations,anddiscussaplausi-
bletreatmentofsomelogicalleapsinthetheory.However,wedidnottrytoestablish
newrigorousresults.Generally,ourexpositionisnotmathematicallyrigorous.For
example, we do not distinguish between Hermitian symmetric and the selfadjoint
operators,eventhoughthespectralresolutionisusedrepeatedly.
Inappendices(Chaps.12and13),weexplainrelateddetailsfromClassicalElec-
trodynamicsandSpecialRelativity,GeometricalOptics,theHamilton–Jacobitheo-
rem,anupdatedversionoftheNoethertheoryofcurrents,andthelimitingamplitude
principle.
InChap.14,wecollectclassicalcalculationslyinginthebaseofthe‘oldquan-
tummechanics’.
FurtherReading Ourmaingoalistogiveaconciseexplanationofmathematical
principles of Quantum Mechanics. More technical details and a systematic com-
parison with experimental data can be found in [7, 11, 12, 20, 23, 34, 63, 75, 81,
130, 131, 145, 160, 171, 179, 191]. The books [46, 79, 89, 143] and [93, 185],
respectively,explainbasicconceptsofQuantumMechanicsandClassicalElectro-
dynamics. A suitable introduction to the mathematical theory of the Schrödinger
equationiscontainedin[10].
Wedevelopthemethodsofquantummechanicsforthehydrogenatomwhichcan
be extended to other one-electron atoms (lithium, sodium, potassium atoms, etc),
anddonottouchmulti-electronproblemsofquantumchemistry[34,41,178,187].
WealsodonottouchtheStabilityofMatter[31,134],theQuantumElectrodynam-
icsandQuantumFieldTheory[13,14,33,64,77,85,137,138,158,159,163,189,
195,196].
Acknowledgements Over a period of years the author has been pleased to have
manystimulatingdiscussionswithA.ShnirelmanandH.Spohn.Theauthorisgrate-
fultoE.Kopylovaforreadingandcheckingthemanuscript,andtoC.Adamforhis
helpinwritingpreliminaryversionsoftheExercises(Sects.14.1–14.6).Theauthor
is indebted to the Faculty of Mathematics of Vienna University, and the Institute
fortheInformationTransmissionProblemsoftheRussianAcademyofSciencesfor
providingcongenialfacilitiesforthework.
Preface ix
The author is also grateful to the Max Planck Institute for Mathematical Sci-
ences(Leipzig),theMünchenTechnicalUniversity,andtheTeamONDES/POEMS
(INRIA,Rocquencourt)fortheirhospitality.
TheworkwassupportedinpartbytheDepartmentofMechanicsandMathemat-
icsoftheMoscowStateUniversity,bytheAlexandervonHumboldtFoundation,by
the Austrian Science Fund (FWF) (projects Nos. P19138-N13 and P22198-N13),
andbygrantsfromtheDeutscheForschungsgemeinschaftandtheRussianFundfor
BasicResearch.
Leipzig–Moscow–München–Rocquencourt–Vienna A.Komech
Contents
1 GenesisofQuantumMechanics . . . . . . . . . . . . . . . . . . . . . 1
1.1 AtomsandSpectra . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 ConceptofAtom. . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 EarlySpectraObservations . . . . . . . . . . . . . . . . . 2
1.1.3 ResonanceNatureofSpectra . . . . . . . . . . . . . . . . 2
1.1.4 CombinationPrinciple . . . . . . . . . . . . . . . . . . . . 3
1.2 Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 AmpèreTheoryofMagnetization . . . . . . . . . . . . . . 4
1.2.2 MaxwellElectrodynamics . . . . . . . . . . . . . . . . . . 4
1.2.3 CathodeRays:Thomson’sDiscoveryofElectron . . . . . . 5
1.2.4 ElementaryElectricCharge . . . . . . . . . . . . . . . . . 6
1.2.5 TheZeemanEffect. . . . . . . . . . . . . . . . . . . . . . 7
1.2.6 Lorentz’sTheoryofElectrons . . . . . . . . . . . . . . . . 7
1.2.7 Abraham:Mass–EnergyIdentification . . . . . . . . . . . 8
1.3 Thermodynamics. . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 EquilibriumRadiation . . . . . . . . . . . . . . . . . . . . 10
1.3.2 ExperimentsandTheory . . . . . . . . . . . . . . . . . . . 11
1.3.3 TheRayleigh–JeansLaw:UltravioletDivergence . . . . . 13
1.3.4 Planck’sLaw:QuantizationofEnergy . . . . . . . . . . . 15
1.3.5 PhotoelectricEffect:Einstein’sRules . . . . . . . . . . . . 17
1.3.6 EinsteinandDebye:SpecificHeatofSolids . . . . . . . . 17
1.4 ‘OldQuantumMechanics’. . . . . . . . . . . . . . . . . . . . . . 18
1.4.1 PlanetaryModelandInconsistencyofClassicalPhysics . . 18
1.4.2 Bohr’sPostulates . . . . . . . . . . . . . . . . . . . . . . 19
1.4.3 DebyeQuantumRules . . . . . . . . . . . . . . . . . . . . 19
1.4.4 BoltzmannDistributioninOldQuantumMechanics . . . . 20
1.4.5 EinsteinTheoryofRadiation:CountingPhotons . . . . . . 21
1.4.6 Bohr’sCorrespondencePrinciple . . . . . . . . . . . . . . 23
xi
xii Contents
2 Heisenberg’sMatrixMechanics . . . . . . . . . . . . . . . . . . . . . 25
2.1 Heisenberg’sMatrixFormalism . . . . . . . . . . . . . . . . . . . 25
2.1.1 ClassicalOscillator . . . . . . . . . . . . . . . . . . . . . 25
2.1.2 QuantumOscillator . . . . . . . . . . . . . . . . . . . . . 27
2.2 EarlyApplicationsofHeisenbergTheory . . . . . . . . . . . . . . 30
2.2.1 EigenvalueProblem . . . . . . . . . . . . . . . . . . . . . 30
2.2.2 IntensityofSpectralLines . . . . . . . . . . . . . . . . . . 32
2.2.3 TheNormalZeemanEffect . . . . . . . . . . . . . . . . . 32
2.2.4 QuantizationofMaxwellFieldandPlanck’sLaw . . . . . 33
3 Schrödinger’sWaveMechanics . . . . . . . . . . . . . . . . . . . . . 35
3.1 Wave-ParticleDuality:deBroglieandSchrödinger . . . . . . . . . 35
3.1.1 DeBroglieWavesofMatter . . . . . . . . . . . . . . . . . 35
3.1.2 DeBroglieWavelengthandDispersionRelations. . . . . . 37
3.1.3 CanonicalQuantizationI:FreeParticles . . . . . . . . . . 39
3.1.4 CanonicalQuantizationII:BoundParticles . . . . . . . . . 39
3.1.5 QuantumStationaryStates:EigenvalueProblem . . . . . . 41
3.1.6 StationaryPerturbationTheory . . . . . . . . . . . . . . . 42
3.2 QuasiclassicalAsymptotics . . . . . . . . . . . . . . . . . . . . . 43
3.2.1 GeometricalOptics . . . . . . . . . . . . . . . . . . . . . 43
3.2.2 ApplicationtoSchrödingerEquation . . . . . . . . . . . . 44
3.2.3 Generalizations . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.4 HamiltonOptical-MechanicalAnalogy . . . . . . . . . . . 45
3.2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 QuantumObservablesandConservationLaws . . . . . . . . . . . 46
3.3.1 QuantumObservables . . . . . . . . . . . . . . . . . . . . 47
3.3.2 ConservationLaws . . . . . . . . . . . . . . . . . . . . . 47
3.3.3 CorrespondencePrinciple . . . . . . . . . . . . . . . . . . 49
3.4 ChargeContinuityEquation . . . . . . . . . . . . . . . . . . . . . 50
3.5 EquivalenceofHeisenberg’sandSchrödinger’sTheories . . . . . . 52
3.5.1 HeisenbergObservables . . . . . . . . . . . . . . . . . . . 52
3.5.2 Heisenberg’sPicture . . . . . . . . . . . . . . . . . . . . . 53
3.5.3 Heisenberg’sEquation . . . . . . . . . . . . . . . . . . . . 54
3.5.4 CorrespondenceBetweenHeisenberg’sandSchrödinger’s
Pictures . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4 LagrangianFormalism . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 HamiltonianandLagrangianFormalism . . . . . . . . . . . . . . 57
4.1.1 HamiltonianFormalism . . . . . . . . . . . . . . . . . . . 57
4.1.2 LagrangianFormalism . . . . . . . . . . . . . . . . . . . . 58
4.1.3 VariationalPrincipleandEuler–LagrangeEquations . . . . 58
4.2 Maxwell–SchrödingerEquations . . . . . . . . . . . . . . . . . . 60
4.2.1 LagrangianDensity . . . . . . . . . . . . . . . . . . . . . 60
4.2.2 GaugeInvarianceandChargeContinuity . . . . . . . . . . 62
4.2.3 GaugeTransformation . . . . . . . . . . . . . . . . . . . . 62
4.2.4 PerturbationTheory . . . . . . . . . . . . . . . . . . . . . 63
4.3 Klein–GordonEquation . . . . . . . . . . . . . . . . . . . . . . . 64
Contents xiii
5 Wave-ParticleDuality . . . . . . . . . . . . . . . . . . . . . . . . . . 67
5.1 ElectronBeamandUncertaintyPrinciple . . . . . . . . . . . . . . 67
5.1.1 PlaneWaveasElectronBeam . . . . . . . . . . . . . . . . 67
5.1.2 TheHeisenbergUncertaintyPrinciple. . . . . . . . . . . . 68
5.2 DiffractionofElectronBeams . . . . . . . . . . . . . . . . . . . . 69
5.2.1 ExperimentalObservations . . . . . . . . . . . . . . . . . 69
5.2.2 TheDavisson–GermerExperiment . . . . . . . . . . . . . 70
5.2.3 TheDouble-SlitExperiment . . . . . . . . . . . . . . . . . 71
5.2.4 TheAharonov–BohmEffect. . . . . . . . . . . . . . . . . 71
5.2.5 DiffractionofElectronsviaSchrödingerTheory . . . . . . 72
5.3 ProbabilisticInterpretation. . . . . . . . . . . . . . . . . . . . . . 74
5.3.1 ReductionofWavePacketsandProbabilistic
Interpretation. . . . . . . . . . . . . . . . . . . . . . . . . 74
5.3.2 Soliton-TypeAsymptotics . . . . . . . . . . . . . . . . . . 74
5.3.3 SolitonsandReductionofWavePackets . . . . . . . . . . 75
5.3.4 TheAharonov–BohmParadox . . . . . . . . . . . . . . . 76
5.3.5 TheKnownResultsonSolitonAsymptotics . . . . . . . . 76
5.3.6 Particle-LikeBehaviorofSolitons . . . . . . . . . . . . . 77
6 TheEigenvalueProblem . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.1 TheHydrogenSpectrum . . . . . . . . . . . . . . . . . . . . . . . 79
6.1.1 TheEigenvalueProblem . . . . . . . . . . . . . . . . . . . 79
6.1.2 SphericalSymmetryandSeparationofVariables . . . . . . 80
6.1.3 SphericalCoordinates . . . . . . . . . . . . . . . . . . . . 82
6.1.4 TheRadialEquation . . . . . . . . . . . . . . . . . . . . . 83
6.1.5 EigenfunctionsandQuantumNumbers . . . . . . . . . . . 85
6.2 TheSphericalSpectralProblem . . . . . . . . . . . . . . . . . . . 86
6.2.1 Hilbert–SchmidtArgument . . . . . . . . . . . . . . . . . 86
6.2.2 LieAlgebraofAngularMomenta . . . . . . . . . . . . . . 87
6.2.3 IrreducibleRepresentations . . . . . . . . . . . . . . . . . 87
6.2.4 SphericalHarmonics. . . . . . . . . . . . . . . . . . . . . 89
6.2.5 AngularMomentainSphericalCoordinates . . . . . . . . 91
7 AtomRadiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.1 AtominThermodynamicalEquilibrium. . . . . . . . . . . . . . . 93
7.1.1 RelaxationtoEquilibriumDistribution . . . . . . . . . . . 93
7.1.2 PerturbationTheory . . . . . . . . . . . . . . . . . . . . . 94
7.1.3 TheDirac‘InteractionPicture’ . . . . . . . . . . . . . . . 95
7.1.4 ThermodynamicalEquilibriuminSchrödingerTheory . . . 96
7.2 AtomRadiation . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.2.1 RadiatedMaxwellField . . . . . . . . . . . . . . . . . . . 97
7.2.2 TheRydberg–RitzCombinationPrinciple . . . . . . . . . 98
7.2.3 TheDipoleApproximation . . . . . . . . . . . . . . . . . 99
7.2.4 EnergyFlowatInfinity . . . . . . . . . . . . . . . . . . . 101
7.2.5 IntensityofSpectralLines . . . . . . . . . . . . . . . . . . 101
7.2.6 TheCorrespondencePrinciple. . . . . . . . . . . . . . . . 103