Lectures on Quantum Mechanics (nonlinear PDE point

arXiv:math-ph/0505059v428May20051LecturesonQuantumMechanics(nonlinearPDEpointofview)A.I.KomechWien2002/2004AbstractWeexposetheSchr¨odingerquantummechanicswithtraditionalapplicationstoHydrogenatom:thecalculationoftheHydrogenatomspectrumviaSchr¨odinger,PauliandDiracequations,theHeisen-bergrepresentation,theselectionrules,thecalculationofquantumandclassicalscatteringoflight(Thomsoncrosssection),photoeffect(Sommerfeldcrosssection),quantumandclassicalscatteringofelectrons(Rutherfordcrosssection),normalandanomalousZeemanneffect(Land´efactor),polariza-tionanddispersion(Kramers-Kronigformula),diamagneticsusceptibility(Langevinformula).WediscusscarefullytheexperimentalandtheoreticalbackgroundfortheintroductionoftheSchr¨odinger,PauliandDiracequations,aswellasfortheMaxwellequations.Weexplainindetailallbasictheoreticalconcepts:theintroductionofthequantumstationarystates,chargedensityandelectriccurrentdensity,quantummagneticmoment,electronspinandspin-orbitalcouplingin“vectormodel”andintheRussel-Saundersapproximation,differentialcrosssectionofscattering,theLorentztheoryofpolarizationandmagnetization,theEinsteinspecialrelativityandcovarianceoftheMaxwellElectrodynamics.Weexplainalldetailsofthecalculationsandmathematicaltools:LagrangianandHamiltonianformalismforthesystemswithfinitedegreeoffreedomandforfields,GeometricOptics,theHamilton-JacobiequationandWKBapproximation,Noethertheoryofinvariantsincludingthetheoremoncurrents,fourconservationlaws(energy,momentum,angularmomentumandcharge),Liealgebraofangularmomentumandsphericalfunctions,scatteringtheory(limitingamplitudeprincipleandlimitingabsorptionprinciple),theLienard-Wiechertformulas,LorentzgroupandLorentzformulas,PaulitheoremandrelativisticcovarianceoftheDiracequation,etc.Wegiveadetailedoveviewoftheconceptualdevelopmentofthequantummechanics,andexposemainachievementsofthe“oldquantummechanics”intheformofexercises.Oneofourbasicaiminwritingthisbook,isanopenandconcretediscussionoftheproblemofamathematicaldescriptionofthefollowingtwofundamentalquantumphenomena:i)Bohr’squantumtransitionsandii)deBroglie’swave-particleduality.Bothphenomenacannotbedescribedbyau-tonomouslineardynamicalequations,andwegivethemanewmathematicaltreatmentrelatedwithrecentprogressinthetheoryofglobalattractorsofnonlinearhyperbolicPDEs.Namely,wesuggestthati)thequantumstationarystatesformaglobalattractorofthecoupledMaxwell-Schr¨odingerorMaxwell-Diracequations,inthepresenceofanexternalconfiningpotential,andii)thewave-particledualitycorrespondstothesoliton-likeasymptoticsforthesolutionsofthetranslation-invariantcoupledequationswithoutanexternalpotential.Weemphasize,inthewholeofourexposition,thatthecoupledequationsarenonlinear,andjustthisnonlinearityliesbehindalltraditionalperturbativecalculationsthatisknownastheBornapproximation.Wesuggestthatbothfundamentalquantumphenomenacouldbedescribedbythisnonlinearcoupling.ThesuggestionisconfirmedbyrecentresultsontheglobalattractorsandsolitonasymptoticsformodelnonlinearhyperbolicPDEs.2Contents0Preface............................................81Introduction:QuantumChronology1859-1927......................151.1Missing“MatterEquation”.............................151.2Thermodynamics,OpticsandElectrodynamics..................151.3AtomicPhysics....................................17ILagrangianFieldTheory232Euler-LagrangeFieldEquations...............................252.1Klein-GordonandSchr¨odingerEquations.....................252.2LagrangianDensity.................................252.3FreeEquations....................................262.4TheEquationswithMaxwellField.........................272.5ActionFunctional..................................272.6HamiltonLeastActionPrinciple..........................283FourConservationLawsforLagrangianFields.......................293.1TimeInvariance:EnergyConservation.......................293.2TranslationInvariance:MomentumConservation.................313.3RotationInvariance:AngularMomentumConservation.............313.4PhaseInvariance:ChargeConservation......................324LagrangianTheoryfortheMaxwellField.........................344.1MaxwellEquationsandPotentials.LagrangianDensity.............344.2LagrangianforChargedParticleinMaxwellField................364.3HamiltonianforChargedParticleinMaxwellField................38IISchr¨odingerEquation395GeometricOpticsandSchr¨odingerEquation........................415.1StraightLinePropagationfortheFreeEquations.................415.2WKBAsymptoticsforSchr¨odingerEquationwithaMaxwellField.......446Schr¨odingerEquationandHeisenbergRepresentation...................476.1ElectronsandCathodeRays............................476.2QuantumStationaryStates.............................476.3FourConservationLawsforSchr¨odingerEquation................486.4QuantumObservablesandHeisenbergRepresentation..............497CouplingtotheMaxwellEquations.............................527.1ChargeandCurrentDensitiesandGaugeInvariance...............527.2ElectronBeamsandHeisenberg’sUncertaintyPrinciple.............537.3QuantumStationary