Advanced Visual Quantum Mechanics.pdf

AdvancedVisualQuantumMechanicsthallerFM:PageBerndThallerAdvancedVisualQuantumMechanicsWithIllustrationsINCLUDESCDROMthallerFM:PageBerndThallerInstituteforMathematicsandScientificComputingUniversityofGrazAGrazAustriaberndthallerunigrazatLibraryofCongressCataloginginPublicationDataThaller,Bernd,AdvancedvisualquantummechanicsBerndThallerpcmIncludesbibliographicalreferencesandindexISBN(acidfreepaper)QuantumtheoryQuantumtheoryComputersimulationITitleQCTdcMathematicaisaregisteredtrademarkofWolframResearch,IncQuickTimeisaregisteredtrademarkofAppleComputer,Inc,registeredintheUnitedStatesandothercountriesUsedbylicenceMacromediaandMacromediaDirectorareregisteredtrademarksofMacromedia,Inc,intheUnitedStatesandothercountriesISBNPrintedonacidfreepaperSpringerScienceBusinessMedia,IncAllrightsreservedThisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrittenpermissionofthepublisher(SpringerScienceBusinessMedia,Inc,SpringStreet,NewYork,NY,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysisUseinconnectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbiddenTheuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,eveniftheyarenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubjecttoproprietaryrightsPrintedintheUnitedStatesofAmerica(HAM)SPINspringeronlinecomthallerFM:PagethallerFM:PagePrefaceAdvancedVisualQuantumMechanicsisasystematicefforttoinvestigateandtoteachquantummechanicswiththeaidofcomputergeneratedanimationsButdespiteitsuseofmodernvisualizationtechniques,itisaconventionaltextbookof(theoretical)quantummechanicsYoucanreaditwithoutacomputer,andyoucanlearnquantummechanicsfromitwithouteverusingtheaccompanyingCDROMBut,theanimationswillgreatlyenhanceyourunderstandingofquantummechanicsTheywillhelpyoutogettheintuitivefeelingforquantumprocessesthatissohardtoobtainfromthemathematicalformulasaloneAfirstbookwiththetitleVisualQuantumMechanics(“BookOne”)appearedintheyearTheCDROMforBookOneearnedtheEuropeanAcademicSoftwareAward(EASA)foroutstandinginnovationinitsfieldThetopicscoveredbyBookOnemainlyconcernedquantummechanicsinoneandtwospacedimensionsAdvancedVisualQuantumMechanics(“BookTwo”)setsouttopresentthreedimensionalsystems,thehydrogenatom,particleswithspin,andrelativisticparticlesItalsocontainsabasiccourseofquantuminformationtheory,introducingtopicslikequantumteleportation,theEPRparadox,andquantumcomputersTogether,thetwovolumesconstituteafairlycompletecourseonquantummechanicsthatputsanemphasisonideasandconceptsandsatisfiessomemodestrequirementsofmathematicalrigorNevertheless,BookTwoisfairlyselfcontainedReferencestoBookOnearekepttoaminimumsothatanyonewithabasictraininginquantummechanicsshouldbeabletoreadBookTwoindependentlyofBookOneAppendixAincludesashortsynopsisofquantummechanicsasfarasitwaspresentedinBookOneTheCDROMincludedwiththisbookcontainsalargenumberofQuickTimemoviespresentedinamultimedialikeenvironmentThemoviesillustratethetext,addcolor,atimedimension,andacertainlevelofinteractivityThecomputergeneratedanimationswillhelpyoutoexplorequantummechanicsinasystematicwayThepointandclickinterfacegivesyouquickandeasyaccesstoallthemoviesandlotsofbackgroundinformationYouneednospecialcomputerskillstousethesoftwareInfact,itisnomorevviPREFACEdifficultthansurfingtheInternetYouarenotrequiredtoproducesimulationsbyyourselfThegeneralideaisthatyoushouldfirstthinkaboutquantummechanicsandnotaboutcomputersThemoviesprovidesomephenomenologicalbackgroundTheywilltrainandenhanceyourintuition,andthedesiretounderstandthemoviesshouldmotivateyoutolearnthe(sometimesnasty,sometimeselegant)theoryComputervisualizationsareparticularlyrewardinginquantummechanicsbecausetheyallowustodepictobjectsandeventsthatcannotbeseenbyothermeansHowever,onehastobeawareofthefactthattheanimationsdepictthemathematicalobjectsdescribingreality,notrealityitselfUsually,oneneedssomeexplanationandinterpretationtounderstandthevisualizationsThevisualizationmethodusedheremakesextensiveuseofcolorItdisplaysallessentialinformationaboutthequantumstateinanintuitivewayWatchingthenumerousanimationswillthuscreateanintuitivefeelingforthebehaviorofquantumsystemssomethingthatishardlyachievedjustbysolvingtheSchrodingerequationmathematicallyIwouldevensaythatthemoviesallowustoseethewholesubjectinanewwayInanycase,the“visualapproach”hadagreatinfluenceontheselectionoftopicsaswellasonthestyleandthelevelofthepresentationForexample,VisualQuantumMechanicsputsanemphasisonquantumdynamics,becauseamovieaddsanaturaltimedimensiontoanillustrationWhereasothertextbooksstopwhentheeigenfunctionsoftheHamiltonianareobtained,thisbookwillgoontodiscussdynamicaleffectsItdependsonthesituation,butalsoonthepersonalityofthestudentoroftheteacher,howthemoviesareusedInsomecases,themoviesarecertainlyusefultostimulatethestudent’sinterestinsomephenomenonTheanimationthusservestomotivatethedevelopmentofthetheoryInothercases,itis,perhaps,moreappropriatetoshowamovieconfirmingthetheorybyanexamplePersonally,IpresentthemoviesbyvideoprojectionasasupplementtoanintroductorycourseonquantummechanicsItalkaboutthemoviesinaratherinformalway,andsoonthestudentsstartaskinginterestingquestionsthatleadtofruitfuldiscussionsanddeeperexplanationsOften,themoviesmotivatestudentstostudyrelatedtopicsontheirowninitiativeOnecouldarguethatinadvancedquantummechanics,visualizationsarenotveryusefulbecausethestudenthastolearnabstractnotionsandthatheorsheshouldthinkintermsoflinearoperators,Hilbertspaces,andsoonItiscertainlytruethatasolidfoundationofthesesubjectsisindispensableforadeeperunderstanding,andyouwillhaveoccasiontolearnmuchaboutthemathematicaltheoryfromthistextBut,Iclaimthatdespiteagoodtrainingintheabstracttheory,youcanstillgainalotfromthevisualizationsPREFACEviiTalkingaboutmyownexperience,IfoundthatIlearnedmuch,evenaboutsimplesystems,whenIpreparedthemoviesforVisualQuantumMechanicsForexample,havingdoneresearchonthemathematicalaspectsoftheDiracequationforseveralyears,IcanclaimtohaveagoodbackgroundconcerningthequantummechanicalabstractionsinthisfieldButnevertheless,Iwasnotabletopredicthowawavepacketperforminga“Zitterbewegung”wouldappearuntilIstartedtodosomevisualizationsofthatphenomenonMoreover,whenonetriestounderstandthevisualizationsoneoftenencountersphenomena,thatoneisabletoexplainwiththetheory,butthatonesimplyhasn’tthoughtofbeforeThemainthingthatyoucangainfromthevisualizationsisagoodfeelingforthebehaviorofsolutionsofthequantummechanicalequationsThoughtheCDROMpresentsafewsimpleinteractivesimulationsinthechapteraboutqubits,theoverwhelmingcontentconsistsofprefabricatedmoviesAtruecomputersimulation,thatis,alivecomputationofsomeprocess,wouldofcourseallowahigherdegreeofinteractivityThereaderwouldhavemoreflexibilityinthechoiceofparametersandinitialconditionsButinmanycases,thisapproachisforbiddenbecauseoftheinsufficientspeedofpresentdaycomputersMoreover,inordertoproduceausefulvisualization,onehastoanalyzethephysicalsystemverycarefullyForeverysituation,onehastodeterminethescaleofspaceandtimeandsuitablerangesoftheparameterswheresomethinginterestingisgoingtohappenInquantummechanics,thenumberofpossibilitiesisverylarge,andifonechoosesthewrongparametervalues,itisverylikelythatnothingcanbeseenthatiseasilyinterpretedorthatshowssomeeffectinaninterestingwayTherefore,IwouldnotrecommendtolearnbasicquantummechanicsbydoingtimeconsumingcomputersimulationsProducingsimulationsanddesigningvisualizationscan,however,bringenormousbenefittotheadvancedstudentwhoisalreadyfamiliarwiththefoundationsofquantummechanicsManyoftheanimationsontheCDROMweredonewiththehelpofMathematicaWiththeexceptionoftheMathematicasoftware,allthenecessarytoolsforproducingsimilarresultsareprovidedontheCDROM:Thesourcecodeforallmovies,MathematicapackagesbothforthenumericalsolutionoftheSchrodingerequationandforthegraphicalpresentationoftheresults,andOpenGLbasedsoftwareforthethreedimensionalvisualizationofwavefunctionsMyrecommendationistostartwithsomesmallprojectsbasedontheexamplesprovidedbytheCDROMItshouldnotbedifficulttomodifytheexistingMathematicanotebooksbyslightlyvaryingtheparametersandinitialconditions,andthenwatchingandinterpretingtheresultsYoucouldthenproceedtolookforotherexamplesofquantumsystemsthatmightbegoodforaviiiPREFACEphysicallyormathematicallyinterestingvisualizationWhenyouproduceavisualization,oftensomenaturalquestionsaboutthesystemwillariseThismakesitnecessarytolearnmoreaboutthesystem(oraboutquantummechanics),andbyknowingthesystembetter,youwillproducebettervisualizationsWhenthevisualizationfinallybecomesuseful,youwillunderstandthesystemalmostperfectlyThisis“learningbydoing”,anditwillcertainlyenhanceyourunderstandingofquantummechanics,asthemakingofthisbookhelpedmetounderstandquantummechanicsbetterBewarned,however,thatpersonalcomputersarestilltooslowtoperformsimulationsofrealisticquantummechanicalprocesseswithinareasonabletimeManyofthemoviesprovidedwiththisbooktypicallytookseveralhourstogenerateConcerningthemathematicalprerequisites,ItriedtokeepthetwobooksonanintroductorylevelHence,ItriedtoexplainallthemathematicalmethodsthatgobeyondbasiccoursesincalculusandlinearalgebraBut,thisdoesnotmeanthatthecontentofthebookisalwayselementaryItisclearthatanytextthatsetsouttoexplainquantumphenomenamusthaveacertainlevelofmathematicalsophisticationHere,thislevelisoccasionallyhigherthaninotherintroductions,becausethetextshouldprovidethetheoreticalbackgroundforthemoviesDoingvisualizationsismorethanjustobtainingnumericalsolutionsAsurprisingamountofmathematicalknowhowisinfactnecessarytoprepareananimationWithoutpresentingtoomanyunnecessarydetails,ItriedtoincludejustwhatIthoughtwasnecessarytoproducethemoviesMyapproachtoteachingquantummechanicsthusmakesnoattempttotrivializethissubjectTheanimationsdonotreplacemathematicalformulasButinordertofacilitatetheapproachforthebeginner,Imarkedsomeofthemoredifficultsectionsas“specialtopics”andplacedthesymbolΨinfrontofparagraphsintendedforthemathematicallyinterestedreaderThesepartsmaybeskippedatfirstreadingThoughthebookthusaddressesstudentsandscientistswithsomebackgroundinmathematics,themovies(togetherwiththemoviesofBookOne)cancertainlybeusedinfrontofawideraudienceThesuccess,ofcourse,dependsonthestyleofthepresentationImyselfhavehadtheoccasiontousethemoviesinlecturesforhighschoolstudentsandforscientificallyinterestedpeoplewithoutanytraininginhighermathematicsBasedonthisexperience,IhopethatthebooktogetherwithCDROMwillhavebroaderapplicationsthaneachcouldhaveifusedaloneAccordingtoitssubtitle,BookTwocanbedividedroughlyintothreeparts:atomicphysics(Chapters–),quantuminformationtheory(Chapters–),andrelativisticquantummechanics(Chapters,)Thisdivision,however,shouldnotbetakentooseriouslyForexample,ChapteronPREFACEixqubitscompletesthediscussionofspinparticlesinChapterandservesatthesametimeasanintroductiontoquantuminformationtheoryChapterdiscussescompositequantumsystemsbycombiningtopicsrelevantforquantuminformationtheory(forexample,twoqubitsystems)withtopicsrelevantforatomicphysics(forexample,additionofangularmomenta)Together,BookOneandBookTwocoverawiderangeofthestandardquantumphysicscurriculumandsupplementitwithaseriesofadvancedtopicsForthesakeofcompleteness,someimportanttopicshavebeenincludedintheformofseveralappendices:theperturbationtheoryofeigenvalues,thevariationalmethod,adiabatictimeevolution,andformalscatteringtheoryThoughmostofthesemattersareverywellsuitedforanapproachusinglotsofvisualizationsandexamples,Isimplyhadneithertimenorspace(theCDROMisfull)toelaborateonthesetopicsasIwouldhavelikedtodoTherefore,theseappendicesareratherinthestyleofanordinarytextbookonadvancedtheoreticalphysicsIwouldbegladifthismaterialcouldserveasabackgroundforthereader’sownventuresintothefieldofvisualizationIfthereshouldeverbeanothervolumeofVisualQuantumMechanics,itwillprobablycenteronthesetopicsandonothersliketheThomasFermitheory,periodicpotentials,quantumchaos,andsemiclassicalquantummechanics,justtonameafewfrommylistoftopicsthatappeartobesuitableforamodernizedapproachinthestyleofVisualQuantumMechanicsThisbookhasahomepageontheinternetwithURLhttp:wwwunigrazatimawwwvqmAnoccasionalvisittothissitewillinformyouaboutsoftwareupgrades,printingerrors,additionalanimations,etcAcknowledgementsIwouldliketothankmysonWolfgangwhoquicklywrotetheprogram”QuantumGL”whenitturnedoutthattheavailablesoftwarewouldn’tservemypurposesThankstoManfredLiebmann,GeraldRoth,andReinholdKainhoferforhelpwithMathematicarelatedquestionsIamverygratefultoJerryBatzelwhoreadlargepartsofthemanuscriptandgavemevaluablehintstoimprovemyEnglishThisbookowesalottoMichaelAMorrisonHestudiedthemanuscriptverycarefully,madealargenumberofhelpfulcomments,askedlotsofquestions,andeliminatednumerouserrorsMostimportantly,hekeptmegoingwithhisenthusiasmThanks,MichaelFinancialsupportfromSteiermarkischeLandesregierung,fromtheUniversityofGraz,andfromSpringerVerlagisgratefullyacknowledgedGraz,JanuaryBerndThallerthallerFM:PageContentsPrefacevChapterSphericalSymmetryANoteonSymmetryTransformationsRotationsinQuantumMechanicsAngularMomentumSphericalSymmetryofaQuantumSystemThePossibleEigenvaluesofAngularMomentumOperatorsSphericalHarmonicsParticleonaSphereQuantizationonaSphereFreeSchrodingerEquationinSphericalCoordinatesSphericallySymmetricPotentialsChapterCoulombProblemIntroductionTheClassicalCoulombProblemAlgebraicSolutionUsingtheRungeLenzVectorAlgebraicSolutionoftheRadialSchrodingerEquationDirectSolutionoftheRadialSchrodingerEquationSpecialTopic:ParabolicCoordinatesPhysicalUnitsandDilationsSpecialTopic:DynamicsofRydbergStatesChapterParticleswithSpinIntroductionClassicalTheoryoftheMagneticMomentTheSternGerlachExperimentTheSpinOperatorsSpinorWaveFunctionsThePauliEquationSolutioninaHomogeneousMagneticFieldSpecialTopic:MagneticGroundStatesTheCoulombProblemwithSpinxixiiCONTENTSChapterQubitsStatesandObservablesMeasurementandPreparationEnsembleMeasurementsQubitManipulationsOtherQubitSystemsSingleParticleInterferenceQuantumCryptographyHiddenVariablesSpecialTopic:QubitDynamicsChapterCompositeSystemsStatesofTwoParticleSystemsHilbertSpaceofaBipartiteSystemInteractingParticlesObservablesofaBipartiteSystemTheDensityOperatorPureandMixedStatesPreparationofMixedStatesMoreAboutBipartiteSystemsIndistinguishableParticlesSpecialTopic:MultiparticleSystemswithSpinSpecialTopic:AdditionofAngularMomentaChapterQuantumInformationTheoryEntangledStatesofTwoQubitSystemsLocalandNonlocalTheEinsteinPodolskyRosenParadoxCorrelationsArisingfromEntangledStatesBellInequalitiesandLocalHiddenVariablesEntanglementAssistedCommunicationQuantumComputersLogicGatesQuantumAlgorithmsChapterRelativisticSystemsinOneDimensionIntroductionTheFreeDiracEquationDiracSpinorsandStateSpacePlaneWavesandWavePacketsSubspaceswithPositiveandNegativeEnergiesKinematicsofWavePacketsZitterbewegungCONTENTSxiiiSpecialTopic:EnergyRepresentationandVelocitySpaceRelativisticInvarianceChapterTheDiracEquationTheDiracEquationRelativisticCovarianceClassificationofExternalFieldsPositiveandNegativeEnergiesNonrelativisticLimitandRelativisticCorrectionsSphericalSymmetryTheDiracCoulombProblemRelativisticHydrogenAtomAppendixASynopsisofQuantumMechanicsAppendixBPerturbationofEigenvaluesAppendixCSpecialTopic:AnalyticPerturbationTheoryAppendixDVariationalMethodAppendixEAdiabaticandGeometricPhasesAppendixFFormalScatteringTheoryAppendixGBooksAppendixHMovieIndexIntroductionSphericalSymmetryCoulombProblemSpinQubitsCompositeSystemsRelativisticSystemsListofSymbolsIndexthallerFM:PageChapterSphericalSymmetryChaptersummary:InthefirstbookofVisualQuantumMechanics,weconsideredmainlyoneandtwodimensionalsystemsNowweturntotheinvestigationofthreedimensionalsystemsThischapterisdevotedtotheveryimportantspecialcaseofsystemswithsphericalsymmetryInthepresenceofsphericalsymmetry,theSchrodingerequationhassolutionsthatcanbeseparatedintoaprodu