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首页 Phase Space Formulation of Quantum Mechanics. In…

Phase Space Formulation of Quantum Mechanics. Insight into the Measurement Problem.pdf

Phase Space Formulation of Quan…

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2018-11-22 0人阅读 举报 0 0 0 暂无简介

简介:本文档为《Phase Space Formulation of Quantum Mechanics. Insight into the Measurement Problempdf》,可适用于自然科学领域

PHASESPACEFORMULATIONOFQUANTUMMECHANICSINSIGHTINTOTHEMEASUREMENTPROBLEMDDragoman*–UnivBucharest,PhysicsDept,POBoxMG,Bucharest,RomaniaAbstract:AphasespacemathematicalformulationofquantummechanicalprocessesaccompaniedbyandontologicalinterpretationispresentedinanaxiomaticformTheproblemofquantummeasurement,includingthatofquantumstatefiltering,istreatedindetailUnlikestandardquantumtheorybothquantumandclassicalmeasuringdevicecanbeaccommodatedbythepresentapproachtosolvethequantummeasurementproblem*Correspondenceaddress:ProfDDragoman,POBox,Bucharest,Romania,email:danieladragomanyahoocomIntroductionAtmorethanacenturyafterthediscoveryofthequantumanddespitetheindubitablesuccessofquantumtheoryincalculatingtheenergylevels,transitionprobabilitiesandotherparametersofquantumsystems,theinterpretationofquantummechanicsisstillunderdebateUnlikerelativisticphysics,whichhasbeenfoundedonanewphysicalprinciple,ietheconstancyoflightspeedinanyreferenceframe,quantummechanicsisratherasuccessfulmathematicalalgorittinteractionscanbefoundinTheexpressionofthistransitionprobabilityanditsmeaningasphasespaceoverlapshouldbecontrastedtothephasespacetreatmentoffiltering,whichisdevelopedinthefollowingPostulate:Theinteractionbetweenaquantumsystemandameasuringdevicecanbetreatedinphasespaceirrespectiveofthenature(quantumorclassical)ofthemeasuringdeviceTheresultoftheinteractiondependsonthetypeofthemeasuringdevice(filterordetector)PostulatewillbefirstdetailedforthecaseofaquantummechanicalmeasuringdevicethatcanbedescribedbyaWDFHowever,itwillbecomeapparentthatclassicalmeasuringapparatuscanalsobeaccommodatedbythistheoryaslongastheycanbecharacterizedbyaWDFNotethatthevariablesofthequantumandclassicalphasespacesarethesame,andthereforeaclassicalmeasuringapparatuscanbeformallytreatedinthesamewayasaquantumoneTheonlydifferencebetweenthetwocasesishowtocalculatetheWDFForsizeabledevicesitmightbedifficulttocalculateaquantumwavefunctionoraWDFfromquantumconsiderationsandthusthelibertytouseaclassicaldescriptioninwhichtheWDFcanbedeterminedmuchmoreeasilyisinvaluablePhasespacedescriptionsofbothclassicallightbeamsandclassicalensemblesofparticlesarewelldevelopedsothatthereisnoimpedimenttotreatinphasespaceamacroscopicmeasuringdeviceItisalsoimportanttospecifythattheWDFofaclassicalstatechangesitsontologicstatus:iftherearealargenumberofquantumparticlesthatsharethesamestate,theWDFisnolongeraprobabilitydistributionbutadistributionofthenumberofcomponentquantumparticlesinphasespace,ieitrepresentstheobjectWDFswithnegativevalueshavelongbeenknownandevenmeasuredinclassicaloptics,forexample(seethereviewin)AccordingtotheinterpretationinthenegativevaluesoftheWDFcanevendisappearforclassicalobjectsifthereisnocorrelationbetweenadjacentquantumblobsCorollary:NomeasuringapparatusisabletomeasureaneigenstateofthepositionormomentumoperatorsThiscorollaryfollows,forquantummeasuringapparatus,fromcorollaryandpostulateaquantummechanicalapparatuscannotmeasureaneigenstateofthepositionormomentumoperatorsbecausethecorrespondingstateoftheapparatuscannotexistOntheotherhand,amacroscopicmeasuringapparatusisformedfromalargenumberofquantumconstituentsifaquantumsystemcannotoccupyaphasespaceregionsmallerthanaquantumblob,itiscertainthatamacroscopicmeasuringapparatuscannoteitherMoreprecisely,theprojectionareaAoccupiedbytheclassicaldeviceonanyoftheconjugateplanesjq,jpscalesashNn,whereNisthenumberofquantumstateswithenergysmallerthanorequaltotheenergyshellNishugeforanysizeabledeviceFollowingthesamelineofreasoningasthatinthequantumcaseitisapparentthattheWDFofanextendeddevicecannotbeaδlineandthereforeitcannotmeasureaneigenstateofthepositionormomentumoperatorsSimilarconclusionsholdforoperatorsthatdependlinearlyonpositionormomentumTheapparentlackofquantummechanicaleffectsintheclassicalrealmandhenceforamacroscopicmeasuringapparatuswithalargenumberofdegreesoffreedomndoesnotautomaticallyimplythatquantummechanicsisnotapplicablebutonlythatquantummechanicaleffectsarenotnoticeablesincethephasespaceuncertaintyhismuchsmallerthantheprojectionareaAoccupiedbytheclassicaldeviceonanyoftheconjugateplanesjq,jpItisimportanttostressthatmeasurementdeviceisunderstoodinthiscontextasanydevicethatinfluencesthestateofaquantumsystemitcanbeeitherafilteroradetector,thesetwocasesbeingtreatedseparatelyinthefollowingsectionsInthefirstcasetheincidentquantumstatecharacterizedbyaWDFinWcanbefilteredineitherthecoordinateormomentumspacebythemeasuringapparatusthatisdescribedbyaWDFmW,theoutgoingquantumstatebeingcharacterizedinphasespacebyaWDFoutWInthesecondcasetheresultofthemeasurementisthesquaredmodulusofthequantumwavefunctionthatresultsaftertheincidentwavefunctionisfilteredinsomewaybythemeasuringdevicePhasespaceeffectofafilteringdeviceAfilteringdeviceisanydevicethatinfluencestheevolutionofaquantumstatesuchthattheoutgoingwavefunctionhassome“memory”ofitsoriginalformAfilterinfluencestheresultsofasubsequentmeasurementsinceitalterstheincidentwavefunctionandthereforeactivelymanipulatestheresultofthemeasurementLetusconsiderfirstaquantumfilterIftheincidentquantumwavefunction)(qinΨisfilteredinthecoordinatespacebythetransmissionfunction)(qmΨofafilter,theoutputwavefunctionis)()()(qqqminoutΨΨ=ΨThisfilteringactioncanbedescribedinphasespaceas')',()',(),(pppqpqpqdWWWminout∫−=,()ieasameremultiplicationalongthecoordinateaxisinphasespaceandaconvolutionalongthemomentumdirectionSimilarly,afilterwithatransmissionfunctioninthemomentumspace)(pmΨtransformsanincidentquantumwavefunctioninthemomentumrepresentation)(pinΨinto)()()(pppminoutΨΨ=Ψ,transformationthatcanberepresentedinphasespaceas'),'(),'(),(qpqqpqpqdWWWminout∫−=()Theconvolutionisnowperformedalongthecoordinateaxisofthephasespace,thetransformationalongthemomentumaxisbeingasimplemultiplicationAmoregeneralfilteringprocesscangenerateanoutputwavefunctionoftheform∫−ΨΨ=Ψ−')'exp()'()'()(qqpqqqqdihminnout�()Thisfilteringprocesscanberepresentedinphasespaceas∫−−='),'(),'(),(qpqqppqpqdWWWminout,()thesimilarexpression∫−−=')',()',(),(pppqpqqpqdWWWminout()describingthefilteringprocess∫−−ΨΨ=Ψ−')'exp()'()'()(ppqppppdihminnout�()Notethatinallphasespaceexpressions(),(),()and()onlytheWDFofthefilterappearsitisirrelevantifthisiscalculatedornotwiththeuseofquantummechanicalprinciplessincetheWDFofquantumandclassicalphysicsarethesameTherefore,theeffectofaclassicalfilteronaquantumstatecanequallybedescribedbyequations(),(),()and()AfilteringprocessisdifferentfrominterferencesincetheWDFsoftheincidentquantumstateandthefilteringdevicemust(atleastpartially)overlapItdiffersalsofromtransitionsincethefilteringisnotjustaphasespaceoverlapmoreover,inquantumtransitionsthereisnointermediatefilterThereisonlytheincidentwavefunctionandanoutgoingwavefunction,totallydifferentfromthefirstUnlikefiltering,quantumtransitionhasno“memory”oftheoriginalwavefunctionexceptfortransitionprobabilityExamplesItisworthwhiletoexemplifytheinfluenceofafilterinthespatialdomainforthecasen=Letusconsiderthattheinputwavefunction)exp()()(iiinqqqq−=Ψ−piisaGaussianwithaspatialextentiqandthatthequantumfilterisaslitwithanormalizedtransmissionfunction)exp()()(mmmqqqq−=Ψ−piThecorrespondingWDFs,calculatedasexemplifiedinrelationwithpostulate,arerepresentedinFigforthecasemiqq>darkerareascorrespondtohighervaluesoftheWDFsNotethat,sincetheWDFofaGaussianislocalizedinaphasespaceareaequaltoh,theextentoftheWDFontheqaxisisinverselyproportionaltotheextentonthepaxisTheWDFoftheoutputstateisgiven,accordingto(),by−−−=expexp)(),(mimimimioutqqqqpqqqqqqhpqWpi()From()itfollowsthatinthelimitmiqq>>theWDFoftheoutputstateisascaledversionoftheWDFofthefilter,whileifmiqq<<thefilterdoesnotsignificantlyinfluencetheinputstateTheseresultsareintuitivewhenweviewthefilteringprocessinthecoordinaterepresentation,butaresomehowcounterintuitivewhenlookingatFigAsimplefilteringinphasespacealongbothqandpwouldresultinanoutputWDFgivenbytheintersectionoftheWDFsoftheinputstateandfilterhowever,theconvolutiontransformalongpresultsinanoutputWDFthatcanhaveanextentalongpmuchlargerthantheextentoftheintersectionofthetwoWDFsAsimilarphasespaceapproachtowardsthestudyofthequantumcoordinatemeasurementwithahardslitwastakeninalthoughtheform()doesnotexplicitlyappearinthisreferenceTheresultthatifmiqq<<thefilterdoesnotsignificantlyinfluencetheinputstatebecomesextremelysuggestiveifweinterpolateitintheclassicallimitofbothstatesandapparatusAswehavealreadymentionedaclassicalobjectcanalsoberepresentedinphasespacebyaprobabilitydistributionorevenbyanonpositivevaluedWDFifitisaclassicalfieldAfiltercannotletaclassicalobjectpassunlessithaslargerdimensionsAccordingto(),aslit(much)largerthantheobjectleavesitpracticallyunchangedAnotherinterestingexampleisthatofaslitputinfrontofacatlikequantumstate,whichischaracterizedbytheonedimensionalwavefunction−−−=Ψ)(exp)(exp)(iiinqdqqdqNq,()wherethenormalizationconstant)exp()(−−−=iiqdqNpiTheWDFofacatstate,givenby−−−−−−=dpqqqdqqdqqdhqppqWiiiiiincosexp)(exp)(exp)exp()exp(),(,()consists(forasufficientlylargeseparationd)oftwoouterterms,whichrepresenttheWDFsoftheindividualGaussianconstituents,andofaninterferenceterm(thelasttermin())FigshowstheWDFofacatstateforiqd=darkerareasrepresentshighervaluesoftheWDFNotethatthemiddle,oscillatoryinterferencetermhaslargeramplitudethantheoutertermsandattainsitsmaximumvalueatthephasespaceorigin=q,=pIfthisWDFisfilteredbyaquantumGaussianslitthatiscenteredatDq=theoutputWDFcalculatedaccordingto()isgivenby−−−−−×−−−=cosexpexp)(exp)(expexp)(exp),(mimmiiiimimimoutqqqdpqqdqqqdqqdqqqqqpqDqKDpqW��()Thisformulatellsusthat,asabove,thefilterdoesnotsignificantlyinfluencetheinputstateifmiqq<<andmqd<<However,thedifferenttermsininWaredifferentlytransformedasaresultofthefilteringifmiqq≅Inourspecificexample,foriqd=,aslitwithimqq≅cannotfilterallthreetermsoftheincidentWDFTostudytheinfluenceofthefilteringonthesetermsitisinstructivetomodifytheoffaxisfilterpositionDacrossthewidthofthecatlikestateTheresultingfunction),(DqWout,displayedinFig,showsthataslitwithimqq=reallyfilterstheoutertermsonlyonlythecontributionsoftheincidentWDFthatcorrespondtotheprobabilityoffindingparticlesandnottheinterferencetermare“sensed”bytheslitTheinterferenceterm,whichdoesnotcorrespondtoanysignificantprobabilityoffindingquantumparticlesbutonlysignalsthepotentialityofinterference,isignoredbytheslitduetotheconvolutionalongpperformedinthefilteringprocessalthoughitsamplitudeinFigislargerthanthatoftheoutertermsThisexampleillustratesbetterthananyotherthatphasespacefilteringreferstofilteringoftheregionsinwhichtheprobabilityoffindingtheincidentquantumwavefunction(incoordinateorpositionrepresentation)issignificantThisistobeexpectedfromourdefinition)()()(qqqminoutΨΨ=ΨoftheslitactionPhasespacerepresentationofquantumstatesmustbehandledwithcareinordertodistinguishbetween“true”termsand“phony”interferencetermsthesehavedifferentsignificancesandareessentialindifferentcasesAsimilarconclusionwasestablishedinconnectionwithphasespaceinterferenceObservationCanafilteractinamoresophisticatedmannerthanjustatransmissionfunctionincoordinateormomentumrepresentationsPerhapsitcan,butitisquitedifficulttoimaginetheactionofsuchafilterCoordinateandmomentumarecomplementaryvariablestheexternalmotionofquantumstatescanbedescribedonlyintermsoftheseOtherinternalquantumvariablesmanifesttheirexistencealsointhecoordinateormomentumspaceitisthispossibilitytodiscernvariousinternalquantumstatesatdifferentpositionormomentumcoordinatesafterpassingthroughappropriatelydesignedsetupsthatmakeinternalquantumvariablesobservableForexample,differentspincomponentvaluesarespatiallyseparatedduetothedifferentdeflectiondirectionsfromaSternGerlachexperimentalsetupAnintroductionofinternalquantumvariablesinthedefinitionofthequantumWDFhasnotbeencarriedoutuptonowbutitmaybeconsiderednecessaryasthephasespaceformulationofquantummechanicsbecomesmorewidespreadUptothatmomentthefilteringtheoryinphasespacepresentedinthispapercoversthemajorityofpracticalsituationsPhasespaceeffectofadetectorAconvolutionoftheWDFofthefilteredquantumstatealongbothcoordinateandmomentumdirectionscannotgenerallyrepresentaWDFofaquantumwavefunctionsince*|')'exp()'()'(|'')','()','(∫−−ΨΨ=∫−−−qpqqqqpqppqqpqdihddWWminnmin�()isapositivedefinitefunctionHowever,itcanrepresenttheresultofameasurementoftheinputquantumstatewithaquantumdetectiondevicethatfiltersitinbothpositionandmomentumcoordinatesThisphasespacerepresentationofthequantumdetectionprocessisnotnew,However,asadvocatedintheprevioussection,thelefthandsideof()describesequallywelltheresultofdetectionofaquantumstatewithaclassicaldetectorcharacterizedbyaWDFmWNotethat,accordingto(),thedetectionprocessexpressedby()isageneralizedtransitionprocessThequantumparticleperformsatransitionfromtheinputstatetothedetector,itspresencebeingobservedbya“click”orsomeothermanifestationEvenatdetectionthemeasuredphasespacedistributionofquantumparticlesisnotindependentoftheinherentfilteringprocessperformedbyanydetectorItisnotpossibletodetectquantumstateswithoutperturbingthemDiscussionsandconclusionsWehavedevelopedaquantummechanicalphasespaceformalismthatincorporatesnotonlyanestablishedmathematicalformalismbutalsoontologicalinterpretationsofthephysicalreality,focusingonthecharacterizationinphasespaceoftheresultoffilteringanddetectionuponanincidentquantumstateAlthoughsomeofthemathematicalformulationspresentedinthispaperarewellknown,thephasespacetreatmentofthefilteringprocessaswellastheaxiomaticformofthetheoryandtheontologicalinterpretationarenovelAtthepresentlevelofthedevelopmentofthephasespaceformalismofquantummechanicsthereappeartobenodifferencesbetweenthepredictionsofthetheoryinthispaperandstandardquantummechanicsaslongasquantumconceptsareusedHowever,differencesexist,oneofthemostimportantbeingthatofthepossibilityofintroducinganequaltreatmentofquantumandclassicalmeasuringapparatusAsubsequentdevelopmentofthetheory,whichisonlydraftedhere,mightrevealotherdifferencesAlthoughonlythemeasuringapparatuscanbeclassicalintheformofthetheorydevelopedhere,alltheresults,includingthedescriptionoftheinputstate,canbeextendedtotheclassicaldomainTheparticularformofthetheorywasexpressedwiththegoaloftryingtobringaninsightintheunsolvedquantummechanicalmeasurementproblemtheclassicalmeasurementproblemisfarless(ifany)controversialThequantumclassicalcorrespondenceismostrelevantinthephasespaceformulation,sinceboththeorieshavesimilarmathematicalformulations(seeforadetailedtreatmentofquantumclassicalcorrespondenceinphasespace)ThefactthattheconversionfromquantumtoclassicalterminologydoesnotaffectthedescriptionofameasuringapparatusifthedynamicalvariableisanoperatoroftheWeylWignertypehasalreadybeenshowninreferenceAfewremarksontherelationbetweentheformalismpresentedinthispaperandotherinterpretationsofquantummechanics,especiallyofthemeasurementprocess,areinorderInthephasespaceformulationofthequantumtheoryofmeasurementitisnoneedtoassumeanyreductionofthewavefunctionoftheinitialquantumstate,sincethiswavefunctiondoesnotneedtobedecomposedineigenstatesofsomeoperatorTheproblemchangesfromfindingtheeigenstatestothatofhowtodesignthefilterdetector(whatWDFshouldithave)togatherinformationaboutaspecificpropertyTherefore,thereisnoneedforacutinthemeasurementprocessbetweenthequantumsystemandthe(classical)measuringdevicethisiswelcomesincethelocationofthecutintheCopenhageninterpretationistoalargeextentarbitraryAsdiscussedabove,evenclassicaldevicescanbeaccommodatedbythepresentphasespacetheoryiftheirWDFcanbededucedThesubsequenttheoryofinterference,measuringandtransitionprobabilitiesinphasespacecanbeappliedirrespectiveofthemannerinwhichtheWDFaredeterminedThetheorypresentedinthispaperrespondstomostofdesideratesofthesocalledIthacainterpretationofquantummechanicsNamely,itisbasedonthefactthat(i)quantummechanicsdescribesanobjectivereality,(ii)itisbasedonthenotionofobjectiveprobability,(iii)describesindividualsystems,notjustensembles,(iv)doesnothavetoinvokeinteractionswithenvironment(ortheexistenceofaclassicaldomain)whendescribingsmallisolatedsystems,and,mostimportant(v)theconceptofmeasurementplaysnofundamentalroleAlthoughnotexplicitlystatedthesixthdesiderate,that(vi)objectivelyrealinternalpropertiesofanisolatedindividualsystemdonotchangewhenanothernoninteractingsystemisperturbed,followsimplicitlyfromthetheorypresentedinthispaperTheprobabilisticinterpretationofthemeasurementresults,inwhichtheWDFisseenastheprobabilityamplitudeofaquantumparticle(orparticles)thatcannotbeconfinedtoaphasespaceregionsmallerthanaquantumblob,canexplainthesocalledwaveparticledualityAsdiscussedwithregardtothephasespaceinterpretationofdelayedchoiceexperimentsthe“choice”ofwaveorparticleisdonebytheplacewherethedetectiondeviceismadeandbythetypeofthedetectiondevicesincethedetectionprocessislocaltheoutcomeoftheresultisonlydeterminedbytheformoftheWDFofthetotalsystem(twoquantumstates)atthatparticularplace(andmoment)ofdetectionAllpossibletypesofinteractions:interference,filtering,transitionsanddetectionhavedifferent“signatures”inthephasespacetreatmentandthereforethisapproachtoquantummechanicsisbestsuitedReferencesDFStyer,MSBalkin,KMBecker,MRBurns,CEDudley,STForth,JSGaumer,MAKramer,DCOertel,LHPark,MTRinkoski,CTSmith,andTDWotherspoon,“Nineformulationsofquantummechanics”,AmJPhys,()JAWheelerandWHZurek,(eds),QuantumTheoryandMeasurements,PrincetonUnivPress,Princeton,NJ()JvonNeumann,MathematicalFoundationsofQuantumMechanics,PrincetonUnivPress,Princeton,NJ()WHZurek,“Decoherence,einselection,andthequantumoriginsoftheclassical”,RevModPhys,()HEverettIII,“’Relativestate’formulationofquantummechanics”,RevModPhys,()NPLandsman,“Observationandsuperselectioninquantummechanics”,StudHistPhilModPhys,()EWigner,“Onthequantumcorrectionforthermodynamicequilibrium”,PhysRev,()GABaker,Jr,“Formulationofquantummechanicsbasedonthequasiprobabilitydistributioninducedonphasespace”,PhysRev,()MHillery,RFO’Connell,MOScully,andEPWigner,“Distributionfunctionsinphysics:Fundamentals”,PhysRep,()YSKimandMENoz,PhaseSpacePictureofQuantumMechanics,WorldScientific,Singapore,HWLee,“Theoryandapplicationofthequantumphasespacedistributionfunctions”,PhysRep,()WPSchleich,QuantumOpticsinPhaseSpace,WileyVCH,Berlin,,ppAEGlassgoldandDHolliday,“Effectofacoordinatemeasurementonthestatisticalensembleofaquantummechanicalsystem”,PhysRev,()MAdeGosson,“Phasespacequantizationandtheuncertaintyprinciple”,PhysLettA,()DDragoman,“PhasespaceinterferenceasthesourceofnegativevaluesoftheWignerdistributionfunction”,JOptSocAmA,()WHZurek,“SubPlanckstructureinphasespaceanditsrelevanceforquantumdecoherence”,Nature,()MdeGosson,“Thesymplecticcamelandphasespacequantization”,JPhysA,()MAdeGosson,“The‘symplecticcamelprinciple’andsemiclassicalmechanics”,JPhysA,()DDragoman,“Quantuminterferenceasphasespacefiltering”,Optik,()DDragoman,“TheformulationofFermi’sgoldenruleinphasespace”,PhysLettA,,()DDragoman,“TheinterferencetermintheWignerdistributionfunctionandtheAharonovBohmeffect”,PhysLettA,()JAWheeler,inMathematicalFoundationsofQuantumMechanics,ARMarlow(ed),AcademicPress,NewYork,,ppDDragoman,“TheWignerdistributionfunctioninopticsandoptoelectronics”,ProgOpt,()AJLichtenberg,PhaseSpaceDynamicsofParticles,JohnWileySons,NewYork,ARoyer,“MeasurementoftheWignerfunction”,PhysRevLett,()KWódkiewicz,“Operationalapproachtophasespacemeasurementsinquantummechanics”,PhysRevLett,()DDragoman,“Phasespacecorrespondencebetweenclassicalopticsandquantummechanics”,ProgOpt,()OHayandAPeres,“Quantumandclassicaldescriptionsofameasuringapparatus”,PhysRevA,()NDMermin,“TheIthacainterpretationofquantummechanics”,Pramana,()FigurecaptionsFig(a)TwostateswithindividualWDFsWandWthathavecommonprojectionsalongtheqandpaxesinterferealongbothspatialandangularcoordinates(b)TransitionsareexpectedtooccuriftheindividualWDFsoverlapatleastpartiallyFigWDFsofanincidentGaussianwavefunction,inW,andafilterwithaGaussiantransmission,mW,alignedwithrespecttotheincidentwavefunctionFigWDFofacatlikequantumstateDarkerareasrepresentlargervaluesoftheWDFFigFilteringeffectofanarrowmisaligned,offaxisGaussianslituponacatlikestateDarkerareasrepresentlargervaluesoftheWDFqpWW(a)qpWW(b)FigureqpWm(q,p)Win(q,p)FigureqqipqiWin(au)FigureWout(au)qqiDqiFigur
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