Quantum-mechanical motion and the stillness of exp

arXiv:quant-ph/0305096v119May2003Quantum-mechanicalmotionandthestillnessofexperimentalrecordsF.HadiMadjid82PowersRoad,Concord,Massachusetts01742JohnM.MyersGordonMcKayLaboratory,DivisionofEngineeringandAppliedSciences,HarvardUniversity,Cambridge,Massachusetts02138(Dated:19May2003)AbstractExperimentingwithmetastabilityinrecordingdevicesleadsustowonderaboutaninterfacebetweenequationsofmotionandthestillnessofexperimentalrecords.HerewedelineateaninterfacebetweenwavefunctionsaslanguagetodescribemotionandTuringtapesaslanguagetodescribeexperimentalrecords.Afterextendingquantumformalismtomakethisinterfaceexplicit,wereportonconstraintsandfreedomsinchoosingquantum-mechanicalequationstomodelexperimentswithdevices.Weprovethatchoosingequationsofwavefunctionsandoperatorstoachieveafitbetweencalculatedprobabilitiesandexperi-mentalrecordsrequiresreachingbeyondbothlogicandtheexperimentalrecords.Althoughinformedbyexperience,a“reachbeyond”canfairlybecalledaguess.Recognizingthatparticlesasfeaturesofwavefunctionsdependonguesswork,weintroducetheirusenotasobjectsofphysicalinvestigationbutaselementsofthoughtinquantum-mechanicalmodelsofdevices.Wemakeinformedguessestoofferaquantum-mechanicalmodelofa1-bitrecordingdeviceinametastablecondition.Probabilitiescalculatedfromthemodelfitanexperimentalrecordofanoscillationinatime-varyingprobability,showingatemperature-independentroleforPlanck’sconstantinwhatheretoforewasviewedasa“classical”electronicdevice.PACSnumbers:03.65.-w,03.65.Nk,03.65.Ta,84.30.Sk1I.INTRODUCTIONUbiquitousinphysicsistheneedtodecidewhichoftwosignalscamefirst.Besidesdirectmeasurements,suchaswhetheraparticlehasbeenemittedpriortoaclocktick,decisionsaboutwhichoftwosignalscomefirstpervadecomputers.Inthisconnectionweexperimentedsomeyearsagowithaflip-flopasitconfrontedaracebetweentwosignals.Asweseeitnow,theflip-flopisatoncea1-bitrecordingdeviceanda1-bitdecision-makingdevice.Insomeessentialuses,aflip-flop,afterbeingwrittenin,isreadtwice.Itisknownthatexpo-sureofaflip-floptoaracebetweenasignalandaclockticksometimesresultsinadisagreementbetweenthetworeadings:onereadingseesa0whiletheotherseesa1.Thefrequencyofdis-agreementsisreportedtodecreaseexponentiallywiththewaitingtimetbetweentheraceandthepairofreadings[1];oscillationoftheoutputvoltageofaflip-flophasalsobeenreported[2],buttothebestofourknowledgetheeffectofthisoscillationondisagreementshasnotbeendiscussed.Inanexperimentwithflip-flops(Texasinstruments7400and74S00)wefoundboththedecayinfrequencyofdisagreementsandanoscillationofthatfrequency.Becauseoftheoscillation,thedecreaseindisagreementfrequencyisnotmonotonic:insomecasesreadingsoonerbringslesschanceofdisagreementthanreadinglater.Aftermorestudyoftherecordingofdecisionsaboutwhichoftwoeventshappensfirst,were-alizedthatthistopicstraddlesaninterfacebetweenequationsofmotionandexperimentalrecords.Herewedescribethisinterfaceandsomeoftheconsequencesofrecognizingit.Amongtheseareaquantum-mechanicalmodeloftheflip-flopinitsdualroleofdecidingandrecording,showingaroleforPlanck’sconstantincomputercircuitry,evenatzerotemperature.A.AlanguageinterfaceTheoreticalphysicsemploysthemathematicallanguageofcontinuousfunctionstomodelmo-tion,anexamplebeingaquantum-mechanicalwavefunction.Experimentalevidence,however,showsdirectlyneithermotionnorcontinuousfunctions.Tobecomparedwiththeoreticalcal-culations,experimentsmustproducenumbers,andtheseresideinexperimentalrecords,aboutwhichonespeaksinadifferentlanguagealtogether.Thelanguageofrecords,sofarastheoryisconcerned,isthelanguageintroducedbyTuring,whichchopsmotionintodiscretemoves,interspersedbystillmomentsinwhicharecord—picturemarksonaTuringtape—canberead2unambiguously,exactlybecauseitisdevoidofmotion.Byinvokingprobabilities,quantummechanicsmakesroomforboththelanguageofwavefunctionsandthatofrecords:interpretedintermsofquantummechanics,experimentalrecordsexpressrelativefrequencies,andthecomparisonbetweenequationscontainingwavefunctionsandexperimentalrecordsisacomparisonbetweenprobabilitiescalculatedfromthewavefunctionsandexperimentalnumbersinterpretedasrelativefrequencies.B.OrganizationofthepaperSectionIIdisplaystheinterfacebetweenwave-functionlanguageandthelanguageofTuringtapesthatunderliesdescriptionsofexperimentalrecords.Asacategorydistinctfromprobabilitiescalculatedfromwavefunctionsandoperators,experimentalrecordswillbeexpressedmathemat-ically,soastodisplayrelativefrequenciesofdetectionsasthesedependonhowexperimenterssetvariousknobsoninstruments.Thenweextendquantumformalismtomakeexplicitthede-pendenceofwavefunctions—or,moregenerally,quantumstatesandoperators—onparameterslinkablebyassumptionstoexperimentalknobsettings.Asaconsequence,probabilitiescalcu-latedfromquantumstatesandoperatorsdependonknobparameters.Inthiswaytheinterfacebetweenwavefunctionsandexperimentalrecordsismadeintoaninterfacebetweenacategoryoffunctionsofknobsettingsexpressingprobabilitiescalculatedfromwavefunctionsandacate-goryoffunctionsofknobsettingsexpressingexperimentalrelativefrequencies.Puttingtheseinseparatecategoriesmakesprobabilitiescalculatedfromwavefunct