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1、PHYSICALREVIEWAVOLUME43,NUMBER215JANUARY1991Quantum-classicalcorrespondenceandquantumchaosintheperiodicallykickedpendulumBoonLeongLanandRonaldF.FoxSchoolofPhysics,GeorgiaInstituteofTechnology,Atlanta,Georgia30332(Received27August1990)Thebehavioroftheperiodicallykickedpendulumisdeterminedclass
2、icallyandquantummechanically.Ourearlieranalyticstudy[Phys.Rev.A41,2952(1990)]isconfirmedandextendedbythenumericalresultspresentedhere.Weareabletoconstructminimumuncertainty,Gaussianwavepacketsthatmaintaintheirform,andshadowthecorrespondingclassicalmotionforlongtimeswhentheclassicalmotionisnon
3、chaoticoronlyweaklychaotic.However,whentheclassicalmotionisrobustlychaotic,thequantum-mechanicalprobabilitydistributiongrowsmacroscopicallybroadexponentiallyfastandthequantum-classicalcorrespondencebreaksdown.I.INTRODUCTIONvaluesofthepositionandmomentumarestudiedalongwiththeirclassicalcounter
4、partsastheytimeevolvefromItiswellestablishedthatclassicalmechanicsonlythesameinitialconditions.Initially,themeanvaluesworkswellforparticlesmovingatspeedsmuchlessthanneedtobelargecomparedtotheirrespectiveroot-mean-c,thespeedoflightinfreespace,whereasrelativisticsquaredeviations(squarerootofvar
5、iances)sothatthemechanicsisvalidforallparticlespeedsboundedbyc.quantumstateisoneofwell-definedpositionandmomen-Therelativistic-classicalcorrespondenceholds;i.e.,thetum,mimickingtheinitialclassicalstate.InordertoquantitativepredictionsofthetwotheoriesagreewhensatisfyHeisenberg'suncertaintyprin
6、ciple,thebestoneU&&c;thesmallerUiscomparedtoc,thebetterthecandoisconstructaminimumuncertaintyGaussianagreement.Itisalsowellestablishedthatclassicalwavepacketfortheinitialquantumstate.Formechanicsprovidesavaliddescriptionwhenevertheac-correspondence,themeanvaluesshouldfollowthesolu-tionofasyst
7、emislargecomparedtoA,whilequantumtiontoHamilton'sequationsofmotiontoagoodapprox-mechanicsistheunderlyingtheory.Inthedomainwhereimation(weshallrefertothisasshadowing)andtheirtheclassicaldescriptionisvalid,i.e.,actionmuchgreaterrespectivermsdev
