Dissipation in Quantum Mechanics, Scalar and Vecto

arXiv:quant-ph/0610133v117Oct2006DissipationinQuantumMechanics,ScalarandVectorFieldTheoryF.Kheirandish1∗andM.Amooshahi1†1DepartmentofPhysics,UniversityofIsfahan,HezarJaribAve.,Isfahan,Iran.February9,2008AbstractAnewminimalcouplingmethodisintroduced.Ageneraldissipa-tivequantumsystemisinvestigatedconsistentlyandsystematically.Somecouplingfunctionsdescribingtheinteractionbetweenthesys-temandtheenvironmentareintroduced.Basedoncouplingfunc-tions,somesusceptibilityfunctionsareattributedtotheenvironmentexplecitly.Transitionprobabilitiesrelatingthewayenergyflowsfromthesystemtotheenvironmentarecalculatedandtheenergyconserva-tionisexplecitlyexamined.Thisnewformalismisgeneralizedtothedissipativescalarandvectorfieldtheoriesalongtheideasdevelopedforthequantumdissipativesystems.Keywords:DissipativeQuantumSystems,FieldQuantiza-tion,Environment,ScalarandVectorFields,CouplingFunc-tions,TransitionProbabilities1IntroductionTherearebasicallytwoapproachestostudydissipativequantumsystems.Oneisfoundintheinteractionsbetweentwosystemsviaanirreversibleen-ergyflow[1,2].Thesecondapproachisaphenomenologicaltreatmentunder∗fardin−[e-mail]†[e-mail]1theassumptionofnonconservativeforces[3,4].Instudyingnonconservativesystems,itisessentialtointroduceatimedependentHamiltonianwhichde-scribesthedampedmotion.Suchaphenomenologicalapproachforthestudyofdissipativequantumsystems,especiallyadampedharmonicoscillator,hasaratherlonghistory.CaldirolaandKanai[5,6]adoptedtheHamiltonianH(t)=e−βtp22m+eβt12mω20q2,(1)whichleadsexactlytotheclassicalequationofmotionofadampedharmonicoscillator¨q+β˙q+ω20q=0.(2)Thequantumaspectofthismodelhasbeenstudiedinagreatamountofliterature.Inthosestudiessomepeculiaritiesofthismodelandsomefea-turesofithaveappearedtobeambiguous[7]-[15].TherearesignificantdifficultiesinobtainingthequantummechanicalsolutionsfortheCaldirola-KanaiHamiltonian.QuantizationwiththisHamiltonianviolatestheuncer-taintyrelations,i.e.,theuncertaintyrelationsvanishastimegoestoinfinity[16]-[19].BasedonCaldirola-kanaiHamiltonian,ithasbeenconstructedequivalenttheoriesbyperformingaquantumcanonicaltransformationandhasbeenusedthepathintegraltechniquestocalculatetheexactpropagatorsofsuchtheories,alsothetimeevolutionofgiveninitialwavefunctions,havebeenstudiedusingtheobtainedpropagators[20].IntheframeworkofthephenomenologicalapproachLopezandGonza-lez[21]havetakentheexternalnonconservativeforcesthathaslinearandquadraticdependencewithrespecttovelocity.TheyhavededucedclassicalconstantsofmotionandHamiltonianforthesesystemsandeigenvaluesoftheseconstantshavebeenquantizedthroughperturbationtheory.Asimplepseudo-HamiltonianformulationhasbeenproposedbyKupriyanovandetal[22].Startingfromthispseudo-Hamiltonianformulationacon-sistentdeformationquantizationhasbeendevelopedthatinvolveanon-stationarystarproductandanextendedoperatoroftimederivativedif-ferentiatingthestarproduct.Acompleteconsistentquantum-mechanicaldescriptionforanylineardynamicalsystemwithorwithoutdissipationhasbeenconstructedinthisscheme.Inanotherapproachtoquantumdissipativesystems,onetriestobringaboutthedissipationasaresultsofanaveragingoverallthecoordinatesofthebath2system,whereoneconsidersthewholesystemascomposedoftwoparts,ourmainsystemandthebathsystemwhichinteractswiththemainsystemandcausesthedissipationofenergyonit[23]-[30].Themacroscopicdescriptionofaquantumparticlewithpassivedissipationandmovinginanexternalpotentialv(~x)isformulatedintermsofLangevin-Schr¨odingerequation[31,32,33]m¨x+Zt0dt′μ(t−t′)˙~x(t′)=−~∇v(~x)+FN(t).(3)Thecouplingwiththeheat-bathinmicroscopiclevelscorrespondstotwotermsinmacroscopicdescription.Ameanforcecharacterizedbyamemoryfunctionμ(t)andanoperatorvaluedrandomforceFN(t).Thesetwotermshaveafluctuation-dissipationrelationandbotharerequiredforaconsistentquantummechanicaldescriptionoftheparticle.In[33],therearesomemodelsforinteractionofthemainsystemwiththeheat-bathwhichleadtothemacroscopicLangevin-Schr¨odingerequation.Thelayoutofthepaperisasfollows:Insection2,weconsideraparti-clemovinginaone-dimensionalexternalpotentialandtaketheabsorptiveenvironmentoftheparticleasaKlein-Gordonfieldwhichinteractwiththemomentumoftheparticlethroughaminimalcouplingterm.Inthisap-proach,theLangevin-Schrodingerequation(3)isobtainedasthemacroscopicequationofmotionoftheparticleandthenoiseforceFN(t),isderivedintermsofthecouplingfunctionandtheladderoperatorsoftheenvironment.Bychoosingaspecialformforthecouplingfunction,africtionforcepropor-tionaltothevelocityoftheparticleisobtained.Foraninitiallyexitedonedimensionaldampedharmonicoscillator,itisshownthattheentireenergyoftheoscillatorwillbeabsorbedbyitsenvironment.Insection3,wearecon-cernedwithaparticlemovinginthree-dimensionalabsorptiveenvironment.InthissectionwemodeltheenvironmentbytwoindependentKlein-GordonfieldsBand˜B.ThefieldBinteractswiththemomentumoftheparticlethroughaminimalcouplingtermandthefield˜B,interactswiththepositionoperatoroftheparticlesimilartoadipoleinteractionterm.AgeneralizedLangevin-Schrodingerequationisobtainedwhichcontainstwomemoryfunc-tionsdes