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arXiv:quant-ph/0507243v126Jul2005Calculationsoftime-dependentobservablesinnon-Hermitianquantummechanics:TheproblemandapossiblesolutionIdoGilary,AvnerFleischerandNimrodMoiseyevFebruary1,2008DepartmentofChemistryandMinervaCenterforNonlinearPhysicsofComplexSystems,Technion–IsraelInstituteofTechnology,Haifa32000,Israel.AbstractThesolutionsofthetimeindependentSchr¨odingerequationfornon-Hermitian(NH)Hamiltonianshavebeenextensivelystudiedandcalcu-latedinmanydifferentfieldsofphysicsbyusingL2methodsthatorigi-nallyhavebeendevelopedforthecalculationsofboundstates.Theexist-ingnon-Hermitianformalismbreaksdownwhendealingwithwavepack-ets(WP).AnopenquestionishowtimedependentexpectationvaluescanbecalculatedwhentheHamiltonianisNH?UsingtheF-productfor-malism,whichwasrecentlyproposed,[J.Phys.Chem.,107,7181(2003)]wecalculatethetimedependentexpectationvaluesofdifferentobservablequantitiesforasimplewellknownstudytestcasemodelHamiltonian.Wecarryoutacomparisonbetweentheseresultswiththoseobtainedfromconventional(i.e.,Hermitian)quantummechanics(QM)calculations.TheremarkableagreementbetweentheseresultsemphasizesthefactthatintheNH-QM,unlikestandardQM,thereisnoneedtosplittheentirespaceintotworegions;i.e.,theinteractionregionanditssurrounding.Ourre-sultsopenadoorfornewtypeofWPpropagationcalculationswithintheNH-QMformalismthatuntilnowwereimpossible.Inparticularourworkisrelevanttothemanydifferentfieldsinphysicsandchemistrywherecomplexabsorbingpotentialsareintroducedinordertoreducetheprop-agationcalculationsintoarestrictedregioninspacewheretheartificialreflectionsfromtheedgeofthenumericalgrid/boxareavoided.11IntroductionInthelasttwodecadestherehasbeenanincreasinginterestinnon-Hermitian(NH)quantummechanics(QM)inmanydifferentfieldsofphysics.Inmanystud-iedcasesinatomic,molecularandnuclearphysicstheHamiltonianisNHduetotheuseofdifferenttypeofanalyticalcontinuationtransformations.Onesuchmethodiscomplexscaling(CS)whereˆH(r)→ˆH(reıθ)[1][2][3][4].Anothercommonmethodistheintroductionofcomplexabsorbinglocalen-ergyindependentpotentials(CAPs),alsoknownasopticalpotentials,intotheHamiltonian[5][6]suchthatˆH→ˆH+ˆVCAP.ThecomplexeigenvaluesandeigenfunctionsoftheNHHamiltonians,inthesestudies,werecalculatedusingsquare-integrablemethodsthatwereoriginallydevelopedforthecalculationofboundstates(seereferencestoreviewsmentionedabove).TheneedtoreplacethescalarproductwhichisusedinconventionalQMbyageneralizedinnerproductiswellunderstoodandhasbeendiscussedquiteextensivelyinthelit-erature.SeeforexamplethedifferentapproachesforgeneralizedinnerproductsinNH-QMinRef.[7]andRef.[8]andreferencestherein.However,forthesakeofclarityinSection2.1weprovideabriefaccountofthegeneralizedinnerproduct(socalledc-product[9])whichhasbeenuseduntilnowforcalculatingtimeindependentobservablesinNH-QM[4].Itisquitestraightforwardtoap-plythesamegeneralizeddefinitionoftheinnerproducttoNHtimeperiodicHamiltonians(seeforexampleRef.[4]),whichdescribetheinteractionofatomsormoleculeswithoscillatingelectricfield.TheapplicationoftheNHformalismtolaserdrivensystemhasbeenveryusefulinexplainingphysicalphenomena.Forexample,remarkableagreementhasbeenachievedbetweentheresultsob-tainedfromtheoreticalNH-QMcalculationsandtheexperimentalresultsoftheprobabilityofgeneratingoddhigh-orderharmonics(i.e.,generationofhighen-ergyphotonsresultingfromtheabsorbtionofmanylowenergyphotons)[10].However,oneimportantproblemremainsopen.Howtocalculatetheentirehigh-orderharmonicgenerationspectraandnotonlytheprobabilitiestoobtainevenoroddharmonics?Thisquestionisrelatedtoamoregeneralquestion.HowtocalculatetimedependentexpectationvaluesinNH-QM?InNH-QMthepropagatedrightwavepacket(WP)isthesolutionofthetimedependentSchr¨odingerequation(TDSE)whereˆHisaNHHamiltonian.ˆHΨ(r,t)=ı¯h∂∂tΨ(r,t).(1)ExtensiveWPpropagationcalculationswerecarriedoutforcomplexpotentialenergysurfacesbysolvingtheNH-TDSE(seeforexampleRefs.[11,12,13,14,15]).However,inordertocalculatetimedependentexpectationvaluesoneneedstocarryoutWPpropagationcalculationsfromt=0tot=−∞.TheproblemisthatinNH-QMitisimpossibletopropagatetheinitialWPfromt=0tot=−∞.WebrieflyreviewthisknownprobleminSection2.2andrelateittotheconceptoftimeasymmetryinQMwhichisasubjectofnumerousstudiesintheliterature(seeRefs.[16,17,18,19]andreferencestherein).VeryrecentlyapossiblesolutiontothetimeasymmetryprobleminNH-QM2wasproposedbyMoiseyevandLein[20].Thesolutionrequiresmodificationofthecurrentlyusedinnerproduct.AswewillshowinSection3thismodificationofthewellknowngeneralizedinnerproduct(brieflyreviewedinSection2.1)whichwewillterm”F-product”(FP)isapplicableforcaseswheretheHamil-tonianiseitheratime-independentortimedependentNHoperator.NaturallythismodificationofthegeneralizedinnerproductreducestotheconventionalscalarproductwhentheHamiltonianisHermitian.MoiseyevandLein[20]haveshownthatusingthemodifiedgeneralizeddefinitionoftheinnerprod-uctnotonlysolvesthetimeasymmetryprobleminNH-QM,butalsohasaveryimportantphysicalimplications.Theyhaveshownthattheanalyticalexpressionforthehigh-orderharmonicgenerationspectra(HHGS)whichhasbeenderivedwithintheframe