The Finite Difference Time Domain Method for Compu

TheFiniteDifferenceTimeDomainMethodforComputingSingle-ParticleDensityMatrixIWayanSudiarta1,*andDJWallaceGeldart1,21DepartmentofPhysicsandAtmosphericScience,DalhousieUniversity,Halifax,NSB3H3J5,Canada2SchoolofPhysics,UniversityofNewSouthWales,Sydney,NSW2052,AustraliaE-mail:[e-mail]fornumericalcomputationofthethermaldensitymatrixofasingle-particlequantumsystemispresented.TheSchrödingerequationinimaginarytimeτissolvednumericallybythefinitedifferencetimedomain(FDTD)methodusingasetofinitialwavefunctionsat0=τ.Bychoosingthisinitialsetappropriately,thesetofwavefunctionsgeneratedbytheFDTDmethodcanbeusedtoconstructthethermaldensitymatrix.Thetheoreticalbasisofthemethod,anumericalalgorithmforitsimplementation,andillustrativeexamplesinone,twoandthreedimensionsaregiveninthispaper.Thenumericalresultsshowthattheprocedureisefficientandaccuratelydeterminesthedensitymatrixandthermodynamicpropertiesofsingle-particlesystems.Extensionsofthemethodtomoregeneralcasesarebrieflyindicated.Keywords:densityoperator(densitymatrix);numericalmethods;finitetemperature;finitedifferencemethods1.IntroductionAquantumsystemcanbefullydescribedbyasinglestatevectoronlyifthesystemisinapurestate.IsolatedatomsormoleculesineigenstatesoftheirHamiltonianareamongthecommonexamplesofsuchsystems.Inpractice,agreatmanyphysicalsystemsofinterestarenotinpurestatesbutinsteadareinmixedstates.Thisrequiresadescriptionintermsofastatisticaloperatorordensitymatrix[1].Time-independentsystemsinthermalequilibriumwithaheatbathatconstanttemperatureTareimportantpracticalexamplesofmixedstatesrequiringadensitymatrixdescription.Thedensitymatrixofasysteminthermalequilibriumcanbespecifiedbyitsmatrixelementsinthepositionrepresentation[1],nnnnRRRRρφφβρ∑∞=∗′=′0)()(),,((1)*CorrespondingAuthor.E-mail:[e-mail]1CoordinatevectorsaredenotedbyRandspindegreesoffreedomwillbesuppressedforsimplicity.andnE)(Rnφarethetheenergyeigenvaluesandeigenfunctionsofthetime-independentSchrödingerequation,)()(RERnnnφφ=ˆHTkB/1=βwiththeBoltzmannconstant,Bk)(/}][exp{ββρZEn−n=andisthepartitionfunction.∑∞==0)(nZβ−}exp{nEβSolvingtheSchrödingerequationtoobtainallenergyeigenvaluesandwavefunctionsofageneralmany-particlesystemisobviouslynotfeasible.AnalyticalsolutionsoftheSchrödingerequationareavailableonlyforarelativelysmallnumberofidealizedcases.Forpracticalapplications,alternativenumericalmethodsareneededtodeterminethedensitymatrix,evenforone-particlesystems.Themosteffectiveprocedureforaparticularproblemdependsonthenumberofparticlesinthesystem.NForsystemsoflargeparticlenumber,thepathintegralMonteCarlo(PIMC)method[2,3]isveryeffectiveduetothehighefficiencyofstochasticmethodsforsamplinghighdimensionalconfigurationspaceswhencomputingmultidimensionalintegrals[4].However,therearetwodisadvantagesofthePIMCmethod.Thetreatmentofnodesofmany-fermionwavefunctionsisapproximate(thesignproblem)[5].Also,convergencethePIMCmethodisratherslowsincevariancesareoforderCN/1whereisthenumberofsampledconfigurations.Accurateresultsthenrequireaveragingoveralargenumberofconfigurationswithcorrespondinglylongsimulationtimes.Inspiteofthesedisadvantages,PIMCisstillindispensableforverylargeparticlenumberduetotheefficiencyofstochasticmethods.CNHowever,ourpresentinterestisfocusedontheoppositeextremeofverysmallparticlenumber.Wearemainlymotivatedbyfew-electronsystemsinreduceddimensionality;quantumwires,transistordevices,heterojunctions,andquantumdots.Thesesystemsarestronglyinhomogeneousduetoconfiningpotentialsandthenumberofelectronscanbeaslowasoneortwoinsomemodernelectronicdevices.Protonhoppingbetweenquasi-one-dimensionalpotentialwellsinhydrogen-bondedchainsisanotherapplicationofthesamecharacter.Themajorfeaturesharedbythesesystemsisthestronginhomogeneityofthepotentialsconfiningtheparticles.Forsystemswithonlyfewparticles,especiallyinlowdimension,othercomputationalproceduressuchasthediscretizedpathintegral(DPI)method[6]orthefinitedifferencetimedomain(FDTD)method[7-9]becomepractical.ThesecanbemoreaccuratethanthePIMCmethodsincetheyarenotbasedonslowlyconvergentstochasticsampling.TheDPImethodforcomputingthedensitymatrixrequiresmanipulationofadensitymatrixwithelements(whereisthenumberofgridpoints).ThedensitymatrixcanalsobecomputedbyconvertingEq.(1)toadifferentialequation(Blochequation)[10].ThiswellknownBlochequationcanbesolvedbyFDTDmethods.Thisrequiresmanipulationofamatrixwithelements,justasintheDPImethod.ggNN×ggNN×gNThepurposeofthispaperistopresentaFDTDprocedureforthenumericalcomputationofthedensitymatrixusingasetofwavefunctionsobtainedbyintegratingtheSchrodingerequationinimaginarytime,startingfromasuitablesetofinitialfunctionsat0=τ.ThisuseofwavefunctionscanresultinanefficientalternativetotheDPImethodsincecomputationwithadiscretizedwavefunctionrequiresonlyelementsascomparedtogNggNN×elementsforthe2matrixelementsintheDPImethod.Itisimportanttoemphasizethattheinitialwavefunctionsarenottheenergyeigenfunctionsofthesystem.NoinformationonandnE)(Rnφisrequired.Theprocedureforselectingthesetofwavefunctionstobeusedandtheproofthattheres