下载
加入VIP
  • 专属下载特权
  • 现金文档折扣购买
  • VIP免费专区
  • 千万文档免费下载

上传资料

关闭

关闭

关闭

封号提示

内容

首页 [14] An efficient propagation scheme for time-de…

[14] An efficient propagation scheme for time-dependent schroedinger equation in velocity gauge.pdf

[14] An efficient propagation s…

tingwhang
2012-11-20 0人阅读 举报 0 0 暂无简介

简介:本文档为《[14] An efficient propagation scheme for time-dependent schroedinger equation in velocity gaugepdf》,可适用于人文社科领域

LaserPhysics,Vol,No,,pp–OriginalTextCopyright©byAstro,LtdCopyright©byåÄàä“ç‡Û͇Interperiodica”(Russia)INTRODUCTIONNumericallysolvingthetimedependentSchrödingerequationonaspace–timegridhascontributedalottoourunderstandingofthebehaviorofatomsinstronglaserfieldsOneoftheadvantagesofthismethodisthatitmakesnoaprioriassumptionaboutthebehaviorofthewavefunction,andconvergingthecalculationbytakingthespatialgridspacingδrandthetemporalspacingδttoinprincipleproducesexactsolutionsEspeciallyinspacesofhigherdimensionality,however,suchconvergencemightnotbeeasytoobtainThisismuchunliketheonedimensionalcase,wherecurrentcomputingtechnologyprovidessuchanoverwhelmingpowerthatalmostanyalgorithmcanbeeasilyconvergedtomachineprecisionOnesuchpopularalgorithmisCrank–Nicholson,forpropagationonapositiongridAlsooftenusedaresplitoperatormethods,whichuseafastFouriertransformtoalternatebetweenpositionandmomentumrepresentations,wherepotentialandkineticenergybecometrivialandexactoperations,respectivelyTheonedimensionalcasecanbetreatedsoefficientlybecausetheoperationsofwhichthepropagatorisbuiltcanallberepresentedbybandmatricesofverynarrowwidth,usuallytridiagonalorbetterOperationsinvolvingsuchmatricesinvolveanamountofworkthatissimplyproportionaltothenumberofgridpointsProblemsinvolvingmorethanonespatialdimensionarenotsotrivial,andeveninmoderatelylargetwodimensionalproblems(suchasthecomputationofaphotoelectronspectrum),asinglesuboptimalchoiceinthealgorithmcaneasilydriveuptheamountofworkrequiredbyanorderofmagnitudethiscanbeenoughtopushtheproblemoutsidethereachofeventodayspowerfulsupercomputersObviously,thisiseventruerinthreedimensions:thehigherthepowertowhichapossibleinefficiencyfactorisraised,themoreitpaystobecleverTheaimofthispaperistofocusonseveralaspectsofthepropagationofawavefunctiononatwoormoredimensionalgrid,inordertosqueezeoutoptimalperformanceEveniftheindividualtricksleadtoagainofonlyafactortwo,anumberofsuchtrickscombinedcandriveupperformancebyseveralordersofmagnitudeThealgorithmpresentedheremakesitpossibletoruncalculationsthatareconvergedtoaboutfourdigitsonapersonalcomputerinreasonabletimes:TypicalperformanceonaPCwithaMHzPentiumCPUisμspergridpointpertimestep,whichamountstounderhforafullythreedimensionalcalculationtogeneratetheelectronspectrumofargonexposedtoacyclenmpulseofagivenhelicityandintensityItrequiresaboutmintodothesameforlinearpolarization(wheresymmetryreducestheproblemtoatwodimensionalone)GENERALCONSIDERATIONSInthissectionwewilldiscussanumberofissuesthatcanhavealargeimpactonthesizeofthediscreteformulationoftheproblem,andthusonefficiencyThechoiceofcoordinatesystem,representation(positionormomentum),andgaugecan,inprinciple,bemadeindependently,butonlyonecombinationofthepossiblealternativesleadstoaverylargereductionoftheproblemsizeCoordinatesForcalculationsonphotoionizationofatoms,anangularmomentumgridisclosetooptimalThereasonisthatelectronswithhighmomentaarecreatedatorneartheorigin,throughinteractionwiththeCoulombfieldofthenucleusAtlargedistances,theneedforahighdensityofgridpointsintheradialdirectionremains,butthetransverse(tangential)structureofthewavefunctionis“diluted”bytheradialexpansionSuchahighanisotropyinthetypicalscaleofthestructurecanonlybeexploitedtothefullbyorientingasmanycoordinateaxesaspossiblealongthesmoothestAnEfficientPropagationSchemefortheTimeDependentSchrödingerEquationintheVelocityGaugeHGMullerFOMInstituteforAtomicandMolecularPhysics,Kruislaan,Amsterdam,SJNetherlandsemail:mulleramolfamolfnlReceivedAugust,AbstractAnumericalalgorithmispresentedforsolvingthethreedimensionaltimedependentoneparticleSchrödingerequationofanatomsubjecttoelectromagneticradiationThealgorithmisoptimizedforefficiency,andtorunaconvergedcalculationofarealisticsituation,suchasanoblegasatomandanearinfraredlaser,takesaboutminonaPCApplicationtotrulythreedimensionalproblems,suchasphotoionizationbyfieldsofarbitrarypolarization,deliversanacceptableperformance(honatypicalPCforcalculatingthephotoelectronspectrumfromacyclepulse)STRONGFIELDPHENOMENALASERPHYSICSVolNoANEFFICIENTPROPAGATIONSCHEMEdirectionsWithcoordinatesystemsthatruninunnaturaldirections,evenanadaptivestepsizewouldnothelp:IneachcoordinatedirectiononewouldneedatleastenoughstepstorepresentvariationonthescalesetbythecomponentofthelocalwavevectoralongthatcoordinateaxisInthephotoionizationproblemthewavefunctionbecomesverysmoothperpendiculartotheradius,andtheonlywaytoexploitthisoptimallyistoputtwoofthethreecoordinates(atlarger)inthatdirectionThisleadsinanaturalwaytosphericalcoordinateswithradialandangularstepsthatstayconstantthroughtheentiregrid(TheradialstepsizeisdeterminedbythehighestmomentumofthephotoelectronsThisishardlydependentonradius,becausethephotoelectronsaretypicallymuchmoreenergeticthanthedepthoftheatomicpotentialwell)TheangulardimensionthencanusuallybedescribedwithsufficientaccuracywithL=–points,whilefortheradialdimensionthenumberofpointsNismorecommonly–(dependingonhowmuchofthecontinuumelectronwaveshouldberetainedforenergyanalysis)Eveninthetwodimensionalcasethegainistoordersofmagnitudecomparedtoasquarecartesiangridinthreedimensionsgoingtoasphericaldescriptioncancutthegridsizebyastaggeringorders,enoughtooutweighalmostanyconceivabledifficultythatmightbeassociatedwiththesphericaldescription(suchasthenastyformofthederivativeoperatorsinthosecoordinates)RepresentationUnlikeinclassicalmechanics,theentirephasespaceinquantummechanicsisdescribedbythecomplexwavefunctionofeitherpositionormomentumThefreedomtochoosebetweentheserepresentationscanbeusedtosimplifytheoperatorsoccurringintheHamiltonian:thekineticenergypandthepotentialenergyV(r)aretrivial(multiplicative,ie,diagonalinmatrixrepresentationonthegridbasis)operatorsinmomentumandpositionrepresentations,respectivelyHowever,pisaderivativeoperatorinpositionspace,whichcanbeapproximatedassomefinitedifferenceschemeinvolvingnear(est)neighbors(ie,a(tri)diagonalbandmatrix)Ontheotherhand,V(r)is,ingeneral,representedbyadensematrixinmomentumspacethatcannotbesimplifiedinanyway(exceptforsomeveryspecialandunrealisticcasessuchaszerorangepotentialsordeltafunctions)ApplyingsuchamatrixdirectlyinvolvesanamountofcomputationalworkthatgrowsasNand,foraninverse,evenasNThisisnevercompetitivewithtemporarilytransformingtothepositionrepresentationandapplyingV(r)there,sincethecostoftheFouriertransformneededtoshuttlebetweentherepresentationsisonlyNlogN,buteventhistransformhastovisiteachofthepointslogNtimes,leadingtoaboutmultiplicationswithacomplexbidiagonalmatrixfortransformingbothwaysoneachtimestep,eveninasimpleonedimensionalcaseThismoreorlessrulesouttheuseofamomentumrepresentationindimensionswhereapotentialispresent(ie,indimensionsonwhichthepotentialdepends)Intheabsenceofpotentials,amomentumdescriptioniseasier,andtheuseofangularmomentumvariableslandminsteadofanglevariablesϑandφsimplifiestheangularpartofthekineticenergyfromthenastyr(sinϑ)tothemultiplicativecentrifugalbarrierl(l)rIthastheadditionaladvantagethattheangularmomentumquantumnumbersarenaturallydiscrete,sothatnoapproximationshavetobeusedfortheangularoperatorsattheleveloftheHamiltonianWewillthusrepresentthewavefunctionas()wherethefactorriswrittenexplicitlytoreducetheradialpartofthedimensionalLaplaciantoasimplesecondderivativeactingontheψlmThisimpliesaboundaryconditionψlm()=,sothattheoriginisnotpartoftheradialgrid:thecontinuousvariablerisdiscretizedasrn=nδr(n∈�)Below,wewilluseψnasashorthandnotationforψ(rn)Incylindricallysymmetricproblems,misaconstantofthemotion,andthesecondsumin()isabsent,reducingtheproblemtoatwodimensionaloneThreedimensionalproblemswithasymmetryplanehaveareducednumberofindependentmterms:eithertheevenoroddtermsareabsentforsymmetryaboutthexyplaneOnlythepositivemtermsareneededforsymmetrywithrespecttothexzplane,becausetheyimplythevalueofthecorrespondingnegativemterms,whichareeitherequaloroppositeGaugeAnissuethatusuallyreceiveslessattentionisthechoiceofgaugeNevertheless,thischoiceisextremelyimportant,asisimmediatelyclearfromthe(phase)factoreiA(t)r,whichtransformsthewavefunctionfromonegaugetotheotherInaradiationfieldthisfactorishighlyoscillating(typicalpeakvaluesforA(t)are−au),bothasafunctionofspaceandtime,evenasafunctionofangleFormulatingtheprobleminthewronggaugecanthuscompletelyspoilthesimplificationofthesphericaldescription,reintroducingeverincreasingdetailintheangulardependenceofthe(unobservable)phaseofthewavefunctionwithgrowingr,wherethephysicallysignificantmagnitudewassmoothandsimpleCloserexaminationrevealsthattheculpritisthelengthgauge,wherethelaserinteractionisdescribedasE(t)×r,apotentialthatgrowstoarbitraryhighinstantaneousvaluesatlargerThishurts∂θϑcot∂ϑ∂φΨrϑϕ,,()rψlmr()Ylmϑϕ,(),mL–=L∑l=L∑=LASERPHYSICSVolNoMULLERquadratically,becauseboththenumberofrequiredtimestepsandtheangulardimensionofthegridgrowtovaluesthatareunrealisticallylargefromthepointofviewofthephysical(observable)behaviorEstimatesforthenumberofangularmomentaneededcaneasilybemadeinbothgaugesbasedonclassicalargumentsTheangularmomentumisdefinedasl=r×pInthemomentumgauge,pisaconstantofthemotion(andittypicallyrunsuptothemaximumofA(t),sayauforE=andω=)oncetheelectronhasescapedtheatomicpotentialOnlythepositionquiversasr=ptαsinωtwithatypicalquiveramplitudeα=EωofBohrThiscausesaboundedangularmomentumquiverΔl=Δr×pofunitsInthelengthgauge,however,pquiversasA(t),andthereisanothercontributionΔl=r×Δp,whichgrowsproportionaltorForacalculationthatrunsuptor=,forthepurposeofcalculatinganionizationyieldorharmonicspectrum,thisisnotyettoobad,butifwewanttoretainelectronsuptor=,forthepurposeofanalyzingtheirenergy,lhastorunuptoseveralthousandsInaddition,theinstantaneous(potential)energywouldreachuptoHartree,anorderofmagnitudelargerthanthekineticenergyofthephotoelectrons,andwewouldhavetoreducethetimestepoftheintegrationaccordinglyTHEPROPAGATIONSCHEMEAsmentionedintheprevioussection,acalculationthatinvolveslarger(suchasneededforthecalculationofelectronspectra)remainstractableonlyinthevelocitygaugeInthisgaugetheHamiltonianisgivenby()Atanyinstant,thisHamiltonianhascyclindersymmetryaroundthedirectionofthevectorpotentialA(t)Withoutlossofgenerality,wewillsupposethatA(t)generatedbythelaserisparalleltothezaxisofthesphericalcoordinatesystemInthiscase,thesubspacesofdifferentmareindependent,andcanbetreatedseparatelyForthisreason,wewillsometimessuppressthesubscriptmLater,wewilldiscusshowtoapplythecylindricalpropagatortoatrulythreedimensionalsituationwherethewavefunctionlackscylindricalsymmetryTheinteractionpartofHamiltonian()isgivenas()andcontainsaradialderivativeaswellasaϑdependenceTheϑdependenceofbothtermsin()couplesneighboringangularmomentaie,itobeystheselectionrulesΔl=±,Δm=Thevalueoftheangularcouplingsislandmdependent,()and()Theradialderivativeinvolvesatleastcouplingofnearestneighborsinr,(ie,Δn=±,wherenistheradialpointnumber),butisotherwiserindependentThesecondtermin()isdiagonalinr,(Δn=),butdoeshaverdependenceTogetherwiththetwopurelyradialcouplingsduetothekineticenergytermoftheatomicHamiltonian,thevelocitygaugeHamiltonianthereforecoupleseachgridpointtoeightneighbors(Figb)ItisnotpracticaltoapplytheCrank–NicholsonpropagatorUC–N(tτ,t)=(iHτ)–(–iHτ)()(whereτ=δt)directlyThegridpointsofamultidimensionalgridcannotbemappedintoa(onedimensional)vectorinsuchawaythatcouplingwithdistantneighborsisavoided,andcalculatingtheinverseofanythinglargerthanatridiagonalmatrixisfatalfromtheviewpointofefficiencyMappingofagridintoavectorsuchthattheHamiltonianbecomestridiagonalisonlypossibleifeachgridpointinteractswith,atmost,twoneighborsHpVr()At()p×=At()pziAt()∂z–iAt()ϑ∂rcosϑsinr∂ϑ–⎝⎠⎛⎞–==clmYlmϑcosYlm〈〉l()m–l()l()==slmYlmϑcos∂ϑYlm〈〉lclm==rlabFigTopologyofthecouplingsonan(n,l)grid(a)inthelengthgauge,and(b)inthevelocitygaugeLASERPHYSICSVolNoANEFFICIENTPROPAGATIONSCHEMELetus,forcomparison,digressforamomenttothelengthgauge(Foranexcellentdescriptionofthepropagationinthisgauge,see)There,theinteractionisgivenasE(t)z=E(t)rcosϑ,()andcouplingexistsbetweenneighborsthatdifferinrorl,butnotinboth(Figa)ThenumberofcoupledneighborsisthenonlyfourUndertheseconditions,theHamiltonianHcanbewrittenasthesumoftwotridiagonalmatrices,theatomicHamiltonianHatandtheradiationHamiltonianHint,containingtheradialandangularcoupling,respectivelyThisleadstothePeaceman–Rachfordapproximationforthepropagator,()Justlike(),thelocaltruncationerrorof()isoforderτ,(andtheglobalerrorforstepwiseintegrationoveragivenintervalisthereforeoforderτ)ThisiseventruefortimedependentHamiltonian,providedthetermsareevaluatedsymmetricallyaroundthecenterofthestepInthiscaseonlyHintistimedependent,andforthegivenorderingoftheoperatorsitisbesttoevaluateeachoperatoratthecorrespondingendpointofthetimeintervalie,Hint(tδt)onthefirstlineof(),Hint(t)onthesecondTherightmostfactorthencombineswiththeleftmostoftheprevioustimesteptoformaunitaryoperator(thetwocenterfactorsalreadydosoinsideasinglepropagationstep),sothatoverallpropagationisunitaryexceptfortheveryfirstandlastfactorsThevelocitygaugeHamiltonian,withitseightneighbortopology,mustbewrittenasasumof(atleast)fourtridiagonalmatricesLetusfirstremarkthatsplittingtheoperatorasdonein()isingeneralbeneficialforaccuracyAlthoughitintroducesanerrorcomparedtotheCrank–Nicholsonapproximationofthepropagator,thiserrorusuallycancelspartoftheerrorthisapproximationalreadycontainedwithrespecttothetruepropagatorThisisimmediatelyobviousifwesplitanoperatorintotwoequalparts:thisamountstointegrationwithhalfthestepsize,andreducestheerrorbyafactorofapproximately(determinedfromtheorderofthemethod)Theonlycaseinwhichabaderrorisintroducedoccurswhenwesplitanoperatorinlargebutoppositeparts(suchasinkineticandpotentialenergynearaCoulombsingularity,ordifferenttermsofonediscreteapproximationofaderivative)Splittingintopartsthatactindisjunctsubspacesisevenexact(theproductoftwosuchoperatorsvanishes)TherethusisnoneedtoshyawayfromsplittingtheHamiltonianintoalargenumberofpartsApartfromHat,(purelyradial)letusfirstdistinguishthepurelyangularcouplingHangduetorsinϑ∂ϑ,andthemixedcouplingHmixduetocosϑ∂rThenextstepistowriteeachofthemasasumovertheangularmomentumlofthosepartsoftheoperatorthatcoupleangularUP–Rtτt,()iHintτ()–iHatτ()–=×iHatτ–()iHintτ–()momentumltolThusHmix==−iA(t),whereLlisa×subblock()ofthematrixrepresentingcosϑintheangularmomentumbasis(ie,=PlHmixPlPlHmixPl,wherePlistheprojectoronangularmomentumsubspacel)ThestraightforwardgeneralizationofthePeaceman–Rachfordschemeforthepropagator,()isindeedaccurateuptoorderτThenumberoffactorsin()seemslarge,buteachofthefactorsactsinasubspaceofNpoints,against(L)NpointsforafullsizeoperatorsuchasHat(whichcouldbethoughtofasLindependentoperatorsactingonNpointspaces)orHang(Noperatorsactingon(L)pointspaces)Nevertheless,eachgridpointistouchedbytwodifferent,againstonlyoncebyHatorHang,whichisconsistentwiththeearlierremarkthatthetopologyofcouplingsforcesasplitintofouroperatorsFornoncommutingoperators()isnotnecessarilyunitary,sincetheHmixfactorscannotbecommutatedthroughtheotherfactorstobuildunitarycombinationsUnitarityisanextremelyimportantproperty,sinceitprotectsusfromexponentialblowupofunphysicalcomponentsinthewavefunctionthatareintroducedbyroundingerrors,nomatterhowlargethetimesteporhowstifftheequationsThereforeweareforcedtousetheslightlymoreinvolvedscheme()Thisschemehasthesameorderinτas(),andisactuallymoreaccurate,bec

用户评价(0)

关闭

新课改视野下建构高中语文教学实验成果报告(32KB)

抱歉,积分不足下载失败,请稍后再试!

提示

试读已结束,如需要继续阅读或者下载,敬请购买!

文档小程序码

使用微信“扫一扫”扫码寻找文档

1

打开微信

2

扫描小程序码

3

发布寻找信息

4

等待寻找结果

我知道了
评分:

/11

[14] An efficient propagation scheme for time-dependent schroedinger equation in velocity gauge

VIP

在线
客服

免费
邮箱

爱问共享资料服务号

扫描关注领取更多福利