Perepelitsa MIT Department of Physics 70 Amherst Ave Cambridge MA 02142 Abstract We present the path integral formulation of quantum mechani cs and demon strate its equivalence to the Schr57512odinger picture We appl y the method to the free particl ID: 22613
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PathIntegralsinQuantumMechanicsDennisV.PerepelitsaMITDepartmentofPhysics70AmherstAve.Cambridge,MA02142AbstractWepresentthepathintegralformulationofquantummechanicsanddemon-strateitsequivalencetotheSchrodingerpicture.Weapplythemethodtothefreeparticleandquantumharmonicoscillator,investigatetheEuclideanpathintegral,anddiscussotherapplications.1IntroductionAfundamentalquestioninquantummechanicsishowdoesthestateofaparticleevolvewithtime?Thatis,thedeterminationthetime-evolutionj (t)iofsomeinitialstatej (t0)i.Quantummechanicsisfullypredictive[3]inthesensethatinitialconditionsandknowledgeofthepotentialoccupiedbytheparticleisenoughtofullyspecifythestateoftheparticleforallfuturetimes.1Intheearlytwentiethcentury,ErwinSchrodingerderivedanequationspeci eshowtheinstantaneouschangeinthewavefunctiond dtj (t)idependsonthesysteminhabitedbythestateintheformoftheHamiltonian.Inthisformulation,theeigenstatesoftheHamiltonianplayanimportantrole,sincetheirtime-evolutioniseasytocalculate(i.e.theyarestationary).Awell-establishedmethodofsolution,aftertheentireeigenspectrumof^Hisknown,istodecomposetheinitialstateintothiseigenbasis,applytimeevolutiontoeachandthenreassembletheeigenstates.Thatis, 1Intheanalysisbelow,weconsideronlythepositionofaparticle,andnotanyotherquantumpropertysuchasspin. 2D.V.Perepelitsaj (t)i=n=1Xn=0exp[ iEnt=~]hnj (t0)ijni(1)This(Hamiltonian)formulationworksinmanycases.Inclassicalmechanics,however,theLagrangianformulationisknowntobeequivalenttotheHamiltonianone.Thus,weseekananswertotheabovequestionthatreliesonsomeanalogueoftheLagrangianaction.In1920,P.A.M.Diracmadeamysteriouscommenttothise ect,whichlaterinspiredRichardFeynman.Consideratrajectoryx(t)betweenaninitialpoint(x0;t0)andpossiblefuturepoint(x0;t0).Letthetransitionprobabilityamplitudeh (x;t)j (x0;t0)ibetheinnerproductofthewavefunctionintheSchrodingerpictureoftheparticleevaluatedatthesetwopoints.Feynmanhintedatthe\equivalence"oftheprobabilityamplitudeandtheexponentoftheclassicalactionofthetrajectoryexp[iS[x(t)]=~],where\equivalence"isnotyetwell-de ned.Itwasnotuntil1948thatFeynman,asapost-doctoralstudentatPrinceton,formalizedthisconnection.Inhislandmarkpaper[4],Feynmanpresentedaformulationofquantummechanicsbasedonthisprinciple.Letagiventrajectoryx(t)beassociatedwithatransitionprobabilityamplitudewiththesameformasthatgivenbyDirac.Ofcourse,byquantummechanics,wecannotspeakoftheparticletakinganywell-de nedtrajectorybetweentwopoints(x0;t0)and(x0;t0).Instead,wecanonlyspeakoftheprobabilityof ndingtheparticleattheselocations,whichisrelatedtowavefunctionsj (x0;t0)iandj (x0;t0)i.Thatis,allthatcanbedeterminedistherelativeprobabilityoftheparticletakingonepathoranother.Feynman'sinsightwasthis-thetotaltransitionprobabilityamplitudecanbeobtainedbysummingtheamplitudesoftheparticlehavingtakenanyindividualpath.Ifthequantityh (x0;t0)j (x0;t0)icanbecalculatedinthemethodsuggestedbyFeynman,thetime-evolutionofthestatecanbedeterminedbyconsideringcon-tributionsfromallpossiblefuturestates,andtheproblemissolved.Below,theketsareeigenstatesofthepositionoperator,suchthatintegrationoverallxspanstheentirebasis.j (x;t0)i=Z1 1h (x0;t0)j (x0;t0)idx0j (x0;t0)i(2)Knownasthepathintegralformulationofquantummechanics,thismethodgivesthesameresultsasthosedictatedbytheSchrodingerpicture,butalsoilluminatessomeofthedeeperaspectsofquantummechanics.Inthispaper,wewillpresentthemethodusedbyFeynman.Thoughitispedagogicallybackward,wewillthendemonstrate PathIntegralsinQuantumMechanics3theuseofthemethodbeforeshowingitsequivalencetotheSchrodingerpicture.Wewilltheninvestigatethemethodasappliedtotheharmonicoscillator.Followingthis,wewillintroducetheconceptofEuclideanpathintegralsanddiscussfurtherusesofthepathintegralformulationinthe eldofstatisticalmechanics.2PathIntegralMethodDe nethepropagatorofaquantumsystembetweentwospacetimepoints(x0;t0)and(x0;t0)tobetheprobabilitytransitionamplitudebetweenthewavefunctionevaluatedatthosepoints.U(x0;t0;x0;t0)=h (x0;t0)j (x0;t0)i(3)IftheHamiltoniancarriesnoexplicittime-dependence,wecanrelabelthe rsttime-valuet0=0andworkonlywithelapsedtimet=t0 t0.Wewilloftenwrite(3)asU(x0;t;x0)toillustratethis.Thepropagatorabove,alongwithaninitialstateket,fullydescribestheevolutionofasystemovertime.Itisalsocustomary,asisdoneinSakurai[2],touseherethesymbolKinsteadofUandrefertoasthe\kernel"or\Feynmankernel".Thepathintegralmethod,asweareabouttosee,isanexplicitwaytoconstructthispropagator.Weconsiderpossibletrajectoriesx(t)ofaparticlemovingthroughatime-independentpotentialV(x)withendpoints xedat(x0;t0)and(x0;t0).2Anin nitecontinuumofsuchtrajectoriesarepossible,eachwithclassicalactionS[x(t)].Feynman[4]positsthatthecontributiontothepropagatorfromaparticulartrajectoryisexp[iS[x(t)]=~].Thatis,everypossiblepathcontributeswithequalamplitudetothepropagator,butwithaphaserelatedtotheclassicalaction.Summingoverallpossibletrajectories,wearriveatthepropagator.ThenormalizationconstantA(t)isindependentofanyindividualpathandthereforedependsonlyontime.U(x0;t;x0)=A(t)Xalltrajectoriesexpi ~S[x(t)](4)Equation(4)istheheartofthepathintegralformulation.Howthissummation(whichhasyettobewell-de ned)istoarriveatthecorrectpropagatorisfarfrom 2Atthispoint,x(t)coulddescribeatrajectorythroughspacetimewithanynumberofdimensions.Forourpurposes,however,theparticlemovesalongonespatialdimension,thoughageneralizationtomoreisstraightforward. 4D.V.Perepelitsaobvious.Wenowdiscusssomesalientfeaturesof(4)anddiveintothetechnicaldetailsofasimpleexample.2.1TheClassicalActionHowcanitbethatthein nitesumabovedoesnotdiverge?Thedi erentphasesarethekeytothis.Fortrajectoriesbetweenwhichtheactiondi ersbyS~,thecorrespondingcontributionswillcancel.Contributionstothepropagatorfrompossibletrajectoriesthrougharegionfarawayfromtheclassicalpathwould,intheaggregate,cancel.Denotetheclassicaltrajectoryxcl(t)asthetrajectorywiththeminimumvalueoftheactionS[xcl],whichisstationaryto rstorderwithregardtodeviations.Onthemacroscopicsystem,thisisthetrajectoryobservedwithverylittleuncertainty.InvestigationofEquation(4)givesareasonforthis.Trajectoriesclosetotheclassicalonecauseno rst-orderdeviationintheaction,andcontributewithcoherentphasetotheintegral.Trajectorieswithaction~morethantheclassicalactionareoutofphase,andinterferedestructivelywitheachother.Integratingovermoreandmoresuchtrajectoriesshouldcausetheircontributiontoaverageouttozero.Inthisway,theclassicaltrajectoryisqualitativelyimportant.Ingeneral,theregionofcoherenceisrelatedtothe\classical"natureofthesystem.Onthe(macro-scopic)classicalscale,~isafrighteninglysmallamount,makingtheprincipalcon-tributingtrajectoriesthoseinanarrowbandaroundtheclassicalone.Onthequan-tumscale,however,theactionissmallenoughthat~isenoughtoallowsigni cantquantumdeviationsfromtheclassicaltrajectory.Intuitively,thiscorrespondstothefundamentaluncertaintyintheparticle'spositionatanygiventime.Shankar[3]brie\rygivesamorequantitativeargumentofthedi erencesinthetwocases.2.2FreeParticleForconcreteness,weusethemethodabovetodeterminethepropagatorforthesim-plestofsystems-aparticlemovinginfreespacealongonedimension.Inthiscase,wewillactuallyevaluatetheintegralgivenabove\overallpossiblepaths",although,asMacKenzie[5]notesafterasimilarderivation,oftenmuchofthestudyofthepathintegralformulationisconcernedwithhowtoavoidjustthis.Letx(t)describeapotentialtrajectoryfrom(xa;ta)to(xb;tb).Wediscretizethetrajectorybydividingitintochunksoftimet=(tb ta) N,suchthattheintermediate PathIntegralsinQuantumMechanics5pointsare(x1;t1),:::,(xN 1;tN 1).WedothiswiththehopethatinthelimitasN!1,thismodelsacontinuouspath.3AsV(x)=0forafreeparticle,theactiondependsonlyonthevelocity,whichbetweenanytiandti+1=ti+tisaconstant.Wedenotetheactionbetweentiandti+1bySi=Zti+1tim 2_x(t)2dt=m 2xi+1 xi ti+1 ti2(ti+1 ti)=m 2t(xi+1 xi)2(5)Todescribeeverypathfrom(x0;t0)to(xN;tN),wesimplyvaryeachintermediatepointxiateachtiovertheentiredomain.Asthefxigeachtakeonacontinuumofvaluesfrom 1to1,thesumofthecontributionsfromeverypathish (xN;tN)j (x0;t0)i=A(t)Z1 1Z1 1exp"i ~m 2ti=NXi=1(x2i x2i 1)#dx1dxN 1(6)HereA(t)isthenormalizationconstantthatdependsonlyontheelapsedtimetN t0.Wecanevaluate(6)byintegratingoveronevariableatatime.Forconvenience,letk=im 2~t.Integrationoverthe rstvariableyieldsZ1 1expk(x22 x21)+k(x21 x20))dx1(7)=Z1 1exp2kx21+kx1( 2x2 2x0)+k(x22+x20)dx1(8)=p p 2p kexpk 2(x2 x0)2(9)Weignoretheconstantterm,sinceitisabsorbedintothenormalizationconstantA.Afterafewmoreintegrations,apatternemergeswhichFeynmanandHibbs[1]illustratemoreexplicitly:integratingoverthe rstnspacetimepointsleavesafactorofk n+1(xn x0)2intheexponentafterevaluation.WeevaluateN 2moreintegrationsafterthe rstone,andusethefactthatNt=(tN t0)torewriteexpression(6)conciselyas 3Notethattheconstructedaccelerationisadiscontinuousimpulsetrainofinstantaneousjumpsinthevelocity.SincetheLagrangiandoesnotvaryexplicitlywithx,andthatinthelimitthisvalueiscontinuousanyway,weignorethis. 6D.V.PerepelitsaU(xN;tN;x0;t0)=A(tN t0)expk N(xN x0)2(10)U(x;t;x0)=A(t)expim 2t~(x x0)2(11)ThevalueforAisobtainedbynormalizing(11)overallxwhileholdingtconstant.The nalresultisU(x;t;x0)=r m 2i~texpim 2t~(x x0)2;(12)whichisinfactthepropagatorforafreeparticle.AfascinatingfeatureofthisresultisthatUiscomposedofthecontributionfromtheclassicalpathalone.Thisisnotalwaysthecase,andwewilldiscussthisphenomenoninSection3.2.3EquivalencetoSchrodingerPictureButisthisnewformulationreallyanequivalentpictureofnon-relativisticquantummechanics?Schrodinger'sequationindi erentialformde nesthetime-derivativeofagiveninitialstatej (x0;t0)i.i~d dtj (x0;t0)i=^Hj (x0;t0)i= ~2 2md2 dx2+V(x0)j (x0;t0)i(13)Weattempttorecreatethisdi erentialequationusingthepathintegralformulation.Consideraninitialstatej (x0;t0)i.Somein nitesimaltime later4,andatsomein nitesimalposition\raway,thestatecanbedescribedasfollows.j (x;t0+ )i=Z1 1d\rj (x0+\r;t0+ )iU(x0+\r; ;x0)(14)TocalculatethepropagatorU=h (x0+\r;t0+ )j (x0;t0)i,weneedtodeterminetheaction.Foragivenpositioncoordinatex0+\rattime ,theparticle'svelocityandpositioncanbeapproximatedas1 (\r)andx+\r 2,respectively.Thepropagatoris 4Oureventualintentionistoallow !0torecreatetheinstantaneousderivative.Thus,wefeelnoqualmmakinganyjustifyingassumptionsthatareexactinthelimit. PathIntegralsinQuantumMechanics7U(x0+\r; ;x0)=A( )expi ~Z 0dtm 2_x(t)2 V(x(t))(15)A( )expi ~m 2 \r2 Vx0+\r 2 (16)Inthelimitas (andtherefore\r)becomezero,theapproximationabovebecomesexact.Therefore,weexpandseveralofthequantitiesin(16)inpowersof and\r.Theexponentialofthepotentialenergy,andthepotentialenergyitselfarei ~exph Vx0+\r 2i=1 i ~Vx0+\r 2+:::(17)Vx0+\r 2=V(x0)+\rd d\rV(x0)+:::(18)Wekeepthe rsttwotermsof(17)andonlythe rsttermof(18),whichhasthee ectofdiscardingalltermssecondorderin( ;\r)andhigher.Additionally,wecanexpandtheketintheintegrandj (x0+\r;tiaroundx0.The rstandthirdtermsaresigni cant,butthesecondtermresultsinanoddfunctionwithinthe(symmetric)integrand,sowediscarditasin[1].j (x0+\r;t0)i=j (x0;t0)i+\rdj (x0;t0)i dx+1 2\r2d2j (x0;t0)i dx2+:::(19)Theapproximateexpressionfor(14)becomesj (x;t0+ )i=A( )1 i ~V(x0)Z1 1d\rj i+1 2\r2d2j i dx2expi ~m 2 \r2(20)ItisstraightforwardtoevaluatethisGaussian.Integrating,andkeepingthe rstordertermsinthein nitesimalsgivesj (x0;t0+ )i=A( )r 2~ im1 i ~V(x0)+~ 2mid2 dx2j (x0;t0)i(21)Forsmall ,weseethatthe rsttermontherightmustequaltheexpressionontheleft,andthatthenormalizationconstantisnecessarilyA( )=p m 2~i .Torecoverthetimederivative,wecanrearrange(21)andtakethelimitas !0ofi[j (x;t0+ )i j (x;t0)i]=(~ ). 8D.V.Perepelitsai~d dtj (x;t0)i=V(x0) ~2 2md2 dx2j (x0;t0)i(22)Thisisjust(13),thetime-dependentSchrodingerEquation,derivedfromapathintegralformulationofquantummechanics.Thereareotherwaystoshowthisequiv-alence.Forexample,MacKenzie[5]beginswiththeSchrodingerpictureandshowsthatitcanbemassagedintotheFeynmanformulation.Ourdirection,whichisthesameasthatofFeynmanandHibbs[1],istoshowthatthestandardquantumtheorycanbereconstructedfromthe\ rstprinciples"ofthepathintegralmethod.3HarmonicOscillatorNowlettheparticlebeinapotentialgivenbyV(x)=1 2m!2x2.Inprinciple,wecouldexplicitlycomputethepropagatorbyconsideringagenericinitialstateandapplyingtime-evolutiontoeachoftheeigenstates.Thiswayisalgebraicallybrutal,requiringthesummationofanin niteseriesofexpressionsinvolvingHermitepolynomials.Usingthetoolswehavedevelopedabove,wewillderivethepropagatorfortheone-dimensionalharmonicoscillatorwithasurprisinglackofintensecomputation,relyinginsteadonthetheoreticalbasisofthepathintegralformulation.3.1DerivationofthePropagatorWeapproachtheprobleminaslightlydi erentmanner.Letxcl(t)and_xcl(t)betheclassicalpathanditsvelocity,respectively.Wecanrepresentanyotherpathasadeviationfromthisone,xcl(t)+y(t)and_xcl(t)+_y(t),withthecorrectboundarycon-ditions(yand_yarezeroattheendpointsofthepath).Let'srewritetheLagrangianforagenericpathinawaythatisolatestheclassicalLagrangian.Werelaxthetimedependence,sinceitisimplicit.L(xcl+y;_xcl+_y)=1 2m(_xcl+_y)2+1 2m!2(xcl+y)2=1 2m_x2cl+1 2m!2x2cl+(m_xcl_y+m!xcly)+1 2m_y2+1 2m!2y2(23)The rsttermin(23)isjusttheLagrangianfortheclassicaltrajectory.ThesecondtermiszerobytheEuler-Lagrangeequations(sinceonastationarypathxcl,this PathIntegralsinQuantumMechanics9termmustbezeroforanywell-behaveddeviationy).Wewillreturntothislater.ThethirdtermistheLagrangianforthedeviation.Theactionisseparableinthesensethatthepropagatorhasthefollowingform:U=A(t)expi ~S[xcl]Zallpathsexpi ~Zt=tNt=t01 2m_y2+1 2m!2y2(24)Itcanbeshownthattheintegraloverallpathsdependsonlyonelapsedtimetandnotonx(t0)orx(tN).Shankar[3]arguesthatthismustbetruesinceeverydeviationhasnoknowledgeoftheendpointsofthetrajectory(sinceyand_yarezerothere).Wethereforeabsorbthistime-dependentfunctionintothenormalizationfactor,andareleftwithjusttheclassicalaction.5Normalizingthisexpressionisnottrivial.FeynmanandHibbs[1]considerthedecompositionofadeviationy(t)intoitsFourierserieswithperiodtN t0,andthenintegrateoverthecomponents.Weusetheirresult,thoughthereareothermethods.U=r m! 2i~sin!texpim! 2~sin!t(x2+x20)cos!t 2xx0(25)Thisisthecorrectanswer.Later,whenweinvestigatetheconnectionbetweenthepathintegralformulationandstatisticalmechanics,wewillusethisresulttorecovertheenergyeigenvalues.3.2SeparableLagrangiansWehavediscoveredthatinsomecases,onlytheclassicaltrajectorycontributestothesummationin(4).Thereisastraightforwardreasonforthis.TheLagrangianforanygenericpathcanbeexpressedasatwo-dimensionalTaylorexpansionabouttheLagrangianoftheclassicalaction.L(xcl+y;_xcl+_y)=Lcl+1Xn=11 n!yd dx+_yd d_xnLcl=Lcl+ydLcl dx+ydLcl d_x+y2d2Lcl dx2+2y_yd dxdLcl d_x+_y2d2Lcl d_x2+:::(26) 5Thisisaknownresult,sowedonotderiveithere. 10D.V.PerepelitsaTheEuler-Lagrangeequationsontheclassical(stationary)trajectorydictatethatthen=1termiszero.Inanon-relativisticframework,thekineticenergytermintheLagrangiancontributesam 2_y2term.Additionally,inapotentialisquadraticinxand_x(asinthecaseoftheharmonicoscillatorandmanyotherelementarypotentials),theLagrangianseparatesexactly.Relation(26)showsthatinthesecases,thepropagatorissimplyrelatedtotheimaginaryexponentoftheclassicalactiondividedbythequantumofaction.UA(t)expi ~S[xcl](27)Thus,therearemanyproblemswhichcanbereducedtodeterminingtheclassicalaction,oncetherelationaboveisshowntobetrueforthesystem.Afterthis,thenormalizationofthepropagatorbecomestheonlychallengingtask.Although(27)isoftenagoodapproximation(andisevenexactinsomeimportantsystems),wemustbeawareofitslimitationsanduses.Thirdorhigherpowersofxor_xintheLagrangianwillannulit.However,itworksreadilyiftheLagrangiandependsonmorethanonetrajectory,andcanevenbesolvedinthecaseofatime-dependentinputintothesystemf(t).FeynmanandHibbs[1]provideamoreindepthdiscussionofthissolutionmethodandwhereitisapplicable.4EuclideanPathIntegralsOurapproachinthefollowingsectionispragmatic,focusingmoreonthefunctionalityofamethodthanonarigorousjusti cation.Inprevioussections,wehaverelatedthetransitionprobabilityamplitudeofthewavefunctionbetweentwopointsinspacetime.Thisnextsectionconcernsrevisingnotthemethodbutthesenseofspacetime.Farfromtheclassicaltrajectory,therapidlyoscillatingtermsin(2)cancausecon-vergenceissuesandaregenerallyunpleasanttodealwith.ObservethatinMinkowskispacetimewithonephysicalcoordinate,theproperdistance2goesasthenegativesquareofthetime t2.However,inEuclideanspacetime,thesignofthet2termispositive;theydi erbyaphasefactori.Considerwhatwouldhappenifweintroducedafactorofiintotheexponentials-theoscillatingtermswouldturnintodecayingexponentials,whichhaveanentirelydi erentphysicalmeaning.Withthisasourmotivation,letusanalyticallyextendthetimeparameterintothecomplexnumbersC.Inparticular,lettimehavenorealcomponent,andidentify PathIntegralsinQuantumMechanics11t= i witharealparameter calledimaginarytime.ThisisformallyknownasaWickrotation,andrigorousjusti cationofthisstepisnottrivial.Thefruitofsuchagambitwillhopefullysoonbemadeclear.Nevertheless,letusexaminetheformpropagatoroverasmalltimeinterval i .U(x0; i ;x)Xallpathsexpi ~m 2(x0 x)2 i i Vx+x0 2(28)Weseethatthepotentialenergyhas\rippedsignrelativetothekineticenergyterm!Werede netheEuclideanactionSE[x( )]ofatrajectoryinimaginarytimex( )below.SE[x( )]=Z 2 1m 2_x2( )+V(x( ))(29)TheEuclideanactionisavaluerelatedtothequantumtunnelingprocess.Wewillnotenterintomoredetailonthis;bothGrosche[6]andMacKenzie[5]giveamorein-depthtreatment.Withthis,wecande netheimaginarytimepropagatorU(x0; ;x)aswell.U(x0; ;x)=A( )Zallpathsexp 1 ~SE[x( )](30)Thoughthephysicalmeaningbehindthisnewpropagatorisstillnotobvious,wecangainsomesenseofitsinterestingproperties.Whatsortofvaluecanthesumoverallpathsin(30)have?Nolongerdoestheclassicalactioncausethecontributiontothesummationtooscillate;eachpathisnowgivenanegativeexponentialweightbasedonitsEuclideanaction.Theclassicalpathhasthelargestcontribution,sincetheactionisaminimumthere.TrajectoriesfarawayfromtheclassicaloneseeanexponentiallydecreasedcontributionasSEgrows.AparticleobeyingtheEuclideanequationsofmotionexperiencesthepotentialintheotherdirection,whichisintuitivelycorrect-apaththrougharegionofhighpotentialdampensthecontributionofthatpaththroughthenegativeexponentialdependenceoftheaction.4.1StatisticalMechanicsWebrie\ryexploretheconnectionbetweentheEuclideanpathintegralandstatisticalmechanics.Consideranensemblesystematthermodynamicequilibriumwithordered 12D.V.PerepelitsaenergymicrostatesfEngforn=1;2;:::.ThepartitionfunctionZofstatisticalmechanicsencodesprobabilisticinformationaboutthesystem.Z=1Xn=1e En(31)Above, =1=kbTbetheinversetemperatureofthesystematagiventemperatureTwithBoltzmann'sconstantkb.Theformof(31)isreminiscentofthetime-evolutionconstructedfromenergyeigenstatesbackin(1),whichwehaveshownhasanequiv-alentexpressionastheintegraloverpropagatorstoallpossiblepointsin(2).Thenaturalquestioniswhetherwecanderivethepartitionfunctionfromapathintegralstandpoint.Itwouldseemthatreplacingthetimevariabletwiththequantity i wouldbeagood rststep.Aswehaveseenabove,thismovesusintotherealmofEuclideanpathintegrals.TheanalysisbelowisbasedonthatofGrosche[6]andFeynmanandHibbs[1].Consideraquantumsystemwiththediscreteenergyspectrumgivenabove.Werewritetheimaginarytimepropagatorbetweenx0andxbydecomposingthebraandketintoabasisofeigenstatesandapplyingtime-evolutiontoeachone.U(x0; ;x)=Xnhx0jnie iEn( i )=~hnjxi=Xnhnjxie iEn( i )=~hx0jni(32)Now,settingx0=xand = ~andintegratingoverallxyieldsZdxU(x; ~;x)=XnhnjZxdxe Enjxihxjni=Xne En=Z(33)Thisistherelationbetweentheimaginarytimepropagatorandthepartitionfunctionthatwearelookingfor.Asavividillustrationofthisequality,wereturntothecaseoftheharmonicoscillator,whosepropagatorwederivedinSection3:1.U(x; ~;x)=r m! 2~sinh ~!exp mwx2 ~sinh ~!(cosh ~! 1)(34)ThepartitionfunctionfortheharmonicoscillatorZistheintegralof(34)overallx,whichiseasytoevaluatesincethefunctionisaGaussian. PathIntegralsinQuantumMechanics13Z=ZdxU(x; i ;x)=r m! 2~sinh ~!Zdxexp24 x2 ~sinh ~! m!(cosh ~! 1)35(35)=r m! 2~sinh ~!p s ~sinh ~! m!(cosh ~! 1)=1 p 2(cosh ~! 1)(36)Expandingthishyperbolicfunctiongivesthedesiredresult.(p 2(cosh ~! 1)) 1=(p e ~! 2+e ~!) 1=(e ~!=2 e ~!=2) 1(37)=e ~!=21 1 e ~!=Xn=0exp n+1 2~!(38)Thisistheexpectedpartitionfunction,completewiththeenergyeigenvaluesoftheharmonicoscillator!5DiscussionThismanuscripthasbeenintendedasabriefintroduction;theliteratureisricherinapplication,scopeandrigor.FeynmanandHibbs'[1]textbookclassicintroducesallofquantumtheoryfromthisperspective.MacKenzie[5]notesthatapathintegralformulationnaturallyleadstoaninvestigationoftheAharanov-Bohme ectusingthesemethods,andgoesontopresentstationaryperturbationtheoryusingFeynmankernels.Shankar[3]delvesintopathintegralsthroughphasespaceandcoherentstatespace,andappliesthemtoanumberofsophisticatedtopicssuchastheBerryphase.Sakurai[2]writesmoregenerallyaboutGreen'sfunctions.Grosche[6]presentsapaperstartlinginitsrigorandthoroughnessonthetopic.Othersourcesevendemonstratetheuseofpathintegralsinquantum eldtheory.Still,wehavemanagedtocoverquiteabitofgroundonthetopicofpathintegralsinquantummechanics.Ourformulationofquantummechanicshasbeendrivenbytwokeyprinciples.Startingfromtheprinciplethataparticletakesnowell-de nedtrajectorybetweentwopointsatwhichitisobserved,wede nethetransitionprobabilityamplitudebetweentwopointsasasummationoverallpaths.ThiswasFeynman'sinsight.Then,weconnectthecontributiontotheamplitudefromanygivenpathwiththecorrespondingclassicalactionalongthepathinthemannersuggestedbyDirac. 14D.V.PerepelitsaWehavederivedthefreeparticlepropagatorinamannerthatfullydemonstratingthe\integrateoverallpossiblepaths"principle.WehaveshownequivalencetothestandardSchrodingerformulationofquantummechanics.Then,takingadvantageoftherestrictionsthattheclassicallyobservedLagrangianactionprovides,wehavederivedtheharmonicoscillatorpropagator,anddiscussedpotentialsforwhichthepropagatoriseasilysolvable.Afterthis,wehavede nedthepathintegralinEu-clideanspacetime,andshowedhowitisrelatedtothepartitionfunctioninstatisticalmechanics.Fromthis,wehaverecoveredtheenergyeigenvaluesoftheharmonicoscillator.ItissaidthatRichardFeynmancouldnotallowbelieveanyphysicshehadnotrederivedforhimself.Thoughouranalysishasbeenbrief,andwithlessemphasisonstraightforwardrigorthanonpragmatism,itisclearthatthescopeandsuccessofFeynman'smethodistrulyremarkable.AcknowledgmentsDVPwouldliketothankBrianPepperandEricFitzgeraldfortheiradviceandproofreadingduringthecreationofthismanuscript.References[1]R.P.FeynmanandA.R.Hibbs,QuantumMechanicsandPathIntegrals(McGraw-Hill,NewYork,1965)[2]J.J.Sakurai,ModernQuantumMechanics(Addison-Wesley,Reading,MA,1994)[3]R.Shankar,PrinciplesofQuantumMechanics,2ndEd.(PlenumPress,NewYork,NY,1994)[4]R.Feynman,Space-TimeApproachtoNon-RelativisticQuantumMechanics(1948)Rev.ModernPhysics.20[5]R.MacKenzie,PathIntegralMethodsandApplications(2000)arXiv:quant-ph/0004090v1[6]C.Grosche,AnIntroductionIntotheFeynmanPathIntegral(1993)arXiv:hep-th/9302097v1