Path integral formulation - Wikipedia

The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical ... Pathintegralformulation FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Formulationofquantummechanics Thisarticleisaboutaformulationofquantummechanics.Forintegralsalongapath,alsoknownaslineorcontourintegrals,seelineintegral. PartofaseriesofarticlesaboutQuantummechanics i ℏ ∂ ∂ t | ψ ( t ) ⟩ = H ^ | ψ ( t ) ⟩ {\displaystylei\hbar{\frac{\partial}{\partialt}}|\psi(t)\rangle={\hat{H}}|\psi(t)\rangle} Schrödingerequation Introduction Glossary History Background Classicalmechanics Oldquantumtheory Bra–ketnotation Hamiltonian Interference Fundamentals Complementarity Decoherence Entanglement Energylevel Measurement Nonlocality Quantumnumber State Superposition Symmetry Tunnelling Uncertainty Wavefunction Collapse Experiments Bell'sinequality Davisson–Germer Double-slit Elitzur–Vaidman Franck–Hertz Leggett–Garginequality Mach–Zehnder Popper Quantumeraser Delayed-choice Schrödinger'scat Stern–Gerlach Wheeler'sdelayed-choice Formulations Overview Heisenberg Interaction Matrix Phase-space Schrödinger Sum-over-histories(pathintegral) Equations Dirac Klein–Gordon Pauli Rydberg Schrödinger Interpretations Overview Bayesian Consistenthistories Copenhagen deBroglie–Bohm Ensemble Hidden-variable Local Many-worlds Objectivecollapse Quantumlogic Relational Transactional Advancedtopics Relativisticquantummechanics Quantumfieldtheory Quantuminformationscience Quantumcomputing Quantumchaos Densitymatrix Scatteringtheory Quantumstatisticalmechanics Quantummachinelearning Scientists Aharonov Bell Bethe Blackett Bloch Bohm Bohr Born Bose deBroglie Compton Dirac Davisson Debye Ehrenfest Einstein Everett Fock Fermi Feynman Glauber Gutzwiller Heisenberg Hilbert Jordan Kramers Pauli Lamb Landau Laue Moseley Millikan Onnes Planck Rabi Raman Rydberg Schrödinger Simmons Sommerfeld vonNeumann Weyl Wien Wigner Zeeman Zeilinger vte Thepathintegralformulationisadescriptioninquantummechanicsthatgeneralizestheactionprincipleofclassicalmechanics.Itreplacestheclassicalnotionofasingle,uniqueclassicaltrajectoryforasystemwithasum,orfunctionalintegral,overaninfinityofquantum-mechanicallypossibletrajectoriestocomputeaquantumamplitude. Thisformulationhasprovencrucialtothesubsequentdevelopmentoftheoreticalphysics,becausemanifestLorentzcovariance(timeandspacecomponentsofquantitiesenterequationsinthesameway)iseasiertoachievethanintheoperatorformalismofcanonicalquantization.Unlikepreviousmethods,thepathintegralallowsonetoeasilychangecoordinatesbetweenverydifferentcanonicaldescriptionsofthesamequantumsystem.AnotheradvantageisthatitisinpracticeeasiertoguessthecorrectformoftheLagrangianofatheory,whichnaturallyentersthepathintegrals(forinteractionsofacertaintype,thesearecoordinatespaceorFeynmanpathintegrals),thantheHamiltonian.Possibledownsidesoftheapproachincludethatunitarity(thisisrelatedtoconservationofprobability;theprobabilitiesofallphysicallypossibleoutcomesmustadduptoone)oftheS-matrixisobscureintheformulation.Thepath-integralapproachhasproventobeequivalenttotheotherformalismsofquantummechanicsandquantumfieldtheory.Thus,byderivingeitherapproachfromtheother,problemsassociatedwithoneortheotherapproach(asexemplifiedbyLorentzcovarianceorunitarity)goaway.[1] Thepathintegralalsorelatesquantumandstochasticprocesses,andthisprovidedthebasisforthegrandsynthesisofthe1970s,whichunifiedquantumfieldtheorywiththestatisticalfieldtheoryofafluctuatingfieldnearasecond-orderphasetransition.TheSchrödingerequationisadiffusionequationwithanimaginarydiffusionconstant,andthepathintegralisananalyticcontinuationofamethodforsummingupallpossiblerandomwalks.[2] ThebasicideaofthepathintegralformulationcanbetracedbacktoNorbertWiener,whointroducedtheWienerintegralforsolvingproblemsindiffusionandBrownianmotion.[3]ThisideawasextendedtotheuseoftheLagrangianinquantummechanicsbyPaulDiracinhis1933article.[4][5]Thecompletemethodwasdevelopedin1948byRichardFeynman.SomepreliminarieswereworkedoutearlierinhisdoctoralworkunderthesupervisionofJohnArchibaldWheeler.Theoriginalmotivationstemmedfromthedesiretoobtainaquantum-mechanicalformulationfortheWheeler–FeynmanabsorbertheoryusingaLagrangian(ratherthanaHamiltonian)asastartingpoint. ThesearefiveoftheinfinitelymanypathsavailableforaparticletomovefrompointAattimettopointBattimet’(>t).Pathswhichself-intersectorgobackwardsintimearenotallowed. Contents 1Quantumactionprinciple 2Feynman'sinterpretation 3Pathintegralinquantummechanics 3.1Time-slicingderivation 3.2Pathintegral 3.3Freeparticle 3.4Simpleharmonicoscillator 3.5Coulombpotential 3.6TheSchrödingerequation 3.7Equationsofmotion 3.8Stationary-phaseapproximation 3.9Canonicalcommutationrelations 3.10Particleincurvedspace 3.11Measure-theoreticfactors 3.12Expectationvaluesandmatrixelements 4Euclideanpathintegrals 4.1WickrotationandtheFeynman–Kacformula 4.2Thepathintegralandthepartitionfunction 5Quantumfieldtheory 5.1Thepropagator 5.2Functionalsoffields 5.3Expectationvalues 5.4Asaprobability 5.5Schwinger–Dysonequations 6Localization 6.1Ward–Takahashiidentities 7Caveats 7.1Theneedforregulatorsandrenormalization 7.2Orderingprescription 8Thepathintegralinquantum-mechanicalinterpretation 9Quantumgravity 10Quantumtunneling 11Seealso 12Remarks 13Notes 14References 15Externallinks Quantumactionprinciple[edit] Inquantummechanics,asinclassicalmechanics,theHamiltonianisthegeneratoroftimetranslations.ThismeansthatthestateataslightlylatertimediffersfromthestateatthecurrenttimebytheresultofactingwiththeHamiltonianoperator(multipliedbythenegativeimaginaryunit,−i).Forstateswithadefiniteenergy,thisisastatementofthedeBroglierelationbetweenfrequencyandenergy,andthegeneralrelationisconsistentwiththatplusthesuperpositionprinciple. TheHamiltonianinclassicalmechanicsisderivedfromaLagrangian,whichisamorefundamentalquantityrelativetospecialrelativity.TheHamiltonianindicateshowtomarchforwardintime,butthetimeisdifferentindifferentreferenceframes.TheLagrangianisaLorentzscalar,whiletheHamiltonianisthetimecomponentofafour-vector.SotheHamiltonianisdifferentindifferentframes,andthistypeofsymmetryisnotapparentintheoriginalformulationofquantummechanics. TheHamiltonianisafunctionofthepositionandmomentumatonetime,anditdeterminesthepositionandmomentumalittlelater.TheLagrangianisafunctionofthepositionnowandthepositionalittlelater(or,equivalentlyforinfinitesimaltimeseparations,itisafunctionofthepositionandvelocity).TherelationbetweenthetwoisbyaLegendretransformation,andtheconditionthatdeterminestheclassicalequationsofmotion(theEuler–Lagrangeequations)isthattheactionhasanextremum. Inquantummechanics,theLegendretransformishardtointerpret,becausethemotionisnotoveradefinitetrajectory.Inclassicalmechanics,withdiscretizationintime,theLegendretransformbecomes ε H = p ( t ) ( q ( t + ε ) − q ( t ) ) − ε L {\displaystyle\varepsilonH=p(t){\big(}q(t+\varepsilon)-q(t){\big)}-\varepsilonL} and p = ∂ L ∂ q ˙ , {\displaystylep={\frac{\partialL}{\partial{\dot{q}}}},} wherethepartialderivativewithrespectto q ˙ {\displaystyle{\dot{q}}} holdsq(t+ε)fixed.TheinverseLegendretransformis ε L = ε p q ˙ − ε H , {\displaystyle\varepsilonL=\varepsilonp{\dot{q}}-\varepsilonH,} where q ˙ = ∂ H ∂ p , {\displaystyle{\dot{q}}={\frac{\partialH}{\partialp}},} andthepartialderivativenowiswithrespecttopatfixedq. Inquantummechanics,thestateisasuperpositionofdifferentstateswithdifferentvaluesofq,ordifferentvaluesofp,andthequantitiespandqcanbeinterpretedasnoncommutingoperators.Theoperatorpisonlydefiniteonstatesthatareindefinitewithrespecttoq.SoconsidertwostatesseparatedintimeandactwiththeoperatorcorrespondingtotheLagrangian: e i [ p ( q ( t + ε ) − q ( t ) ) − ε H ( p , q ) ] . {\displaystylee^{i{\big[}p{\big(}q(t+\varepsilon)-q(t){\big)}-\varepsilonH(p,q){\big]}}.} Ifthemultiplicationsimplicitinthisformulaarereinterpretedasmatrixmultiplications,thefirstfactoris e − i p q ( t ) , {\displaystylee^{-ipq(t)},} andifthisisalsointerpretedasamatrixmultiplication,thesumoverallstatesintegratesoverallq(t),andsoittakestheFouriertransforminq(t)tochangebasistop(t).ThatistheactionontheHilbertspace–changebasistopattimet. Nextcomes e − i ε H ( p , q ) , {\displaystylee^{-i\varepsilonH(p,q)},} orevolveaninfinitesimaltimeintothefuture. Finally,thelastfactorinthisinterpretationis e i p q ( t + ε ) , {\displaystylee^{ipq(t+\varepsilon)},} whichmeanschangebasisbacktoqatalatertime. Thisisnotverydifferentfromjustordinarytimeevolution:theHfactorcontainsallthedynamicalinformation–itpushesthestateforwardintime.ThefirstpartandthelastpartarejustFouriertransformstochangetoapureqbasisfromanintermediatepbasis. ...weseethattheintegrandin(11)mustbeoftheformeiF/h,whereFisafunctionofqT,q1,q2,…qm,qt,whichremainsfiniteashtendstozero.Letusnowpictureoneoftheintermediateqs,sayqk,asvaryingcontinuouslywhiletheotheronesarefixed.Owingtothesmallnessofh,weshalltheningeneralhaveF/hvaryingextremelyrapidly.ThismeansthateiF/hwillvaryperiodicallywithaveryhighfrequencyaboutthevaluezero,asaresultofwhichitsintegralwillbepracticallyzero.TheonlyimportantpartinthedomainofintegrationofqkisthusthatforwhichacomparativelylargevariationinqkproducesonlyaverysmallvariationinF.ThispartistheneighbourhoodofapointforwhichFisstationarywithrespecttosmallvariationsinqk.Wecanapplythisargumenttoeachofthevariablesofintegration...andobtaintheresultthattheonlyimportantpartinthedomainofintegrationisthatforwhichFisstationaryforsmallvariationsinallintermediateqs....WeseethatFhasforitsclassicalanalogue∫tTLdt,whichisjusttheactionfunction,whichclassicalmechanicsrequirestobestationaryforsmallvariationsinalltheintermediateqs.Thisshowsthewayinwhichequation(11)goesoverintoclassicalresultswhenhbecomesextremelysmall. Dirac(1933),p.69 AnotherwayofsayingthisisthatsincetheHamiltonianisnaturallyafunctionofpandq,exponentiatingthisquantityandchangingbasisfromptoqateachstepallowsthematrixelementofHtobeexpressedasasimplefunctionalongeachpath.Thisfunctionisthequantumanalogoftheclassicalaction.ThisobservationisduetoPaulDirac.[6] Diracfurthernotedthatonecouldsquarethetime-evolutionoperatorintheSrepresentation: e i ε S , {\displaystylee^{i\varepsilonS},} andthisgivesthetime-evolutionoperatorbetweentimetandtimet+2ε.WhileintheHrepresentationthequantitythatisbeingsummedovertheintermediatestatesisanobscurematrixelement,intheSrepresentationitisreinterpretedasaquantityassociatedtothepath.Inthelimitthatonetakesalargepowerofthisoperator,onereconstructsthefullquantumevolutionbetweentwostates,theearlyonewithafixedvalueofq(0)andthelateronewithafixedvalueofq(t).Theresultisasumoverpathswithaphase,whichisthequantumaction.Crucially,Diracidentifiedinthisarticlethedeepquantum-mechanicalreasonfortheprincipleofleastactioncontrollingtheclassicallimit(seequotationbox). Feynman'sinterpretation[edit] Dirac'sworkdidnotprovideapreciseprescriptiontocalculatethesumoverpaths,andhedidnotshowthatonecouldrecovertheSchrödingerequationorthecanonicalcommutationrelationsfromthisrule.ThiswasdonebyFeynman.[nb1]Thatis,theclassicalpatharisesnaturallyintheclassicallimit. FeynmanshowedthatDirac'squantumactionwas,formostcasesofinterest,simplyequaltotheclassicalaction,appropriatelydiscretized.Thismeansthattheclassicalactionisthephaseacquiredbyquantumevolutionbetweentwofixedendpoints.Heproposedtorecoverallofquantummechanicsfromthefollowingpostulates: Theprobabilityforaneventisgivenbythesquaredmodulusofacomplexnumbercalledthe"probabilityamplitude". Theprobabilityamplitudeisgivenbyaddingtogetherthecontributionsofallpathsinconfigurationspace. ThecontributionofapathisproportionaltoeiS/ħ,whereSistheactiongivenbythetimeintegraloftheLagrangianalongthepath. Inordertofindtheoverallprobabilityamplitudeforagivenprocess,then,oneaddsup,orintegrates,theamplitudeofthe3rdpostulateoverthespaceofallpossiblepathsofthesysteminbetweentheinitialandfinalstates,includingthosethatareabsurdbyclassicalstandards.Incalculatingtheprobabilityamplitudeforasingleparticletogofromonespace-timecoordinatetoanother,itiscorrecttoincludepathsinwhichtheparticledescribeselaboratecurlicues,curvesinwhichtheparticleshootsoffintoouterspaceandfliesbackagain,andsoforth.Thepathintegralassignstoalltheseamplitudesequalweightbutvaryingphase,orargumentofthecomplexnumber.Contributionsfrompathswildlydifferentfromtheclassicaltrajectorymaybesuppressedbyinterference(seebelow). FeynmanshowedthatthisformulationofquantummechanicsisequivalenttothecanonicalapproachtoquantummechanicswhentheHamiltonianisatmostquadraticinthemomentum.AnamplitudecomputedaccordingtoFeynman'sprincipleswillalsoobeytheSchrödingerequationfortheHamiltoniancorrespondingtothegivenaction. Thepathintegralformulationofquantumfieldtheoryrepresentsthetransitionamplitude(correspondingtotheclassicalcorrelationfunction)asaweightedsumofallpossiblehistoriesofthesystemfromtheinitialtothefinalstate.AFeynmandiagramisagraphicalrepresentationofaperturbativecontributiontothetransitionamplitude. Pathintegralinquantummechanics[edit] Time-slicingderivation[edit] Mainarticle:RelationbetweenSchrödinger'sequationandthepathintegralformulationofquantummechanics Onecommonapproachtoderivingthepathintegralformulaistodividethetimeintervalintosmallpieces.Oncethisisdone,theTrotterproductformulatellsusthatthenoncommutativityofthekineticandpotentialenergyoperatorscanbeignored. Foraparticleinasmoothpotential,thepathintegralisapproximatedbyzigzagpaths,whichinonedimensionisaproductofordinaryintegrals.Forthemotionoftheparticlefrompositionxaattimetatoxbattimetb,thetimesequence t a = t 0 < t 1 < ⋯ < t n − 1 < t n < t n + 1 = t b {\displaystylet_{a}=t_{0}