量子力学英文课件：Chapt9 Time-Dependent Perturbations and Radiation

QuantumMechanicsChapter9.Time-DependentPerturbationsandRadiation

Inpreviouschaptersyouhaveseenselectionrulesrelatedtotransitionsbetweenatomicstates.Theserulesareconsistentwithallobservationsofatomicspectra;transitionsinwhichtheseruleswouldbeviolatedarenever(well,hardlyever)seenwhenweobserveatomicspectra.Andwhen"forbidden"transitions(thoseviolatingaselectionrule)areseen,quantum-mechanicalcalculationscanaccuratelypredicthowoftentheyoccur,relativetopossibleallowedtransitionsfromthesameinitialstate.

ArigorouscalculationoftransitionprobabilitiesrequiresthatwegobeyondtheSchroedingerequationandquantizetheclassicalequationsforelectricandmagneticfields(Maxwell'sequations),sothatwecandealwitheventsinvolvingsinglephotons.Suchcalculationsarebeyondthescopeofthisbook,soweshallsettleforanapproximationtreatmentthatgivesinsightintothereasonforselectionrulesandallowsustogainsomeunderstandingofdevicessuchasthelaserandthemaser.

§9.1TransitionRatesforInducedTransitions

Transitionratesforinducedtransitionscanbecalculatedquitewellbymeansoftime-dependentperturbationtheory.ThistheoryfollowsthemethodofSection8.2,time-independentperturbationtheory,inthatitassumesthatthepotentialenergyofthesystemcontainsasmallperturbingtermandthatwithoutthistermtheSchroedingerequationcanbesolvedexactly.

Thedifferenceisthattheperturbingterm,u(x,y,z,t),isassumedtobeappliedforalimitedtime,andtheresultisthatthesystemmaymakeatransitionfromoneunperturbedstatetoanother.Therefore,wewritethetime-independentSchroedingerequation(inonedimensionforconvenience)aswhereasbefore,H0istheHamiltonian(orenergy)operatorfortheunperturbedsystem,andtheequationforanyeigenfunctionψloftheunperturbedsystemis

AsinSection8.3,wenowrewritetheperturbedequation[Eq.(14.1)]byexpandingthefunctionψn’asalinearcombinationoftheunperturbedeigenfunctionsψl,withtheimportantdifferencethatthecoefficientsintheexpansionare,ingeneral,timedependent.ThusandEq.(14.1)thereforebecomesAfterdifferentiatingtheright-handseriestermbyterm,weobtain

andeliminatingthebracketsonbothsidesgivesusWenowseefromEq.(14.2)thateachterminthefirstseriesontheleftisequaltothecorrespondingterminthesecondseriesontheright-handside,sowecaneliminatebothseries,reducingEq.(l4.6)toWenowproceedasinSection8.2

WemultiplyeachsideofEq.(14.7)byaparticularfunctionψm*andthenintegrateoverallvaluesofx(oroverallspaceinthethree-dimensionalworld)toobtain

whichweintegratetermbytermaswedidwithsimilarexpressionsinSection8.3.Becausethewavefunctions{ψl}arenormalizedandorthogonaltooneanother,theonlynonzerotermontheright-handsideistheoneforwhichl=m,namelyiћ.Usingthefactthatthetimedependenceofmisintegratingtheleft-handsideofEq.(14.8)termbyterm,andagainusingthenormalizationandorthogonalitypropertiesofthewavefunctions,wefinallyarriveatanexactequationforthetimedependenceofthecoefficientanm:where,asinSection8.2,weuseanabbreviation:

Equation(14.9)isstillexact,butlikeEq.(12.10),itcontainstoomanyunknownquantitiestobeusefulasitstands.Therefore,weagainassumetheapproximationthattheeigenfunctionsoftheperturbedsystemdifferveryslightlyfromthoseoftheunperturbedsystem.Thispermitsustomaketheapproximationthatallofthecoefficientsanl

areverysmall,exceptforann,whichisapproximatelyequalto1.Ifwesetannequaltol,andallothercoefficientsanlequaltozero,Eq.(14.9)becomes

Ifv(x,t)isknownforallvaluesofxandt,thenitwouldappearthatitispossibletointegrateEq.(14.10)anddeterminethebehaviorofthesystem,withanaccuracythatislimitedbythesizeoftheneglectedcoefficientsanm.

ComparisonofTime-DependentandTime-IndependentPerturbationTheoryWeusetime-independentperturbationtheorywithaknownsetofstates,tocalculateprobabilitiesoftransitionsbetweenlevels.Weknowthepossiblestatesofthesystembecausethetime-dependentperturbationisassumedtocontinueforalimitedtimeinterval,

afterwhichthesystemrevertstooneofitsunperturbedstates.Typically,weconsiderthefollowingsequenceofevents:l.Attimet=0,thesystemisinanunperturbedstate│ψn〉,aneigenstateoftheSchroedingerequationwithenergyeigenvalueEn.2.Theperturbingpotentialisthen"turnedon."Fort>0,thesystemisthendescribedbytheperturbedSchroedingerequation,withadifferentsetofeigenstates│ψn’〉.Iftheperturbationissmalland/orisappliedforaveryshorttime,thenewstateneverdiffersgreatlyfromthestate│ψn〉.3.Theperturbingpotentialisturnedoffattimet=t',andthesystemisagaindescribedbytheunperturbedSchroedingerequation.Theeigenstatemaybetheoriginalstate│ψn〉,oritmaybeadifferentstate.Inthelattercase,wesaythattheperturbationhasinducedatransitiontothenewstate│ψm〉.Theprobabilitythatthesystemwillbefoundinthestate│ψm〉isgivenby│anm│2,whichisthesquareofthecoefficientofthewavefunctionψn’intheexpansionofwavefunctionψminseriesofeigenfunctionsoftheoriginalwavefunction.ExampleProblem14.lAparticleisinitsgroundstate(n=1,kineticenergyE1,potentialenergyzero)inaninfinitelydeepone-dimensionalsquarepotentialwell.

AconstantperturbingpotentialV=δistunedonattimet=0.Findtheprobabilitythattheparticlewillbefoundinthesecondexcitedstate(n=3)attimet=t'.

Solution.

Theprobabilitythattheparticlewillbefoundinthesecondexcitedstateis|a13|2.ThesecondexcitedstateinthiswellhaskineticenergyE3=9E1.

SubstitutionintoEq.(14.10)gives,andsinceal3=0attimet=0,wehave

Butv31=0,becausethefunctionsψ1andψ3areorthogonal.Thustheprobabilityiszero.Toinduceatransitioninthispotential,theperturbingpotentialmustdependonx.

DipoleRadiation

LetusnowapplyEq.(14.10)toatomicradiation,consideringanelectromagneticwaveasaperturbationthatinducesatransitionbetweentwoatomicstates.Webeginwithradiationwhosewavelengthismuchgreaterthanthediameteroftheatomsinvolved,asistrueforvisiblelight.Inthiscase,wecanmaketheapproximationthatatagiventimetheentireatomfeelsthesamefield.Thatis,thefieldvariesintimebutnotinspace.Thisisknownasthedipoleapproximation,forreasonsthatwillbeclearaswedeveloptheequations.Westartwithmonochromatic(singlefrequency)radiation,polarizedalongthexaxis.ThustheresultdependsonthexcomponentoftheelectricfieldE,orEx=Eoxcosωt,whereEoxisconstant.Itisconvenienttorewritethisfieldincomplexformas

Theperturbingpotentialv(x,t)isthepotentialenergyofanelectronofcharge-e(eisnottobeconfusedwith2.71828...)inthisfield,givenbyInsertingthisexpressionintoEq.(14.10)giveswheretheabbreviationxmn

representstheintegral.[Wenowseethereasonfortheexpression"dipole"radiation.

Thedipolemomentofanelectricchargeeatadistancexfromtheoriginisgivenbyex.Iftheelectronwereinastablequantumstate│ψn〉,thedipolemomentwoulddependontheprobabilitydensityfortheelectroninthatstate,andthuswouldbegivenbytheintegral

Whenthereisatransition,theelectron(beforeitisobserved)isinamixedstate,withdipolemomentgivenbytheintegralThisintegraliscalledthedipolemomentbetweenstatesnandm.]WenowassumethattheEfieldis"turnedon"attimet=0and"turnedoff'attimet=t'.ThereforewemustintegrateEq.(14.13)ontbetweenthesetwolimitstofindthetransitionprobabilityfromstatentostatem,whichisgivenby|anm|2.Fromtheinitialconditionanm(0)=0unlessn=m,weobtainanm(t’):Ratherthanattemptingtofindthecomplicatedgeneralexpressionforthetransitionprobability|anm(t’)|2,letusexamineEq.(14.14)togainsomeinsight.

Thefirstdenominator（分母）iszerowhenEm–En=-ћω;thesecondiszerowhenEm–En=+ћω.•ItisreasonabletosupposethatwecanneglectthefirsttermforfrequenciessuchthatEm–En

+ћω.Wecanthensimplify|anm(t’)|2to:orIfwedefinethefrequencynmby

nm=(Em-En)/ћ(14.17)andthefunctionf()bythenEq.(14.16)becomes•Figure14.1showsagraphoff()versus-nm.Themaximumvalueoff()occurswhen=nm,thefrequencyatwhichthephotonenergyћisequaltothedifferencebetweentheenergylevelsEnandEm.•Thisshouldcomeasnosurprise,butthefactthatotherfrequenciesalsocontributetotransitionsappearstoviolatethelawofconservationofenergy.However,whenweconsidertheresultsofSection2.4wefindthatthereisnoviolation.Thefactthattheperturbationexistsforalimitedtimet'makesthefrequencyuncertain,justasconfiningaparticleinalimitedspacemakesitswavelengthuncertain.Ifwelett'approachinfinityinEq.(l4.18)weseethatf(ω)approachesadeltafunction,becomingzeroforallfrequenciesexcept=nm.•(Foreachpointonthehorizontalaxis,thevalueof-nmisamultipleofl/t';whent'becomesinfinite,everypointonthehorizontalaxisrepresentsavalueofzero.•Thuswhent'isinfinitethevalueof-nmiszeroovertheentirecurve.)

UncertaintyRelationforEnergyandTime•Whent'isfinite,aFourieranalysis(Section2.3)ofthelightwavewouldshowasinusoidaldistributionoffrequencieswhichisconsistentwithEq.(14.18).ThereforeFigure14.lagreeswiththelawofconservationofenergyandwiththeconditionthataphotonofangularfrequencyωhasenergyћω.Thisfigureshowsthat,fortheoverwhelmingmajorityoftransitions,

ћω-(Em-En)≤2πћ/t′=h/t'(14.20)LetusnowconsidertheprobableresultsofameasurementoftheenergydifferenceEm-Enbetweentwolevelsinacollectionofidenticalatoms.•Wemightmeasurethisdifferencebyapplyingafieldofangularfrequencyωtotheatomsforatimet'andmeasuringtheamountofenergythatisabsorbed.

•

Byrepeatingthisprocedureatdifferentfrequencies,wecouldplotagraphlikeFigure14.l.

•ButEq.(14.20)showsthatanyobservedphotonenergyћωcandifferfromtheenergydifferenceEm–Enbyasmuchas2πћ/t',orh/t'.•DenotingthisdifferenceastheuncertaintyΔEinthemeasurement,wehave,forthisspecialcase,ΔE2ћ/t'=h/t'ort'ΔEh(14.21)Thetimeintervalt'canbethoughtofastheuncertaintyinthetimeofthemeasurementoftheenergy.

Thuswehaveanuncertaintyrelationinvolvingtimeandenergy,justaswehavearelationinvolvingpositionandmomentum.•Inthegeneralcase,theuncertaintyrelationforenergyandtimeiswrittenEtћ/2 (14.22)whereΔEistheuncertaintyinameasurementoftheenergyofasystem,andΔtisthetimeintervaloverwhichthemeasurementismade.Thisrelation,liketheparallelrelationΔpxΔxћ/2,isbaseduponthefactthatawaveoffinitelengthmustconsistofasuperpositionofwavesofdifferentfrequencies.Inthecaseoftheenergymeasurement,thewaveisthatofaphotonoftheradiationfieldthatinducesthetransitionbetweenenergylevels,butthemathematicsgoverningthiswaveisidenticaltothatofamatterwave.•TransitionProbabilityforaContinuousSpectrumofFrequencies

•Inthegeneralcase,Eq.(14.14)cannotgivethetransitionprobabilitydirectly;itmustbemodifiedsothatitrepresentsacomponentinacontinuousspectrumoffrequencies.

Whenwehaveacontinuousspectrum,therecannotbeanamplitudeforasinglefrequency.Instead,thereisanenergydensityfunction()suchthattheintegralof()dovertherangefrom1to2istheenergydensityofradiationwithfrequenciesbetween1to2.Accordingtoclassicalelectromagnetictheory,thequantity0E0x2/2istheaverageenergydensityintheelectromagneticfieldgivenbyThusforradiationinanarrowrangeoffrequenciesd,wehave

ε0E20x/2=ρ(ω)dω (14.23)andE0x2canbereplacedinEq.(14.1l)by2()d/0.ThentofindthetotaltransitionprobabilityTnmresultingfromtheentirespectrumofradiation,weintegratetheresultingexpressionforanm(t’)2overallfrequencies,obtaining

WecansimplifyEq.(14.24)byassumingthat()variesmuchmoreslowlythanf(),andsincef()issymmetric,withamaximumat=nm,wecanreplace()bytheconstantvalue(nm),withlittlelossofaccuracy.Ifweremove(nm)fromtheintegralandwedefine=(nm-)t’/2,Eq.(l4.24)becomes[ThereadershouldverifythatEqs.(14.24)and(14.25)areequivalent,giventhesubstitutionsthatweremade.]Theintegralisstandard,beingequalto/2,sothetransitionprobability,forradiationthatispolarizedinthexdirection,is

Inthegeneralcase,whentheradiationisrandomlypolarized,Tnmmustincludeequalcontributionsfromxnm2,ynm2,andznm2,andwehavewherewehavedividedby3becausetheintensityisequallydistributedamongthethreepolarizationdirections.Thefactort'inEq.(l4.27)requiresmorescrutiny.Itislogicalthattheprobabilityofatransitionshouldincreasewithtime,butitcannotincreaseindefinitely,becauseaprobabilitycanneverbegreaterthanl.Obviouslytheapproximationbreaksdownattimest'suchthatTnmisnolongersmallrelativetol.Iftheradiationiscoherent(forexample,producedbyalaser;seeSection14.3.7),thentheperturbationismaintainedfortimest'thatarequitelongrelativetoincoherentradiation,suchasthatemittedbythesun.Therefore,atomsthatarebathedinlaserlightcanbeperturbedforsuchalongtimethatEq.(14.37)isnolongervalid.(Analysisofsuchsituationsfallsintotherealmofnonlinearoptics.)Ontheotherhand,incoherentradiationconsistsofbriefpulsesemittedbyindividualatomsatrandom;forexample,the3plevelofthehydrogenatomsurvivesforabout10-8second.Insuchcases,theemittedpulse(onephoton)canperturbanotherhydrogenatomforatimet'ofthesameorderofmagnitude.ThistimeintervalissufficientlysmalltosatisfytheconditionTnm

<<l,andinthosecasesEq.(l4.27)isquiteaccurate.Afterthetimet'.theperturbationends,andthehydrogenatomisinitsoriginal1sstateorisinthe2pstate.Theprobabilitythatitisinthe2pstateisgivenbyEq.(14.37).Thisprobabilitycanbetestedbysimplyobservingthatthesecondatomemitsaphotoninreturningtothe1sstate.

§9.2SpontaneousTransitions

Intheprevioussectionwefoundtheprobabilitythatasysteminonequantumstatenwillbeinducedtochangetoanotherstatem,ifitisacteduponbyaperturbationsuchasradiationattheresonantfrequencynm=En-Em/h.Butwestillneedawaytocomputetheprobabilityofaspontaneoustransitionatransitionthatoccursintheabsenceofaperturbation.Fortunately,thereisasimplewaytoattackthisproblem.Evenbeforequantummechanicswasdeveloped,Einsteinwasabletoderivetherateofspontaneoustransitionsfrombasicthermodynamics,givenonlytheinducedtransitionrate.Heusedthefollowingargument.Einstein'sDerivationConsideracollectionofidenticalatomswhichcanexchangeenergyonlybymeansofradiation.Thecollectionisinthermalequilibriuminsideacavitywhosewallsarekeptataconstanttemperature.

Becausethesystemisinthermalequilibrium,eachatommustbeemittingandabsorbingradiationatthesameaveragerate,ifoneaveragesoverasufficientlylongtime(suchasonesecond).DefinePnmastheprobabilityofaninducedtransitionofagivenatomfromthestatentostateminashorttimeintervaldt.Thisprobabilitymustbeproportionaltotheprobabilitypn,thattheatomisinitiallyinstatenmultipliedbythetransitionprobabilityTnmforanatominthatstate,whichforunpolarizeddipoleradiationisgivenbyEq.(14.27).Thus Pnm=TnmPn(14.28)GuidedbyEq.(l4.27),wecannowwriteageneralequationforPnmasPnm=Anm

(nm)pndt(14.29)•whichexpressesthefactthatTnmisproportionaltotheradiationdensity(nm),tothetimeintervaldt(denotedbyt'inEq.(14.27),andtootherfactors,incorporatedintoAnm,whichdependonmatrixelements.•Equation(14.27)canbeappliedequallytoaninducedtransitionfromstatentostatem,orfromstatemtostaten.Fromthesymmetryoftheequations,weknowthatAnm=Amnandnm=mn

.Therefore,

Pmn=Amn

(nm)pm

dt(14.30)Equation(14.30)givestheprobabilityofaninducedtransitionfromstatemtostaten,whileEq.(14.29)givestheprobabilityofaninducedtransitionintheotherdirection,fromstatentostatem.Theonlydifferencebetweentheseprobabilitiesisintheoccupationprobabilitiespnandpm.

Thesearenotequal,becausetheprobabilitythatastateisoccupieddependsonitsenergy.Letussaythatstatenhasthelowerenergy;thatis,En<Em.Thenpn>pm,andthereforePnm>Pmn.

Therearemoreinducedtransitionsfromntomthantherearefrommton,simplybecausetherearemoreatomsinstatentobeginwith.Buttheatomsareinthermalequilibrium.Thereforetheremustbeothertransitions,spontaneousones,frommton,tomakethetotalprobabilityofatransitionfrommtonequaltotheprobabilityofatransitionfromntom.Thismeansthat Pnm=Pmn+Smn

(14.31)whereSmnisthespontaneoustransitionprobability,whichmaybewritten

Smn=Bmnpmdt (l4.32)

Noticethat,unlikePmnorPnm,Smndoesnotcontainthefactor(nm),becauseaspontaneoustransition,bydefinition,doesnotdependonexternalfields.SubstitutingfromEqs.(14.29),(14.30),and(14.32)into(14.31),wehave

Anm

(nm)pn=Anm

(nm)pm+Bmnpm

(14.33)or

Bmn=Anm

(nm){pn/pm-l} (14.34)RememberthatBmnisassociatedwithaspontaneoustransition,soitdoesnotreallydependontheenergydensityoftheelectricfield.However,wehavederivedthisequationbyrelatingBmntoinducedtransitionsinacavity;thereforetheenergydensityinthecavityhasappearedintheresult.Wecaneliminate(nm)fromtheresultbyusingtheformulafortheenergydensityinacavity(seeAppendixCforthederivationofthisformula):Youcanverifythat()hasthecorrectdimensions(energypersecondperunitvolume).InsertingthisexpressionintoEq.(14.34)yields

TocompletethederivationofBmnweneedtheratiooftheoccupationprobabilities,pn/pm.ThisratioisknownfromBoltzmannstatistics(AppendixBandSection16.l)tobegivenbyandthereforeEq.(14.36)becomessimply•UsingEqs.(14.27)and(l4.29)tofindAnm,.wefindthatthespontaneoustransitionprobabilityinashorttimeintervaldt,fromstatemtostaten,isequaltodt,

where,theprobabilityperunittimeforatransitiontooccur(alsocalledthedecayconstant).isgivenbyA"short"timeintervaldtisoneforwhichdt<<1.Youshouldverifythathastheproperdimensions(reciprocaltime,tomakedtdimensionless).Whenwespeakofdecayrates,wemustrememberthatthetransitionisobservedasadiscontinuousprocess;aphotoninteractswithameasuringinstrumentasadiscreteunitofenergy.Hereisthesamewave-particledualitythathasbeendiscussedbefore. Theterm"measuringinstrument"hasaverybroadmeaning;itisnotnecessarilyanartifactofourownmaking.Forexample,thousandsofyearsagoinAfricaanuclearchainreactionbeganspontaneously.Nomeasuringinstrumentcouldcountthedecays,buttheevidenceremainsatthesiteforalltosee.(Ifatreefallswherenobodycanhearit,doesitmakeasound?Ofcourseitdoes;manyanimalscanhearit.)

Nowconsiderthetimeatwhicheachatommakesatransition.Thisisdeterminedbytheinteractionofaphotonwiththemeasuringinstrument,whichcouldbeanykindofmatteronwhichthephotoncouldleavealastingimprint.Thusnaturemakesthemeasurementwithoutourintervention.Wecanmakeananalogtoalpha-particleemissionbyaradioactivenucleus.(SeeSection12.5.)Thealphaparticleinauraniumnucleustravelsbackandforthandhas1020ormoreopportunitiestoescapeduringeachsecond.•Ifitdoesnotescape,theatomisunchanged;abillion-year-old235Uatomisidenticaltoa235Uatomthatwasjustformedbyanymeanswhatsoever(perhapsbyalphadecayofa239Puatom).Inasimilarway,theoscillatingdipolemomentofahydrogenatominamixedstate,likethatofEq.(l4.13),createsanelectromagneticfieldthat,soonerorlater,willtransferenergytoanotherhydrogenatom.Buttheenergycanonlybetransferredbyaphoton;aslongasnotransferhastakenplace,theoriginalhydrogenatomisunchanged,andthustheprobabilityofdecayinthenextpicosecondisnotchanged.EnergyDependenceofTransitionRatesandDecayofSubatomicParticlesThefactornminEq.(14.39)tellsusthatthedecayconstantisproportionaltothecubeoftheenergydifferencebetweenstatesnandm.•Thisistrueforanytransitionthatisgovernedbytheelectromagneticforce,(wherephotonsareinvolved).•Astrikingexampleofthisisgivenbycomparingthemeanlifetimesoftwosubatomicsystems:theneutralpimeson(pion,0)andpositronium(Ps),whichisaboundstareofapositronandanelectron.•Inbothcasestheentiremassofthesystemdisappearsandtwophotons(gammarays)areemitted.•Thetotalenergyofthesephotonsisequaltoћω,whichinthiscaseisjusttheoriginalrestenergy.TherestenergyofPsistwicetheelectronrestenergyorl.02MeV;therestenergyofthepionis135MeV.Thereforethevalueofωnmforthepionisabout130timesitsvalueforPs.Sincethevalueofλisproportionaltoω3,wewouldexpecttheratioofthemeanlifetimeofPstothemeanlifetimeofthepiontobe,neglectingotherfactors,about1303,orabout2×109.•ThelifetimeofPsisl.24×10-9s;thatoftheπ0is0.83×10-16s.Theratioisaboutl.5×109.

ExponentialDecayLawGivenacollectionofN0identicalatomsinthefirstexcitedstateattimet=0,weexpecttofindthatNoftheseatomswillremainunchanged

whentheyareobservedattimet>0.Giventhevalueofthedecayconstantλ,letuspredictthevalueofN.Inanytimeintervaldt,theprobabilityofdecaytothegroundstatewillbeλdtforeachatom,soforNthenumberofdecayswillbeNλdt.ThusduringanytimeintervaldtthechangeinNwillbe

dN=-Nλdt (14.40)•Wecanintegratethisequationbyseparatingthevariablesasfollows:

dN/N=-λdt

orInN=-λt+constantofintegrationwiththefinalresultthatN=N0e-λt=N0e-t/τ (14.41)•whereN0isthenumberofexcitedatomsattimet=0,andτ=1/λisthemeanlifetimeintheexcitedstate.(SeeAppendixAforaproofthatthearithmeticmeanofalltheatomstobeintheexcitedstateisindeedequalto1/λ.)•Thetimetatwhiche-λt=l/2iscalledthehalf-life,writtent1/2.Thus,bydefinition,•Thehalf-lifeisindependentofthetimet.Nomatterhowlongtheatomshavebeenintheexcitedstate,onecanarbitrarilysettequaltozeroandthenumberatthattimeequaltoN0,andEq.(14.41)willhold,withλ=0.693/t1/2.Figure14.2isasimulationthatillustratesthispoint.Noticetherandomfluctuationsinthenumberofatomsdecayingineachtimeinterval.ThenumbersaregovernedbythelawsofPoissonstatistics(AppendixA).Thesimulationwasdonebyusingarandom-numbergeneratortodeterminethetimeatwhicheachatomdecays(onthebasisofagivenhalf-life),thenplottingtheresults.WidthofanEnergyLevel

Becauseanatomspendsalimitedamountoftimeinanexcitedstate,theuncertaintyrelationforenergyandtimeimposesabasiclimitationontheaccuracywithwhichtheenergyofastatecanbedetermined.Thereforetheatom,inmakingtransitionsbetweenanytwospecificstates,canemitorabsorbphotonsthathavearangeofenergies.Therangeofenergiesisinverselyproportionaltothemeanlifetimeoftheexcitedstate(justasthescaleinFigure14.lisinverselyproportionaltothetimet'duringwhichtheperturbingfieldisapplied).Eachenergylevelinagivenatomisdefinedonlytotheaccuracypermittedbytheuncertaintyrelationforenergyandtime.Thisuncertaintyiscalledthenaturallinewidthofthestate.Whenthemeanlifetimeofastateislessthan10-17secondyoucanbesurethatthelif