Chapter Timedependentperturbation theory Sofarwehavefocusedlargelyonthequantummechanicsofsystemsin whichtheyamiltonianistimeindependentznsuchcasesthetimedepen denceofawavepacketcanbedevelopedthrought
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Chapter Timedependentperturbation theory Sofarwehavefocusedlargelyonthequantummechanicsofsystemsin whichtheyamiltonianistimeindependentznsuchcasesthetimedepen denceofawavepacketcanbedevelopedthrought

1Timedependentpotentialsgeneralformalism tonsiderthentheyamiltonian Uwherealltimedependence entersthroughthepotential Uznthe Schr57512odingerrepresentation the dynamicsofthesystemarespeci64257edbythetimedependentwavefunction throughtheSchr57512odinge

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Chapter Timedependentperturbation theory Sofarwehavefocusedlargelyonthequantummechanicsofsystemsin whichtheyamiltonianistimeindependentznsuchcasesthetimedepen denceofawavepacketcanbedevelopedthrought




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Presentation on theme: "Chapter Timedependentperturbation theory Sofarwehavefocusedlargelyonthequantummechanicsofsystemsin whichtheyamiltonianistimeindependentznsuchcasesthetimedepen denceofawavepacketcanbedevelopedthrought"— Presentation transcript:


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Chapter12 Time-dependentperturbation theory Sofar“wehavefocusedlargelyonthequantummechanicsofsystemsin whichtheyamiltonianistime]independentˆznsuchcases“thetimedepen] denceofawavepacketcanbedevelopedthroughthetime]evolutionoperator Ht/ or“whencastintermsoftheeigenstatesoftheyamiltonian “as Ht/ TaU iE t/ TaU ˆrlthough thisframeworkprovidesaccesstoanyclosedquantummechanicalsystem“it doesnotdescribeinteractionwithanexternalenvironmentsuchasthatim] posedbyanexternalelectromagneticfieldˆznsuchcases“itismoreconvenient todescribetheinducedinteractionsofasmallisolatedsystem

“througha time]dependentinteraction Uˆvxamplesincludetheproblemofmagnetic resonancedescribingtheinteractionofaquantummechanicalspinwithan externaltime]dependentmagneticfield“ortheresponseofanatomtoanex] ternalelectromagneticfieldˆznthefollowing“wewilldevelopaformalismto treattime]dependentperturbations 12.1Time-dependentpotentials:generalformalism tonsiderthentheyamiltonian U“wherealltime]dependence entersthroughthepotential Uˆznthe Schrodingerrepresentation “the dynamicsofthesystemarespecifiedbythetime]dependentwavefunction throughtheSchrodingerequation

ˆyowever inmanycases“andinparticularwiththecurrentapplication“itisconvenient toworkinthe Interactionrepresentation definedby t/ where TaU TaU ˆWiththisdefinition“onemayshowthatthewave] functionobeystheequationofmotionTexerciseU TbcˆbU where Un t/ Ve t/ ˆThen“ifweformtheeigenfunctionexpansion “andcontracttheequationofmotionwithageneral state “weobtain Un mn mn TbcˆcU Notehowthisdefinitiondi ersfromthatofthe Heisenberg representation Ht/ inwhichalltimemdependenceistransferredintotheoperatorsn rdvancedQuantumPhysics
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tuqtqTIMEpDEPENDENTPOTENTIALS:GENERALFORMALISMtws wherethematrixelements mn Un “and mn nT nm ˆTodevelopsomeintuitionfortheactionofatime]dependentpotential itisusefultoconsiderfirstaperiodically]driventwo]levelsystemwherethe dynamicalequationscanbesolvedexactly Info. The twoplevelsystem playsaspecialplaceinthemoderndevelopment ofquantumtheoryˆznparticular“itprovidesaplatformtoencodethesimplest quantumlogicgate“the qubit ˆrclassicalcomputerhasamemorymadeupof bits“whereeachbitholdseitheraoneorazeroˆrquantumcomputermaintainsa

sequenceofqubitsˆrsinglequbitcanholdaone“azero“or“crucially“anyquantum superpositionoftheseˆMoreover“apairofqubitscanbeinanyquantumsuperposition offourstates“andthreequbitsinanysuperpositionofeightˆzngeneralaquantum computerwith qubitscanbeinanarbitrarysuperpositionofuptoc di erent statessimultaneouslyTthiscomparestoanormalcomputerthatcanonlybeinone ofthesec statesatanyonetimeUˆrquantumcomputeroperatesbymanipulating thosequbitswithafixedsequenceofquantumlogicgatesˆThesequenceofgatesto beappliediscalledaquantumalgorithm rnexampleofanimplementationofqubitsforaquantumcomputercouldstart

withtheuseofparticleswithtwospinstatesk and “or and Uˆznfactany systempossessinganobservablequantity whichisconservedundertimeevolution andsuchthat hasatleasttwodiscreteandsu cientlyspacedconsecutiveeigenval] ues“isasuitablecandidateforimplementingaqubitˆThisistruebecauseanysuch systemcanbemappedontoane ectivespin]b csystem Example Dynamicsofadriventwoplevelsystem: ¨etusconsidera two]statesystemwith ,V Un Specifyingthewavefunctionbythetwo]componentvector UnT UU“vqˆTbcˆcU translatestotheequationofmotionTexerciseU 21 21 where 21 nT ˆWiththeinitialcondition TaUnb“and TaUna“this

equationhasthesolution 21 sin t, nb where nTT [T 21 eU isknownasthe Rabifrequency ˆThesolution whichvariesperiodicallyintime“describesthetransferofprobabilityfromstatebto statecandbackˆThemaximumprobabilityofoccupyingstatecisa¨orentzianwith max 21 takingthevalueofunityatresonance 21 Exercise. uerivethesolutionfromtheequationsofmotionfor Uˆyintk eliminate fromtheequationstoobtainasecondorderdi erentialequationfor Info. Thedynamicsofthedriventwo]levelsystemfindspracticalapplication inthe Ammoniamaser kTheammoniamoleculeNy hasapryramidalstructure

withanorientationcharacterisedbythepositionofthe“lone]pair”ofelectronssited rdvancedQuantumPhysics
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tuquqTIMEpDEPENDENTPERTURBATIONTHEORYtwt onthenitrogenatomˆrtlowtemperature“themoleculecanoccupytwopossible states and “involvingsymmetricTSUoranantisymmetricTrUatomiccon] figurations“separatedbyasmallenergysplitting ˆTMoreprecisely“alongthe axisofthree]foldrotationalsymmetry“thee ectivepotentialenergyofthenitrogen atomtakestheformofadouble]wellˆThetunnelingofthenitrogenatomthroughthe doublewellleadstothesymmetricandasymmetriccombinationofstatesˆUznatime]

dependentuniformelectricfieldthemoleculesexperienceapotential where cos “and denotestheelectricdipolemomentˆSince isoddun] derparitytransformation “and and “thema] trixelementsoftheelectricdipolemomentareo ]diagonalk na and 〉& na Charles Hard Townes 1915- (left) isanAmerican Nobelprizey winningphysicist andeducatorz Townesisknown forhisworkon thetheoryand applicationofthemaser–microwave amplificationbystimulatedemission ofradiationxonwhichhegotthe fundamentalpatentxandotherwork inquantumelectronicsconnected withbothmaserandlaserdevicesz HereceivedtheNobelPrizein

Physicsin1964z zfwestartwithallofthemoleculesinthesymmetricgroundstate“wehave shownabovethattheactionofanoscillatingfieldforaparticulartimecancandrive acollectionofmoleculesfromtheirgroundstateintotheantisymmetricfirstexcited stateˆTheammoniamaserworksbysendingastreamofammoniamolecules“traveling atknownvelocity“downatubehavinganoscillatingfieldforadefinitelength“sothe moleculesemergingattheotherendareallToralmostall“dependingontheprecision ofingoingvelocity“etcˆUinthefirstexcitedstateˆrpplicationofasmallamountof

electromagneticradiationofthesamefrequencytotheoutgoingmoleculeswillcause sometodecay“generatingintenseradiationandthereforeamuchshorterperiodfor alltodecay“emittingcoherentradiation 12.2Time-dependentperturbationtheory Wenowturntoconsideragenerictime]dependentyamiltonianforwhichan analyticalsolutionisunavailable–sadlythetypicalsituationMznthiscase wemustturntoaperturbativeanalysis“lookingforanexpansionofthebasis coe cients Uinpowersoftheinteraction Un (0) (1) U[ (2) U[ where Uand (0) issomeTtime]independentUinitialstateˆThe programmetocompletethisseriesexpansionisstraightforwardbuttechnical

Info. zntheinteractionrepresentation“thestate canberelatedto aninitalstate throughthetime]evolutionoperator t,t U“iˆe t,t ˆSincethisistrueforanyinitialstate “fromvqˆTbcˆbU“we musthave t,t Un t,t withtheboundarycondition ,t Un ˆzntegratingthisequationfrom to formallyweobtain t,t Un dt ,t Thisresultprovidesa self-consistent equationfor t,t U“iˆeˆifwetakethisexpres] sionandsubstitute ,t Uundertheintegrand“weobtain t,t Un dt U[ dt dt "" "" "" ,t zteratingthisprocedure“wethusobtain t,t Un =0 dt dt TbcˆdU rdvancedQuantumPhysics
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tuquqTIMEpDEPENDENTPERTURBATIONTHEORYtwu wheretheterm

natranslatesto ˆNotethattheoperators Uareorganisedin atime]orderedsequence“with ˆWiththisunderstanding wecanwritethisexpressionmorecompactlyas t,t UnT dt where“T”denotesthetime]orderingoperatoranditsactionisunderstoodbyvqˆTbcˆdU zfasystemispreparedinaninitialstate attime “atasubsequent time “thesystemwillbeinafinalstate i,t ,t t,t )* t,t MakinguseofvqˆTbcˆdU“andtheresolutionofidentity “we obtain Un (0) ()*+ ni (1) )* dt (2) )* dt dt "" "" Recallingthat t/ Ve t/ “wethusfindthat (1) Un dt ni ni (2) Un dt dt "" nm mi !! nm mi "" TbcˆeU where nm Un and nm nT “etcˆznparticular“the

probabilityofe ectingatransitionfromstate tostate for isgiven by (1) U[ (2) U[ | Example Thekickedoscillator: Supposeasimpleharmonicoscillatoris preparedinitsgroundstate attime ˆzfitisperturbedbyasmalltime] dependentpotential Un eExe “whatistheprobabilityoffindingitinthe firstexcitedstate “at n[ Workingtothefirstorderofperturbationtheory“theprobabilityisgivenby (1) where (1) Un dt 10 10 U 10 Un eE and 10 ˆUsingtheladderoperatorformalism“with and U“wehave ˆTherefore“makinguseoftheidentity dt exp[ ]n exp[ e]“weobtainthetransitionamplitude (1) Un ieE

ˆrsaresult“weobtainthetransitionprobabil] ity eE ˆNotethattheprobabilityismaximizedfor Exercise. tonsideringthesameperturbation“calculatethecorresponding transitionprobabilityfromthegroundstatetothe second excitedstateˆyintknote thatthiscalculationdemandsconsiderationofthesecondorderofperturbationtheory rdvancedQuantumPhysics
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tuqvq“SUDDEN”PERTURBATIONtwv 12.3“Sudden”perturbation Tofurtherexplorethetime]dependentperturbationtheory“weturnnowto considertheactionoffastor“sudden”perturbationsˆyerewedefinesudden asaperturbationinwhichtheswitchfromonetime]independentyamiltonian

toanother takesplaceoveratimemuchshorterthananynatural periodofthesystemˆznthiscase“perturbationtheoryisirrelevantkifthe systemisinitiallyinaneigenstate of “itstimeevolutionfollowingthe switchwillfollowthatof “iˆeˆonesimplyhastoexpandtheinitialstateas asumovertheeigenstatesof ˆThenon]trivialpart oftheproblemliesinestablishingthatthechangeissuddenenoughˆThisis achievedbyestimatingtheactualtimetakenfortheyamiltoniantochange andtheperiodsofmotionassociatedwiththestate andwithitstransitions toneighboringstates 12.3.1Harmonicperturbations:Fermi’sGoldenRule

¨etusthenconsiderasystempreparedinaninitialstate andperturbedby aperiodicharmonicpotential Un Ve whichisabruptlyswitchedonat time naˆThiscouldrepresentanatomperturbedbyanexternaloscillating electricfield“suchasanincidentlightwaveˆWhatistheprobabilitythat“at somelatertime “thesystemliesinstate wromvqˆTbcˆeU“tofirstorderinperturbationtheory“wehave (1) Un dt Theprobabilityofe ectingthetransitionafteratime isthereforegivenby Plotofsin for nb Notethat“as “thisfunc] tionasymptotestoa ]function U (1) sinTT t/ cU Setting nT c“theprobabilitytakestheformsin witha peakat na“withmaximumvalue

andwidthoforderb /t givingatotal weightoforder ˆThefunctionhasmorepeakspositionedat nT [b cU Theseareboundedbythedenominatoratb ˆworlarge theircontribution comesfromarangeoforderb /t also“andas thefunctiontendstowards ]functioncentredattheorigin“butmultipliedby “iˆeˆthelikelihoodof transitionisproportionaltotimeelapsedˆWeshouldthereforedivideby to getthetransitionrate winally“withthenormalisation sin( “wemaye ectthe replacement“lim sin( Unc Tc Uleadingtothefollowing expressionforthetransitionrate Enrico Fermi 1901-1954: AnItalianphysiy cistmostnoted forhisworkon thedevelopment ofthefirst

nuclearreactorx andforhis contributionsto thedevelopment ofquantum theoryxnuclearandparticlephysicsx andstatisticalmechanicszFermiwas awardedtheNobelPrizeinPhysics in1938forhisworkoninduced radioactivityandistodayregarded asoneofthemostinfluential scientistsofthe20thcenturyzHeis acknowledgedasauniquephysicist whowashighlyaccomplishedinboth theoryandexperimentzFermiumxa syntheticelementcreatedin1952is namedafterhimz Unlim TbcˆfU Thisexpressionisknownas Fermi’sGoldenRule Onemightworrythat inthelongtimelimit“wefoundthattheprobabilityoftransitionisinfact

CuriouslylalthoughnamedafterFermilmostoftheworkleadingtotheGoldenRulewas undertakeninanearlierworkbyDiraclgPnAnMnDiracl The quantum theory of emission and absorption of radiation nProcnRoynSocngLondonhA 114 lrtsrvugqyrwhhwhoformulated analmostidenticalequationlincludingthethreecomponentsofaconstantlthematrix elementoftheperturbationandanenergydi erencenItisgivenitsnameduetothefact thatlbeingsuchausefulrelationlFermihimselfcalledit“GoldenRuleNonr”gEnFermil Nuclear Physics lUniversityofChicagoPresslqyuphn rdvancedQuantumPhysics
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tuqvq“SUDDEN”PERTURBATIONtww

diverging—sohowcanwejustifytheuseofperturbationtheorypwora transitionwith “the“longtime”limitisreachedwhen U avaluethatcanstillbeveryshortcomparedwiththemeantransitiontime whichdependsonthematrixelementˆznfact“wermi’sRuleagreesextremely wellwithexperimentwhenappliedtoatomicsystems Info. AlternativederivationoftheGoldenRule: Whenlightfallson anatom“thefullperiodicpotentialisnotsuddenlyimposedonanatomictimescale butbuildsupovermanycyclesToftheatomandofthelightUˆzfweassumethat Un Ve “with verysmall isswitchedonverygraduallyinthepast“and wearelookingattimesmuchsmallerthanb

ˆWecanthentaketheinitialtimeto be “thatis (1) Un fl dt fl iˆe fl ˆrppliedtothetransitionrate dt (1) “the identitylim fl UleadstothexoldenRule wromtheexpressionforthexoldenruleTbcˆfUweseethat“fortransitionsto occur“andtosatisfyenergyconservationk TaUthefinalstatesmustexistoveracontinuousenergyrangetomatch forfixedperturbationfrequency “or TbUtheperturbationmustcoverasu cientlywidespectrumoffrequencyso thatadiscretetransitionwithafixed ispossible wortwodiscretestates“since if “wehavethesemiclassicalresult –astatementof detailedbalance 12.3.2 Info:

Harmonicperturbations:second-ordertransi- tions rlthoughthefirstorderperturbationtheoryisoftensu cienttodescribe transitionprobabilities“sometimesfirstordermatrixelement isiden] ticallyzeroduetosymmetryTeˆgˆunderparity“orthroughsomeselectionrule etcˆU“butothermatrixelementsarenon]zeroˆznsuchcases“thetransition maybeaccomplishedbyanindirectrouteˆWecanestimatethetransition probabilitiesbyturningtothesecondorderofperturbationtheoryTbcˆeU (2) Un dt dt "" !! "" zf“asabove“wesupposethataharmonicpotentialperturbationisgradu] allyswitchedon Un Ve “withtheinitialtime “wehave (2) Un dt dt ""

!! Theintegralsarestraightforward“andyield (2) Then“followingourdiscussionabove“weobtainthetransitionratek dt (2) rdvancedQuantumPhysics
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tuqvq“SUDDEN”PERTURBATIONtwy Thisisatransitioninwhichthesystemgainsenergyc fromtheharmonic perturbation“iˆeˆtwo“photons”areabsorbedinthetransition“thefirsttaking thesystemtotheintermediateenergy “whichisshort]livedandtherefore notwelldefinedinenergy–indeedthereisnoenergyconservationrequirement forthevirtualtransitionintothisstate“onlybetweeninitialandfinalstates

Ofcourse“ifanatominanarbitrarystateisexposedtomonochromaticlight othersecondorderprocessesinwhichtwophotonsareemitted“oroneis absorbedandoneemittedTineitherorderUarealsopossible rdvancedQuantumPhysics