Time_evolution.nb:9/26/04::22:57:04 1 Timeevolutionandquantumdynamics Sakurai 2.1, 2.2 , Merzbacher 14, BJ 5.7, 5.8 ü Timeevolutionoperator&theHamiltonian ü Basisketsandevolvingstates Startbydefiningabasis i whichspansthevectorspaceofpossiblestates.Thesecouldbeeignstatesofsome operator,sayA.ItisoftenusefultoconsidereigenstatesoftheHamiltonian ? H .Thesestatesarestatic.Ifdefinedin termsofanoperator,andtheoperatorisafunctionoftime,thenthedefinitionofthebasisshouldspecifythetime,in whichcasewemaywrite i t .Thisdoesnotimplytimeevolution,itisjustalabelforwhichsetofstatesweare usingasabasis.Forthemomentwewillconsiderabasisthatdoesnotchangewithtim + /? e. Wehavepostulatedthatthestateofthesystemcanbeexpressedasalinearcombinationofthebasiskets. a = cai i ? ? Now, ingeneral,thestatewillevolveintime,sowewrite a, t = cai t i ? + / ? Note thatthetimedependenceisallinthecai t .Thisisthesocalled"Schroedinger"picture. + / ü U+t, t0/ Definethetimeevolutionoperator U t, t0 a, t0 = a0, t + / ? ? Thesubscript0 isintendedtodenotethatthesystemwasinthestate a attimet0 ,andthenevolvedtot.Thisis generallynotthesameasasystemwhichisinstate a attimet. ? ? WedesireafewpropertiesoftheoperatorU † +Unitary:If a, t0 a, t0 = 1,then a, t a, t = a, t0 U t, t0 U t, t0 a, t0 = 1,i.e,wewantU tobeunitary. 2 Note: thisimplies; S« cai t? = 1forall; t. « ? ; + / + / ? i + / +Lawofcomposition:U t2, t0 = U t2, t1 U t1, t0 ,fort2 > t1 > t0 .Thiscompositionlawisassociative U t3, t2 U t2, t1 U t1, t0+ = U/ t3, t2+ U /t2, t+1 U /t1, t0 + + / + // + / + / + + / + // +Identity:U t1, t1 = 1. + / -1 † +IfweaddInverse:U t, t0 = U t, t0 ,thenthetimeevolutionoperationsformagroup.Wewilldiscusssymmetry groupslateroninconnectionwithspatialtranslationsandrotations.+ / + / Time_evolution.nb:9/26/04::22:57:04 2 Unitaritysuggeststheexpansionforinfinitesimaltimetranslations U t0 + dt, t0 = 1 - i W dt. + / TheHermitianoperatorWisthegeneratoroftimetranslations.Theexplicitfactorofiisrequiredtomakethe infiinitesimaltranslationUnitary,i.e.itisrequiredtohaveU -1 = U † = 1 + i W dt,sothattofirstorder U † U = 1 + i W dt 1 - i W dt = 1 + $ dt2 + / + / + / Whasdimensionsoffrequency.TheHamiltonianisdefinedbyH = Wandhasdimensionsofenergy,butisotherwise just thegeneratoroftimetranslations. ü Schroedingerequations TheSchroedingerequationisadifferentialequationwhichdescribesthetimeevolutionofaquantumsystem.Themost basicversiondescribesthetimeevolutionofthetimeevolutionoperatoritself.Ifthisisknown,itmaybeappliedto determinetheevolutionofthesysteminanystate. H ConsiderU t + dt, t0 = U t + dt, t U t, t0 = 1 - i ÅÅÅÅÅÅ dt U t, t0 .Forsmalldt + / + / + / + / + / U+t+dt,t0/-U+t,t0/ ∑ HU t, t0 = i ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅdt = i ÅÅÅÅÅÅ∑t U t, t0 + / + / ü Fora? Alternatively,onecanderiveanequationforthestatevectordirectly. ∑ ∑ HU t, t0 a, t0 = i ÅÅÅÅ∑tÅÅ U t, t0 a, t0 orH a, t = i ÅÅÅÅ∑tÅÅ a0, t .Notethedifferencebetweenthestaticstate a, t0 andtheevolvingstate+ / ? a0, t+ . / ? ? ? ? ? ü Basiccase,H isconstant(noexplicittimedependence) OnecansolvethedifferentialequationwithboundaryconditionU t0, t0 = 1. + / - ÅÅiÅÅ H t-t U t, t0 = e + 0/ + / Onemayalsoconsiderthelimitofmanyshortevolutionsteps. n i i Dt - ÅÅÅÅ H+t-t0/ U t, t0 = Lim 1 - ÅÅÅÅ H ÅÅÅÅÅÅÅ = e nض n + / + / Alternatively,considerasingleenergyeigenstate a, t = ci t i ? + / ? ∑ i ÅÅÅÅ∑tÅÅ ci t i = H ci t i = Ei ci t i or,since i isjustconstantandnotchangingwithtime,wehave + / ? + / ? + / ? ? -i w t-t ci t = e i+ 0/ ci t0 withwi = Ei + / + / s Andforamoregeneralstateuse Time_evolution.nb:9/26/04::22:57:04 3 i i i e- ÅÅÅÅ H t 1 = S e- ÅÅÅÅ Ht i? ;i = S i? e- ÅÅÅÅ Ei t ;i i i orforastate a0, t = S ca j t j? j ? + / -i w t -i w t a0, t = U t, t0 a, t0? = S ca j t0 S i? e i i j = S cai t0 e i i? j i i ? + / + / ; « ? + / ü Eigenvectorsarefixed,butenergiesvarywithtime b)H varieswithtime,but H t1 , H t2 = 0.Thiscorrespondstothecasewheretheeigenvectorsremainthesame,but 1 theenergyeigenvalueschangewithtime.Anexampleofthiswouldbeaspin# + / + /' ÅÅ2ÅÅ particlepropagatingthroughamagnetic fieldwhereBchangesinmagnitude,butnotdirection. ¹¶ - ÅÅÅÅi t H t' dt' ¼t + / U t, t0 = e 0 + / andsimilarlyforci . ü Eigenvectorsarechangingintime 1 c) H t1 , H t2 ∫ 0Thiscaseisnon-trivial.Anexamplewouldbethes = ÅÅÅÅ casejustmentioned,butwhereB 2 ¹¶ changesdirection.Aformalsolutionis(seeSakurai)# + / + /' -i n t t1 tn-1 U t, t0 = 1 + S ÅÅÅÅÅÅÅ dt1 dt2 ... dtn H t1 H t2 ... H tn n t0 t0 t0 + / + / ¼ ¼ ¼ + / + / + / Thisseriesisrelatedtoformalperturbationtheory. ü TransitionAmplitudes Timeevolutioncanbeconsideredasa"transitionamplitude".Att0 thesystemstartsinstate a .Attimet,systemhas evolvedto a0, t = U t, t0 a, t0 .Then,theamplitudetobeinastate b attimetisgivenby ? ? + / ? ? Aab t = b, t a0, t = b, t U t, t0 a, t0 . + / ; « ? ; « + / « ? Theprobabilityforasysteminitiallyinstate a toevolvetostate b isjustthesquareofthisamplitude. ? ? 2 Pab t = Aab t + / + / Note thattheinnerproductisonlydefinedatequaltimes- b, t a, t0 isnotdefined.Tocomparetwostatesfrom differenttimes,onemusttimeevolveoneofthestatestothetimeoftheother.; « ? ü Examplefortwostatesystem Consideratwostatesystemwhich,att = 0,isinthestate a ª a, 0 = c1 1 + c2 2 . ? ? ? ? where 1 and 2 areenergyeigenstates.Then,atsomelatertimet,thesystemwillevolveto ? ? Time_evolution.nb:9/26/04::22:57:04 4 -iw t -iw t a0, t = c1 e 1 1 + c2 e 1 2 ? ? ? Theamplitudeforthestatetostillbeinstate a attimetis ? Aaa t = a U t, 0 a * * -iw1 t -iw1 t + / =; «1 c+1 + /2« c?2 c1 e 1 + c2 e 2 2 -iw t 2 -iw t = +c;1 e 1; + c/2+ e 2 ? ?/ Theprobabilitytoremaininstate a is ? 2 Paa t = Aaa t 4 4 2 2 + / = c1 + /+ c2 + 2 c1 c2 cos wt + / wheretheoscillationfrequencyisw = w1 - w2 . Herearethreeoscillationplots.Notethefullymixedcaseinblack.Thesmallmixingcaseinred.Thebluecurvehas intermediateamountofmixing,andissettooscillateattwicethefrequency.Thisisadescriptionofneutrinooscillation withtwoflavors.Startoutinstateof ne andoscillateintoanother"flavor".Adetectoratdifferentdistancesfromthe sourcewillmeasuredifferentfluxesof n? e . ? paa #c1_, c2_, w_, t_ ' : Abs #c1 '^4 Г Abs #c2 '^4 Г 2 Abs #c1 '^2 Abs #c2 '^2 Cos #w t'; Plot #paa #Sqrt #.5 ', Sqrt #.5 ', 1, t', paa #Sqrt #.01 ', Sqrt #.99 ', 1, t', paa #Sqrt #.2 ', Sqrt #.8 ', 2, t', t, 0, 10 '; 1 0.8 0.6 0.4 0.2 0 0 2 4 6 8 10 Youmaywishtoconsiderthedifferencebetween"appearance"and"disappearance"experiments,especiallyinthecase wherewt + 1. ü commentoncorrelationamplitude ü Evolutionofexpectationvalues&Conservedquantities ü expectationvalues ConsidertheevolutionoftheexpectationvalueforanoperatorO.Ifattimet = 0 Time_evolution.nb:9/26/04::22:57:04 5 O t=0 = a O a ; ? ; « « ? Thenatsomelatertime i i † ÅÅÅÅ Ht - ÅÅÅÅ Ht O t = a, t O a, t = a U t, 0 OU t, 0 a = a e O e a . ; ? ; ? ; + / + / ? ; ? ü ConservationandcommutationwithH i - ÅÅÅÅ Ht ∑ If O, H = 0,then O, e = 0,and O t = O t=0 ,oralternatively, ÅÅÅÅ∑tÅÅ O = 0.IfOcommuteswithH thenO correspondstoaconservedquantity,butnototherwise.# ' $ ( ; ? ; ? ; ? ü Thetwostatesystemagain Forexample,considerthetwostatesystemfromabove. * iw1 t * iw2 t -iw1 t -iw2 t a0, t O a0, t = 1 c1 e + 2 c2 e O c1 e 1 + c2 e 2 ; ? +; ; / + ? ?/ If O, H = 0thentheenergyeigenstatesarealsoeigenstatesoftheoperatorO,soO i = oi i and # ' ? ? * iw1 t * iw2 t -iw1 t -iw2 t a0, t O a0, t = 1 c1 e + 2 c2 e c1 e o1 1 + c2 e o2 2 2 2 ; = c1 ? o1 +; c 2 o2, ; / + ? ?/ O isindependentoftimeandisconserved. ; ? Ontheotherhand,if O, H ∫ 0thenonemustconsideramoregeneralOwhichmixesenergyeigenstates.Adopting o o # 11 '12 * matrixnotationO = .IfOcorrespondstoanobservableitisHermitian,ando21 = o12 .Then L o21 o22 \ M ] N ^ -iw t 1 * iw t * iw t o11 o12 c1 e 1 2 c1 e c2 e -iw t = L o21 o22 \ L c e 2 \ + / M ] M 2 ] N N -iw1 t -iw2 t ^o11 c1 e ^ + o12 c2 e * iw1 t * iw2 t 2 2 * iwt c1 e c2 e = c1 o1 + c2 o2 + 2 Re c1 c2 e o12 L o c e-iw1 t + o c e-iw2 t \ + / M 21 1 22 2 ] + / N ^ ü spinprecesion e 1 0 Asanexample,considerspinprecesion.TheHamiltonianisH = ÅÅÅÅ sz ,wheresz = ,e = wandstate 1 2 L 0 -1 \ M ] ? correspondstothe z+ polarizationstate.PrecessionisdescribedbytheevolutionofN ^ 0 1 ? 0 i sx = , sy = and sz .Firstconsiderthecasewherethesystemstartsinaspinupstate, L 1 0 \ L -i 0 \ ; ? >M ]B ; ? >M ]B ; ? c1 = 1, cN2 = 0. ^Then sx =N sy =^ 0sincefortheseoperatorso11 = 0.Also, sz t = 1.Nextconsiderthecasewhere ` 1 1 thesystemstartspolarizedinthe; ? ; ?xdirection,c1 = c2 = 1 2 .Now, sz t ;= ÅÅ?ÅÅ+-/ ÅÅÅÅ = 0.Moreinterestingis r 2 2 t ; ? + / 1 iwt sx t = 0 + 0 + 2 Re ÅÅ2ÅÅ e = cos wt ,and 1 iwt ;sy? +t/ = 0 + 0 + 2 Re + ÅÅ2ÅÅ e / i = sin + w/t . ; ? + / + / + / Thespinprecesses,asitshould. Time_evolution.nb:9/26/04::22:57:04 6 ü "Pictures" ConsideranoperatorOwithnoexplicittimedependence,whichisdefinedforsomevectorspacespannedbythebasis kets i .ThenOisdefinedbythematrixelements i O j .Next,considertheevaluationofObetweentwostates definedattime ? t = 0, b, 0 O a, 0 ,andhowthisquantityevolveswithtime.; ? ; ? † b0, t O a0, t = b0 U t, 0 OU t, 0 a0 ª b0 O t a0 . ; ? ; + / + / ? ; + / ? ThelaststepdefinesanoperatorO t .Thesearethreewaysofwritingthesamething.Thefirstputsthetime dependenceintothestateevolution.Thethirdputsthetimedependenceintotheoperators.In+ / themiddle,boththestates andtheoperatorsarestaticandthetimedependenceisaccountedforbyinclusionoftimeevolutionoperators.Thefirst iscalledtheSchroedingerpicture,thethirdiscalledtheHeisenbergpicture.Thereisabitofrelativityhere-whichis movingthesystemortheoperator?orshouldIsayobserver?Idon'tknowofanameforthemiddlenotationwhereonly U knowsabouttime.