<<

Time_evolution.nb:9/26/04::22:57:04 1

Timeevolutionandquantumdynamics

Sakurai 2.1, 2.2 , Merzbacher 14, BJ 5.7, 5.8

ü evolution&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 ,thenthetimeevolutionoperationsforma.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 ÅÅÅÅÅÅ dt U t, t0 .Forsmalldt + / + / + / + / + /  U+t+dt,t0/U+t,t0/  ∑ HU t, t0 = i ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅdt = i ÅÅÅÅÅÅ∑t U t, t0 + / + / ü Fora?

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 tt U t, t0 = e  + 0/ + / Onemayalsoconsiderthelimitofmanyshortevolutionsteps.

n i i Dt ÅÅÅÅ H+tt0/ U t, t0 = Lim 1 ÅÅÅÅ H ÅÅÅÅÅÅÅ = e nØ¶ n + / + / Alternatively,considerasingleenergyeigenstate

a, t = ci t i  ? + /  ?  ∑ i ÅÅÅÅ∑tÅÅ ci t i = H ci t i = Eici t i or,since i isjustconstantandnotchangingwithtime,wehave + /  ? + /  ? + /  ?  ? i w tt 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 ÅÅÅÅ H t1 = Se ÅÅÅÅ Hti?;i = Si?e ÅÅÅÅ Eit;i i i

orforastate a0, t = Sca j t  j? j  ? + / i w t i w t a0, t = U t, t0 a, t0? = Sca j t0 Si?e i i j = Scai t0 e i i? j i i  ? + / + / ; « ? + /

ü Eigenvectorsarefixed,butenergiesvarywithtime

b)H varieswithtime,but H t1 , H t2 = 0.Thiscorrespondstothecasewheretheeigenvectorsremainthesame,but 1 theenergyeigenvalueschangewithtime.Anexampleofthiswouldbea# + / + /' 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 tn1 U t, t0 = 1 + S ÅÅÅÅÅÅÅ dt1 dt2 ... dtnH 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 thattheinnerproductisonlydefinedatequal- 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 = c1e 1 1 + c2e 1 2  ?  ?  ? Theamplitudeforthestatetostillbeinstate a attimetis  ? Aaa t = a U t, 0 a * * iw1t iw1t + / =; «1 c+1 + /2« c?2 c1e 1 + c2e 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 † ÅÅÅÅ Ht ÅÅÅÅ Ht O t = a, t O a, t = a U t, 0 OU t, 0 a = a e Oe a . ; ? ;   ? ;  + / + /  ? ;   ? ü ConservationandcommutationwithH

i ÅÅÅÅ 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.

* iw1t * iw2t iw1t iw2t a0, t O a0, t = 1 c1e + 2 c2e O c1e 1 + c2e 2 ;   ? +;  ;  / +  ?  ?/ If O, H = 0thentheenergyeigenstatesarealsoeigenstatesoftheoperatorO,soO i = oi i and # '  ?  ? * iw1t * iw2t iw1t iw2t a0, t O a0, t = 1 c1e + 2 c2e c1e o1 1 + c2e o2 2 2 2 ;  = c1 ?o1 +; c 2 o2, ;  / +  ?  ?/     O isindependentoftimeandisconserved. ; ? Ontheotherhand,if O, H ∫ 0thenonemustconsideramoregeneralOwhichmixesenergyeigenstates.Adopting o o # 11 '12 * matrixnotationO = .IfOcorrespondstoanitisHermitian,ando21 = o12 .Then L o21 o22 \ M ] N ^ iw t o11 o12 c1e 1 * iw1t * iw2t c1 e c2 e iw t = L o21 o22 \ L c e 2 \ + / M ] M 2 ] N N iw1t iw2t ^o11 c1e ^ + o12 c2e * iw1t * iw2t 2 2 * iwt c1e c2e = c1 o1 + c2 o2 + 2Re c1c2e o12 L o c eiw1t + o c eiw2t \ + / 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 + 2Re ÅÅ2ÅÅ e = cos wt ,and 1 iwt ;sy?+t/ = 0 + 0 + 2Re + 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.

AcommonnotationistodenotestatesandoperatorsintheSchroedingerpicturebyasubscriptS ,andcorresponding quantitiesintheHeisenbergpicturebyasubscriptH .Thenthetimeevolutionofthematrixelementaboveis

† bS, t OS aS, t = bS, 0 U t, 0 OSU t, 0 aS, 0 ª bH OH t aH , ;   ? ;  + / + /  ? ;  + /  ? Note thatthematrixelementsarethesameineitherpicture-thephysicsshouldnotdependonthecoordinatesystem.

InthecasewhereH isconstantintime,theHeisenbergoperatorisgivenby

ÅÅiÅÅ Ht ÅÅiÅÅ Ht OH = e  OSe 

If OS, H = 0,thenOH = OS . # ' ° AgaintakingH = 0,theSchroedingerstateshaveexplicittimedependence,buttheHeisenbergstatesdonot,

† aH = aS, 0 = U t, 0 aS, t  ?  ? + /  ? ComparedtotheSchroedingerstate,thecorrespondingHeisenbergstateappearstobedriftingbackwardintime.Infact, theHeisenbergstateisstatic.

ü ExampleintheSchroedingerpicture

Considerourfriendthetwostatesystem,andforsimplicitytake b = a .IntheSchroedingerpicture,weapplyU to thestates  ?  ?

;aU † t, 0 OU t, 0 a? = . i w t i w t iw t + / + /e 1 0 o11 o12 e 1 0 c1 o11 o12 c1e 1 c* c* * iw1t * iw2t 1 2 i w t i w t = c1 e c2 e iw t L 0 e 2 \ L o21 o22 \ L 0 e 2 \ L c2 \ L o21 o22 \ L c e 2 \ + / M ] M ] M ] M ] + / M ] M 2 ] N ^ N ^ N ^ N ^ N ^ N ^ ü ExampleintheHeisenbergpicture

i i IntheHeisenbergpicture,wecombinetheoperatorsfirst,O t = U † t, 0 OU t, 0 = e ÅÅÅÅ H tO eÅÅÅÅ H t + / + / + / Time_evolution.nb:9/26/04::22:57:04 7

ei w1t 0 o o ei w1t 0 c o o eiwt c c* c* 11 12 1 c* c* 11 12 1 1 2 i w t i w t = 1 2 iwt ,where L 0 e 2 \ L o21 o22 \ L 0 e 2 \ L c2 \ L o e o \ L c2 \ + / M ] M ] M ] M ] + / M 21 22 ] M ] w = w1 wN 2 . ^ N ^ N ^ N ^ N ^ N ^

ü EquationofmotionforoperatorsintheHeisenbergpicture.

ThetimeevolutionofO t = U †OU isgivenby + / dO+t/ dU† † dU i † † i † † ÅÅÅÅÅÅÅÅÅÅÅÅÅdt = ÅÅÅÅÅÅÅÅÅÅÅdt OU + U O ÅÅÅÅÅÅÅÅÅdt = ÅÅÅÅ U HUU OU ÅÅÅÅ U OUU HU = i † † i i ÅÅÅÅ U HUO t O t U HU = ÅÅÅÅ HO t O t H = ÅÅÅÅ H, O t + + / + / / + + / + / / # + /' whereweusedtheSchr.eqforU ,putin1 = UU † ,convertedbacktoO t ,andusedU †HU = H foraHamiltonian whichisindependentoftime.Cuttingoutthemiddlesteps + /

dO+t/ i ÅÅÅÅÅÅÅÅÅÅÅÅÅdt = ÅÅÅÅ H, O t . # + /' Note thecorrespondencetoPoissonbrackets.

ü Interactionpicture(Dirac)

TheHeisenbergandSchroedingerpicturesareprescriptionsforhowtotreatthetimeevolutionofthephasesina quantumsystem.Theyhavedifferentvaluefortreatingdifferentproblems.TheSchroedingerpicturegivesanintuitive pictureofhowthestateofasystemevolves.Ithasthedrawbackthatifthesystemevolvesforalongtimethatthe calculationofthosephasesmaybeannoying.IntheHeisenbergpicturethestatesarestaticforatimeindependent Hamiltonian,andsotheevolutionofthesystemisalmosttrivial.Ithasthedrawbackthattheevolutionofoperatorsis notasintuitiveaconcept.Inaddition,ifH isnotconstantintimethattheHeisenbergstateshavenon-trivialtime evolution.Athirdmethodforallocatingthetimedependenceofasystem,the"interactionpicture",isoftenusedfor calculatingtransitionamplitudes.

SupposethattheHamiltonianofasystemisgivenbyH = H0 + H1 t ,whereH0 istimeindependentbutH1 isnot. TypicallyH0 hasknowneigenstates,labeledas n .Acommoncaseisthat+ / H1 t isonlynon-zeroforanintervaloftime ti < t < t f .Inthiscase,itwouldbeusefultoconsiderthesystemintheHeisenbergpicturef ? + / ort < ti andfort > t f ,but inaSchroedinger-likemannerforthetimewhenH1 isactive.Supposethesystemstartsinaparticularstate,say i .It remainsinthatstateuntilt = ti atwhichtimeitstartstoevolveundertheinfluenceofH1 untilt f ,atwhichpointthe ? systemstopsevolving(atleastintheHeisenbergpicture).Thetimeevolutionofthesystemiscapturedbycalculating thetransitionamplitudes n i fromstate i toanyotherstate n . ; « ?  ?  ? InsuchacaseitisconvenienttoseparateouttheeffectsofthetimeindependentH0 ,anddefinetheoperatorinthe interactionpicture

ÅÅiÅÅ H t ÅÅiÅÅ H t OI t = e  0 OS e  0 + / Note thatinthisexampletheoperatorOS typicallyhasexplicittimedependence.Inparticular,thetimeevolutionofa stateintheinteractionpictureisdeterminedby

ÅÅiÅÅ H t ÅÅiÅÅ H t H1I t = e  0 H1 t e  0 + / + / Time_evolution.nb:9/26/04::22:57:04 8

SchroedingerWaveEquationandcorrespondencetoclassical mechanics

Forthiscourseitisassumedthatthestudentisfamiliarwithcorrespondenceprinciples.Thediscussionbelowfocuses onderivingtheSchroedingerwaveequationfromthemoreformaldiscussionsalreadygiven,ratherthantakingitasan ad-hocdescriptionofnature.

ü SchroedingerWaveEquation

ü Timedependentversioninonedimension

Considerya x, t = x a, t where x isjustabasisketinthecontinuum x basis,i.e.itisnotatimedependentstate. p2 Wewanttoknowhow+ / ; «yevolvesintimeundertheHamiltonian?  ? H = ÅÅÅÅÅÅÅÅÅ2m +V? X .Theequationofmotionis + /  ∑ i ÅÅÅÅ∑tÅÅ a, t = H a, t  ?  ? Actingontheleftwith x yieldstheusualSchroedingerequationforthewavefunctionya x, t .FortheLHSthereis notimedependenceto;x ,andso + /  ?  ∑  ∑  ∑ x i ÅÅÅÅ∑tÅÅ a, t = i ÅÅÅÅ∑tÅÅ x a, t = i ÅÅÅÅ∑tÅÅ ya x, t ;   ? ; « ? + / OntheRHS

p2 x H a, t = x ÅÅÅÅÅÅÅÅÅ2m + V X a, t ;   ? ; p2 + /  ? = x ÅÅÅÅÅÅÅÅÅ2m a, t + x V X a, t ;   ? ;  + /  ? ForthepotentialtermV X actstotheleft,so x V X a, t = V x x a, t = V x ya x, t .Forthemomentumterm,  ∑  ∑ usep a, t = i ÅÅÅÅ∑xÅÅÅ a,+t /= i a', t ,where;a', t+ isdefinedby/  ? + /x; a« ', t ?ª ÅÅÅÅ∑xÅÅÅ+y/a x,+ t .Althoughthisisabitglib,it/ isgoodenoughforthecurrentdiscussion.Wewilldiscusstherelationof ?  ?  ?  ? ; « xand? pinmoredetaillaterunderthetopicof+ / p2 2 '' translationalsymmetry.Fornowthisallows x ÅÅÅÅÅÅÅÅÅ2m a, t = ÅÅÅÅÅÅÅÅÅ2m ya x, t .Combiningtheleftandrighthandsidesof thetimeevolutionequationgives ;   ? + /

 ∑ 2 '' i ÅÅÅÅÅÅ∑t ya x, t = ÅÅÅÅÅÅÅÅÅ2m ya x, t + V x ya x, t + / + / + / + / ü Timedependentversionin3-d

p2+p2+p2 Allowingfor3spatialdimensions,theHamiltonianisH = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅx y z + V X ,whichleadsto 2m ¹¹¶ + /  ∑ 2 ∑2 ∑2 ∑2 i ÅÅÅÅÅÅ∑t ya x, t = ÅÅÅÅÅÅÅÅÅ2m ÅÅÅÅÅÅÅÅÅ∑x2 + ÅÅÅÅÅÅÅÅÅ∑y2 + ÅÅÅÅÅÅÅÅÅ∑z2 ya x, t + V x ya x, t ¶ ¶ ¶ ¶ + /2 2 , 0 + / + / + / = ÅÅÅÅÅÅÅÅÅ2m “ ya x, t + V x ya x, t +¶ / +¶/ +¶ / ü comments

ü continuityequation,probabilityconservation Time_evolution.nb:9/26/04::22:57:04 9

ü Classicallimit