Tim Dpdt Solutios: Propagators ad Rprstatios Michal Fowlr, UVa 1/3/6 Itroductio W v spt most of th cours so far coctratig o th igstats of th amiltoia, stats whos tim dpdc is mrly a chagig phas W did mtio much arlir a suprpositio of two diffrt rgy stats i a ifiit wll, rsultig i a wav fuctio sloshig backwards ad forwards It s ow tim to cast th aalysis of tim dpdt stats ito th laguag of bras, kts ad oprators W ll tak a tim idpdt amiltoia, with a complt st of orthoormalizd igstats, ad as usual ( x, t) ħ ( x, t ) iħ + V ( x) ( x, t ), t m x Or, as w would ow writ it iħ ( x, t) ( x, t ) t Sic is itslf tim idpdt, this is vry asy to itgrat! ( ) i t t / ħ x, t x, t Th xpotial oprator that grats th tim dpdc is calld th propagator, bcaus it dscribs how th wav propagats from its iitial cofiguratio, ad is usually dotd by U: ( x t) U ( t t ) ( x t ),, It s appropriat to call th propagator U, bcaus it s a uitary oprator: i ( t t ) / ħ i t t / ħ i t t / ħ 1 U t t, so U t t U t t Sic is hrmitia, U is uitary It immdiatly follows that ( x, t) ( x, t) ( x, t ) U U ( t t ) ( x, t ) ( x, t ) ( x, t ) th orm of th kt vctor is cosrvd, or, traslatig to wav fuctio laguag, a wav fuctio corrctly ormalizd to giv a total probability of o stays that way (This ca also b provd from th Schrödigr quatio, of cours, but this is quickr)
This is all vry succict, but ufortuatly th xpotial of a scod ordr diffrtial oprator dos t soud too asy to work with Rcall, though, that ay fuctio of a rmitia oprator has th sam st of igstats as th origial oprator This mas that th igstats of i ( t t ) / ħ ar th sam as th igstats of, ad if E, th ħ i t t / ie t t / ħ This is of cours othig but th tim dpdt phas factor for th igstats w foud bfor ad, as bfor, to fid th tim dpdc of ay gral stat w must xprss it as a suprpositio of ths igkts, ach havig its ow tim dpdc But how do w do that i th oprator laguag? Easy: w simply isrt a idtity oprator, th o costructd from th complt st of igkts, thus: i ( t t ) / ħ ( ) / ( ) ie t t ħ 1 1 t t t Starig at this, w s that it s just what w had bfor: at th iitial tim t t, th wav fuctio ca b writt as a sum ovr th igkts: ( t ) ( t ) ( t ) ( t ) c ( t ) with c, c 1, ad th usual gralizatio for cotiuum igvalus, ad th tim dvlopmt is just giv by isrtig th phass: Th xpctatio valu of th rgy E i, ie ( t t ) / ħ ( t) c t E c E ad is (of cours) tim idpdt Th xpctatio valu of th particl positio x is ( ) ( t ) x ( t ) c c t x t, m * i ( E E m ) t t / ħ m m ad is ot i gral tim idpdt (It is ral, of cours, o addig th (,m) trm to th (m,) trm)
3 This aalysis is oly valid for a tim idpdt amiltoia Th importat xtsio to a systm i a tim dpdt xtral fild, such as a atom i a light bam, will b giv latr i th cours Th Fr Particl Propagator To gai som isight ito what th propagator U looks lik, w ll first aalyz th cas of a particl i o dimsio with o pottial at all W ll also tak t to mak th quatios lss cumbrsom For a fr particl i o dimsio momtum igstats, w labl thm k, so E p / m k / m ħ th rgy igstats ar also dk dk π π it / i t / i k t / m U t ħ ħ k k ħ k k Lt s cosidr (followig Shakar ad othrs) what sms th simplst xampl Suppos that at t t, a particl is at x : ( x, t ) δ ( x x ) x : what is th probability amplitud for fidig it at x at a latr tim t? (This would b just its wav fuctio at th latr tim) dk x U t x x k k x (,) iħ k t / m π iħ k t / m m π ħ it usig th stadard idtity for Gaussia itgrals, dk π ( ) ( x ) ik x im x x / ħ t, dk π a ak + bk b / 4 a O xamiig th abov xprssio, though, it turs out to b oss! Notig that th trm i th xpot is pur imagiary, x, t m / π ħ t idpdt of x! This particl appartly istataously fills all of spac, but th its probability dis away as 1/t Qustio: Whr did w go wrog?
4 Aswr: Notic first that ( x, t ) ormalizatio, ( x, t) is costat throughout spac This mas that th dx! Ad, as w v s abov, th ormalizatio stays costat i tim th propagator is uitary Thrfor, our iitial wav fuctio must hav had ifiit orm That s xactly right w took th iitial wav fuctio ( x, t ) δ ( x x ) x Thik of th δ fuctio as a limit of a fuctio qual to 1/ ovr a itrval of lgth, with goig to zro, ad it s clar th ormalizatio gos to ifiity as 1/ This is ot a maigful wav fuctio for a particl Rcall that cotiuum kts lik x ar ormalizd by x x δ x x, thy do ot rprst wav fuctios idividually ormalizabl i th usual ss Th oly maigful wav fuctios ar itgrals ovr a rag of such kts, such as dx x x I a itgral lik this, otic that stats x withi som tiy x itrval of lgth δ x, say, hav total wight ( x) δ x, which gos to zro as δ x is mad smallr, but by writig ( x, t ) δ ( x x ) x w took a sigl such stat ad gav it a fiit wight This w ca t do Of cours, w do wat to kow how a wav fuctio iitially localizd ar a poit dvlops To fid out, w must apply th propagator to a lgitimat wav fuctio o that is ormalizabl to bgi with Th simplst localizd particl wav fuctio from a practical poit of viw is a Gaussia wav packt, x / d ip x / ħ, 1/ 4 ( x ) ( π d ) (I v usd d i plac of Shakar s hr to try to miimiz cofusio with x, tc) Th wav fuctio at a latr tim is th giv by th opratio of th propagator o this iitial wav fuctio: x / d x / d ipx / ħ m im ( x x ) / ħ t ip x / ħ 1/ 4 1/ 4 ( x, t) U ( x, t; x,) dx dx Not first that sic this is just ( x, t) U ( t) ( x, t ) Schrödigr wav fuctios, it is vidt that U ( x, t; x,) δ ( x x ) statmt that it / ħ I, ( π d ) π ħ it ( π d ) writt xplicitly i trms of (, ) (, ;,) (,) x t U x t x x dx as t This is just quivalt to th oprator th uit oprator, as t
5 Th itgral ovr x is just aothr Gaussia itgral, so w us th sam rsult, ax bx π b / 4 a dx + a Lookig at th xprssio abov, w ca s that im b x ħ t p t m 1 im d ħ t, a This givs im p t 1/ 4 x π imx t m, xp xp ħ iħ t ħ t iħ t d 1 + 1 + md md ( x t ) b / 4 a whr th scod xpotial is th trm As writt, th small t limit is ot vry appart, but som algbraic rarragmt yilds: ( 1 iħ t / md ) ( / ) 1/ 4 x p t m ip π ( x, t) xp xp ( x p t / m ) d + d ( 1 + i t / md ) ħ ħ Writt this way, it is vidt that th xprssio gos to th iitial wav packt for t goig to zro, as of cours it must Although th phas i th abov xprssio for ( x, t ) has cotributios from all thr trms, th mai phas oscillatio is i th third trm, ad o ca s th phas vlocity is o half th group vlocity, as discussd arlir Th rsultig probability dsity: ( x t ) ( d + ħ t / m d ) ( x p t / m ) / 1, xp π ( d + ħ t m d ) This is a Gaussia wav packt, havig a width which gos as ħ t / md for larg tims, whr d is th width of th iitial packt i x spac so ħ / md is th sprad i vlocitis v withi th packt, hc th gradual spradig v t i x spac
6 It s amusig to look at th limit of this as th width d of th iitial Gaussia packt gos to zro, ad s how that rlats to our δ fuctio rsult Suppos w ar at distac x from th origi, ad thr is iitially a Gaussia wav packt ctrd at th origi, width d << x At tim ħ, th wav packt has sprad to x ad has ( x, t ) t mxd / cotius to sprad at a liar rat i tim, so locally ( x, t ) of ordr 1/x at x Thraftr, it must dcras as 1/t to cosrv probability I th δ fuctio limit d, th wav fuctio istatly sprads through a hug volum, but th gos as 1/t as it sprads ito a v hugr volum Or somthig Schrödigr ad isbrg Rprstatios Assumig a amiltoia with o xplicit tim dpdc, th tim dpdt Schrödigr quatio has th form iħ ( x, t ) ( x, t ) t ad as discussd abov, th formal solutio ca b xprssd as: it / ħ ( x t) ( x t ),, Now, ay masurmt o a systm amouts to masurig a matrix lmt of a oprator btw two stats (or, mor grally, a fuctio of such matrix lmts) I othr words, th physically sigificat tim dpdt quatitis ar of th form it / ħ it / ħ ( t) A ( t) ( ) A ( ) ϕ ϕ whr A is a oprator, which w ar assumig has o xplicit tim dpdc So i this Schrödigr pictur, th tim dpdc of th masurd valu of a oprator lik x or p coms about bcaus w masur th matrix lmt of a uchagig oprator btw bras ad kts that ar chagig i tim isbrg took a diffrt approach: h assumd that th kt dscribig a quatum systm did but th oprators volvd accordig to: ot chag i tim, it rmaid at ( ), A t A it / ħ it / ħ () Clarly, this lads to th sam physics as bfor Th quatio of motio of th oprator is: da ( t ) iħ [ A ( t), ] dt
7 Th amiltoia itslf dos ot chag i tim rgy is cosrvd, or, to put it aothr way, it / commuts with ħ But for a otrivial amiltoia, say for a particl i o dimsio i a pottial, p / m + V ( x ) th sparat compots will hav tim dpdc, paralll to th classical cas: th kitic rgy of a swigig pdulum varis with tim (For a particl i a pottial i a rgy igstat th xpctatio valu of th kitic rgy is costat, but this is ot th cas for ay othr stat, that is, for a suprpositio of diffrt igstats) Nvrthlss, th commutator of x, p will b tim idpdt: [ ] [ ] x t p t x p i i it / ħ it / it / it /, (), () ħ ħ ħ ħ ħ (Th isbrg oprators ar idtical to th Schrödigr oprators at t ) A, BC A, B C + B A, C, Applyig th gral commutator rsult [ ] [ ] [ ] so x ( t ), p ( t) iħ p ( t ) m m dx ( t) p ( t ) dt m ad sic [ ] x ( t), p ( t) iħ, p ( t) iħ d / dx ( t ), dp ( t ) 1 [ p ( t), V ( x ( t)) ] V ( x ( t )) dt i ħ This rsult could also b drivd by writig V(x) as a xpasio i powrs of x, th takig th commutator with p Exrcis: chck this Notic from th abov quatios that th oprators i th isbrg Rprstatio oby th classical laws of motio! Ehrfst s Thorm, that th xpctatio valus of oprators i a quatum stat follow th classical laws of motio, follows immdiatly, by takig th xpctatio valu of both sids of th oprator quatio of motio i a quatum stat Simpl armoic Oscillator i th isbrg Rprstatio For th simpl harmoic oscillator, th quatios ar asily itgratd to giv:
8 x ( t) x () cos ωt + ( p () / mω )si ω t p ( t) p () cos ωt mω x ()si ω t W hav put i th subscript to mphasiz that ths ar oprators It is usually clar from th cotxt that th isbrg rprstatio is big usd, ad th subscript may b safly omittd Th tim dpdc of th aihilatio oprator a is: with it ħ ( ) a t a / it / ħ ( ) ħ ω a t a t + 1 Not agai that although is itslf tim idpdt, it is cssary to iclud th timdpdc of idividual oprators withi so d iħ a t a t a t a t a t a t a t a t a t dt ħ ħ ħ, ω, ω, ω i t ( ) a t a ω Actually, w could hav s this as follows: if ar th rgy igstats of th simpl harmoic oscillator, it / ħ iħωt / ħ iω t Now th oly ozro matrix lmts of th aihilatio oprator ˆ a btw rgy igstats ar of th form / ħ / ħ ( 1 ) i t 1 a t 1 a 1 a 1 a it it ω iω t iω t Sic this tim dpdc is tru of all rgy matrix lmts (trivially so for most of thm, sic thy r idtically zro), ad th igstats of th amiltoia spa th spac, it is tru as a oprator quatio Evidtly, th xpctatio valu of th oprator a ( t ) i ay stat gos clockwis i a circl ctrd at th origi i th complx pla That this is idd th classical motio of th simpl ξ + i π 1 harmoic oscillator is cofirmd by rcallig th dfiitio a ( mω x + ip ), so ħ m ω mω x, p phas spac discussd ar th bgiig of th th complx pla corrspods to th
9 lctur o th Simpl armoic Oscillator W ll discuss this i much mor dtail i th xt lctur, o Cohrt Stats Th tim dpdc of th cratio oprator is just th adjoit quatio: i t a t a ω