Lctur 8 Phys 375 A Propagating Wav Packt Group Vlocity Disprsion Ovrviw and Motivation: In th last lctur w lookd at a localizd solution t) to th 1D fr-particl Schrödingr quation (SE) that corrsponds to a particl moving along th x axis (at a constant vlocity). W found an approximat solution that has two vlocitis associatd with it, th phas vlocity and th group vlocity. Howvr, th approximat solution did not xhibit an important fatur of th full solution that th localization (i.., width) of th wav packt changs with tim. In this lctur w discuss this proprty of propagating, localizd solutions to th SE. Ky Mathmatics: Th nxt trm in th Taylor sris xpansion of th disprsion rlation ω ( k) will b cntral in undrstanding how th width of th puls changs in tim. W will also gain practic at looking at som complicatd mathmatical xprssions and xtracting thir ssntial faturs. W will do this, in part, by dfining normalizd, unitlss paramtrs that ar applicabl to th problm. I. Th First-Ordr Approximat Solution (Rviw) In th last lctur w lookd at a localizd, propagating solution that can b dscribd i as a linar combination of travling, normal-mod solutions of th form [ kx ω( k ) t ], ( ) ( k k ) i[ = k x ( k ) t x t dk ω, ]. (1) π If Eq. (1) is a solution to th SE, thn th disprsion rlation is ω ( k) = hk m. In ordr to gain som insight into Eq. (1) w Taylor-sris xpandd th disprsion rlation about th avrag wav vctor k associatd with t), 1 + ω ( k) = ω( k ) + ω ( k )( k k ) + ω ( k )( k k )... () W thn approximatd ω ( k) by th first two trms (th constant and linar- k trms) in th xpansion and obtaind th approximat solution [ ] ik x v ph ( k ) t x vgr k t t), (3) whr v ph ( k) = ω( k) k is known as th phas vlocity and v gr ( k) = ω ( k) is known as th group vlocity. Th phas vlocity is th spd of th normal-mod solution ik [ x ( ω( k ) k ) t ], whil th group vlocity is th spd of th nvlop function [ x ω ( k ) ] t. Bcaus it is also th spd of th probability dnsity function D M Riff -1- /6/9
Lctur 8 Phys 375 x ω k t =, () it can b though of as th avrag spd of th particl that th SE dscribs. Howvr, th approximat solution [Eq. (3)], dos not xhibit an important proprty of th xact solution: th localization (or width) of th xact solution varis with tim whil th localization of th approximat solution is constant (and can b dscribd by th width paramtr.) Th thr vidos, SE Wavpackt.avi, SE Wavpackt5.avi, and SE Wavpackt6.avi illustrat th tim dpndnt broadning of propagation wav-packt solutions. Notic that th narrowr th initial wav packt, th fastr it sprads out in tim. This is on rsult from th analysis blow. II. Th Scond-Ordr Approximat Solution By also including th nxt trm in Eq. (), th Taylor's sris xpansion of th disprsion rlation, w obtain an approximat solution that xhibits th dsird fatur of a width that changs as th wav packt propagats. 1 Including th first thr trms in th Taylor's sris xpansion and substituting this into Eq. (1) producs, aftr a bit of algbra, th approximat solution i ( ) [ ω( k ) ( k ) k ] t ( k k ) [ + i ( k ) t ] ik [ x ( k ) t x t dk ] ω, ω ω (5) π This is xactly th sam as th approximat solution that w obtaind in th last lctur xcpt for th trm containing ω ( k ). But notic whr it appars as an additiv trm to, which controls th width of th puls. Thus w might alrady guss that ω ( ) will affct th width as th puls propagats. k Fortunatly, for purposs of furthr analysis Eq. (5) has an analytic solution t) { + [ ω ( k ) t] } 1 [ ] iω ( k ) t x ω ( k ) t ( i ) arctan ω ( k ) t ik [ x ( ω( k ) k ) t ] { } [ ] + [ ω ( k ) t ] x ω k t 1+ ω ( k ) t. (6) [ ] 1 Actually, all of th highr-ordr trms can contribut to th broadning of th puls. Howvr, if th width paramtr is not too small, thn only th contribution of th quadratic trm to th tim dpndnt broadning nds to b considrd. D M Riff -- /6/9
Lctur 8 Phys 375 OK, so mayb solving th intgral wasn't so fortunat. But lt's s what w can do with it. First notic that, compard to our prvious solution w hav two nw xponntial-function trms, ( ) ( ( ) ) i arctan ω k t iω and ( k ) { t x ω k t + ω k t } [Notic that thy ar both qual to 1 whn ω ( k ) =.] Howvr, bcaus th xponnts in both of ths trms ar purly imaginary, thy contribut nothing to th width of th wav packt (as w will s blow). W will thus not worry about thm. Th third xponntial trm w ar alrady familiar with; it is th harmonic travling wav ik [ x ( ω( k ) k ) t ] solution that propagats at th phas vlocity v ph ( k ). (Bcaus its xponnt is also purly imaginary, it too dos not contribut to th width of th wav packt.) It is th fourth xponntial-function trm that has som nw intrst for us. Notic that it is a Gaussian function that again travls with th group vlocity ω ( k ), but with a tim-dpndnt width 1 ω () ( k ) t t = 1+ (7) that is a minimum for t =. Notic that if ω ( k ) =, as in th cas of th wav quation, thn th width has no tim dpndnc and is simply. Howvr, in th cas of th SE, for xampl, ω ( k ) = h m. Thus th SE wav packt has a tim dpndnt width. For larg tims w s from Eq. (7) that ( t) is approximatly linar vs tim ω () ( k ) t t, (8) which tlls us that () t ω ( k ). (9) d dt That is, for long tims th rat of broadning is proportional to th scond drivativ of th disprsion rlation and invrsly proportional to th width paramtr. That is, th narrowr th puls is at t =, th fastr it broadns with tim, as th vidos abov illustratd. Bcaus all drivativs of th disprsion rlation for th WE highr than first ordr ar zro, Eq. (3) is xact for th wav quation. D M Riff -3- /6/9
Lctur 8 Phys 375 III. Normalizd Variabls To gain som furthr insight into this tim-dpndnt width lt's mak a graph of () t vs t. But lt's b smart about th graph; lt's construct th graph so that it has univrsal applicability. To do this w will graph unitlss, normalizd quantitis that ar scald valus of () t and t. So how do w normaliz ( t) and t to mak thm univrsal to th problm at hand. Th answr is in Eq. (7) itslf. First notic that if w divid () t by thn w will hav a unitlss width that is qual to 1 at t =. So lt's dfin a normalizd width N as ( t ). What about th variabl t? Again, th answr is in Eq. (7). Notic that th quantity ω ( k ) t is also unitlss bcaus its valu squard is addd to 1 in Eq. (7). So lt's dfin a normalizd tim variabl τ = ω ( k ) t. With ths two univrsal variabls Eq. (7) can b r-xprssd as ( τ ) = [ 1 τ ] 1 N + (1) Ah, much simplr! Th figur on th nxt pag plots N ( τ ) vs τ on two diffrnt graphs with diffrnt scals. From th graphs w can visually inspct th bhavior of Eq. (1). For xampl, w s for τ << 1 that N ( τ ) 1. This mans that for τ << 1 th width is approximatly constant vs tim. Th actual tim scal (in sconds) ovr which this is tru will, of cours, dpnd upon th paramtrs that wnt into th dfinition of τ : ω ( k ) and. From th graph w also s that for τ >> 1, N ( τ ) τ, indicating (again) that th width changs linarly vs tim for larg ngativ or positiv tims. IV. Application to th Schrödingr Equation Lastly, lt's considr th probability dnsity, assuming that Eq. (6) dscribs a solution to th SE. From Eq. (6) w calculat 1 { + [ ω ( k ) t] } x ω k t 1+ ω ( k ) t [ ]. (11) Using Eq. (7), th dfinition of ( t), Eq. (11) can b rwrittn mor compactly as [ x ω ( k ) ] [ ()] t. (1) () t t D M Riff -- /6/9
Lctur 8 Phys 375 6 5 N A x ( τ) τ ( τ ) 3 1 6 6 τ N 1.7 ( τ ) A x ( τ) 1.5.999 1.1.5.5.1.11 τ.11 Notic that th probability dnsity rtains its Gaussian shap as th wav packt propagats, but with a tim-dpndnt width paramtr qual to () t. Notic that th amplitud () t that multiplis th Gaussian function is also tim dpndnt; as th width () t incrass this amplitud dcrass. Th amplitud varis with th width such that that th total probability for finding th particl anywhr along th x axis, () t = dx t) ( x t), (13) P, rmains constant in tim. Th solution givn by Eq. (6) and th probability dnsity givn by Eq. (11) ar illustratd for both ngativ and positiv tims in th vidos SE Wavpackt7.avi, SE Wavpackt8.avi, and SE Wavpackt9.avi. As Eq. (1) indicats, th puls bcoms D M Riff -5- /6/9
Lctur 8 Phys 375 narrowr as t = is approachd, and th puls bcoms broadr aftr t =. Notic, spcially in SE wavpackt8.avi, that thr is a tim nar t = during which th puls width is approximatly constant, corrsponding to τ << 1. During this tim th hight of (which is also shown in th vidos) is also approximatly constant. Exrciss 8.1 Equation () is th Taylor's-sris xpansion of th disprsion rlation about th point k = k. For th disprsion rlation appropriat to th SE, find all trms in this xpansion. Thn argu why Eqs. (5) and (6) ar xact solutions (as opposd to approximat solutions) to th SE. 8. A SE fr particl (a) Rwrit Eq. (11), th xprssion for th probability dnsity, with xprssions for ω ( k) and ω ( k) that ar appropriat for a wav dscribd by th Schrödingr quation. (b) Mak svral graphs (at last 3) of th probability dnsity k t) k t) vs x for svral diffrnt valus of t. Th graphs should clarly illustrat th chang in th width of th wav packt as th wav packt propagats. For simplicity you may st h =1 and m = 1. 8.3 Th Dimnsionlss Tim Variabl τ (a) Using dimnsional analysis, show that th variabl τ = ω ( k ) t is unitlss. (b) τ << 1 and τ >> 1 corrsponds to what conditions on t? 8. Th figur in th nots shows that for τ <. 1 th width of th wav packt is narly constant. Lt's apply this rsult to a 1 V fr lctron that is bing dscribd by th SE. To do this find th valu of t (in sconds) that corrsponds to τ =. 1. Do this for valus of = 1 nm and 1 µm. For ths two cass how far dos th lctron travl in th tim corrsponding to τ =. 1? How dos ach of ths distancs compar with th rspctiv initial width? 8.5 SE probability dnsity. (a) Substitut Eq. (1) into Eq. (13), calculat th intgral, and thus show that th rsult dos not dpnd upon t. (b) Gnrally, th constant in Eq. (1) is chosn so that th total probability to find th particl anywhr is qual to 1. Using your rsult in (a), find a valu for that satisfis this condition. D M Riff -6- /6/9