to the SCHRODINGER EQUATION The case of an electron propagating in a crystal lattice

Transcription

1 Lctur Nots PH 411/511 ECE 598 A. La Rosa INTRODUCTION TO QUANTUM MECHANICS CHAPTER-9 From th HAMILTONIAN EQUATIONS to th SCHRODINGER EQUATION Th cas of a lctro propagatig i a crystal lattic 9.1 Hamiltoia for a lctro propagatig i a crystal lattic 9.1.A Dfiig th Bas Stats ad th Hamiltoia Matrix 9.1.B Statioary Stats Ergy bads 9.1.C Tim-dp Stats Elctro wav-packt ad group vlocity Effctiv mass (cas of low rgy lctros) 9. Hamiltoia quatios i th limit wh th lattic spac tds to zro 9..A From a discrt to a cotiuum basis 9..B Th dpdc of amplitud probability o positio. Trasitio from dscribig a quatum stat I i vry gral trms, to a aild accout of th amplitud-probability dpdc o positio x 9..C Equatio dscribig a lctro i a xtral pottial V = V( x, t ) 9.3 Th Postulatd Schrodigr Equatio Rfrcs: Fyma lcturs Vol. III, Chaptrs 13 ad 16. 1

2 CHAPTER-9 From th HAMILTONIAN EQUATIONS to th SCHRODINGER EQUATION Th cas of a lctro propagatig i a crystal lattic O of th purposs of this chaptr is to show that th corrct fudamtal quatum mchaics quatio (th Schrödigr quatio) has th sam form that o obtais for th limitig cas wh b 0 of a lctro movig alog a li of atoms sparatd by a distac b from ach othr. Startig from a problm whr a lctro movs alog a st of discrtly sparatd atoms alog a li, a quatio is obtaid for th cas i which th sparatio distac tds to zro. This procdur ca hlp us udrstad bttr th itrprtatio of th wavfuctio solutios of th Schrödigr quatio. 9.1 Hamiltoia for a lctro propagatig i a crystal lattic 1 I our study of a two-stat systm, w lard that thr is a amplitud probability for th systm to jump back ad forth to th othr rgy stat. Similarly i a crystal, O ca imagi a lctro i a wll at o particular atom ad with som particular rgy. Suppos thr is a amplitud probability that th lctro mov ito aothr wll at o of th arby atom. From its w positio it ca furthr mov to aothr atom or rtur to its iitial wll. W b x 1 This study will allow udrstadig a ubiquitous phomo i atur that if th lattic wr prfct (i.. o dfcts), th lctros would abl to travl through th crystal smoothly ad asily, almost as if thy wr i vacuum. x 9.1.A Dfiig th bas stats ad th Hamiltoia matrix W would lik to aalyz quatum mchaically th dyamics of a xtra lctro put i a lattic (as if to produc o slightly boud gativ io). Som cosidratios first: a) A atom has may lctros. Howvr may of thm ar quit boudd that to affct thir stat of motio a lot of rgy (i xcss of 10 V) is dd. W will

3 assum that th motio of a xtra lctro (wakly boudd to th atom, th dpth of th wll is much smallr tha 10 V.) It is th dyamics of this wkly boudd lctro that w ar itrst i. b) What would b a rasoabl st of bas stats? W could try th followig: 1 crystal stat i which th - is i th atom 1. crystal stat i which th - is i th atom. crystal stat i which th - is i th atom. (1) Ay stat t of th o-dimsioal crystal ca th b xprssd as, ( t ) = A ( t ) = ( t ) () whr th scalars A ( t ) = ( t ) ar th amplitud probabilitis of fidig th stat ( t ) i th bas stats, rspctivly, at th tim t. Th volutio of A ( t ) as a fuctio of tim is rmid by da ( t i ) H j( t) Aj( t) (3) j So w d to fid th compots H j of th Hamiltoia matrix first. c) How dos th Hamiltoia look lik? That is, how to fid th compots H j? W will procd similarly to th cas of th ammoia molcul (tratd i th prvious chaptr.) Sic all th atoms sits i th crystal ar rgtically quivalt, w ca assum H = E o, = 1,, 3, (4) O th othr had, a lctro ca mov from o atom to ay othr atom, thus trasfrrig th gativ io to aothr plac. W will assum howvr that th lctro will jump oly from o atom to th ighbor of ithr sid. That is, for a giv, H (+1) -W. (5) H (-1) -W. ad H j = 0 for j ± 1. Rplacig ths valus i th quatio of motio (3) o obtais, 3

4 da -W A - + E o A -1 - W A i -1 da i - W A -1 + E o A - W A +1 (6) da 1 i -W A + E o A +1 -W A B Statioary Stats From our xpric studyig th ammoia molcul, lt s rcall that aftr iifyig two localizd sats 1 ad, w wr abl to fid two pculiar statioary stats S I ad S II, 1 1 S I ( 1 + ) ad SII ( 1 - ) which had th followig dpdc o tim (s Chaptr 8, xprssio (51)), ( i/ ) E S I (t) = 1 t S I (0), whr E I = E 0 - W ( i/ ) S II (t ) = E II t S II (0), whr E II = E 0 + W That is, S I ad S II, ar stats of dfiit rgy. W wat to highlight th pculiar phas with which th stats 1 ad participat i S I ad S II. Notic, stat S I (t ) ca b r-writt as, S I (t ) 1 ( i/ ) E 1 t 1 + ( i/ ) E 1 t 1 Notic, both amplitud vary with th sam frqucy E /, ad ar i phas. 1 Stat S II (t) look ssimilar, xcpt for a 180 dgrs phas diffrc btw th amplituds, 4

5 S I I ( t ) 1 ( i/ ) E II t 1 - ( i/ ) E II t 1 Notic, both amplitud vary with th sam frqucy E /, but ar 180 dgrs out of phas II That is, S I (t ) ad S II (t ) hav th followig gral form, S (t) a a whr a 1 ad a ar complx umbrs, which rmi th vtual phas diffrc with which th stats 1 ad participat i th wavfuctio S (t ). This obsrvatio mad i th two-stat systm, about th diffrt phas associatd to ach of th two localizd stat, will b xtrapolatd to th cas i which th umbr of localizd stats participatig i th systm is ifiit. I ffct, for solvig Eq. (5), lt s look for a wavfuctio (t) = A (t) with th followig charactristics: To b a statioary solutio. That is, a solutio i which all th amplituds A vary with tim at ( i/ )( E) t th sam frqucy. This will mak th stat (t) to b dscribd with a sigl rgy E (th lattr valu still to b rmid.) Th diffrt amplituds A ar rquird to hav diffrt phas rspctivly. Such phass would hav to b rmid. Ths two rquirmts ar summarizd i th followig xprssio, A t ) = a ( i/ )( E) t Accordigly lt s look for a statioary solutio of th form, (t) = A ( t ) = a ( i/ )( E t (7) ) (7) Th amplituds will vtually hav diffrt phas btw ach othr; that is, th diffrt a ar complx umbrs. All th amplitudprobabilitis oscillat at th sam frqucy ( E/ ). 5

6 W hav to fid valus for E ad th diffrt a compatibl with Eq. (6). Rplacig (7 ) i (6) Ea = - W a -1 + E o a - W a +1 (8) Altrativ otatio: If w labl th atoms by thir positios, x = b whr b is th spacig btw cotiguous atoms i th crystal, th quatio (8) taks th form, Ea(x ) = - W a(x -1) + E o a(x ) - W a(x +1) (9) Trial solutio for Eq. (9), ik a ( x ) = x (10) whr th valu of th sigl valu of k, accompayig th diffrt cofficits a, still to b rmid. Rplacig (10) i (9) o obtais, ik E x ik x = - W - 1 ik x + E ik x o - W 1 E = - W = E o - W [ - i k b + E o - W - i k b + i k i k b ] b E(k) = E o - W Cos (k b) Ergy of th statioary stats (11) W raliz, for ay ral valu of k thr is a rgy E(k) = E o - W Cos(k b) ad a i k ifiit-st of valus a(x ) = x, which, wh rplacd i (7) costitut a statioary solutio of Eq. (6). It would appar that thr ar a ifiit umbr of solutios (o for ach ral valu of k that ca occur to us.) W will s latr that this is ot compltly tru. 6

7 Summary. W hav foud that for ay giv ral valu of k, Thr xists a corrspodig statioary solutio (from xprssio (7 ) ) k ( t) = A k (t) = ik x ( i/ )[ E ( k )] t (1) whr th rgy is giv by (s xprssio (11) ) E(k) = E o - W Cos(k b) Notic th amplitud probability at th sit varis with tim as, A k ( t ) = ik x ( i/ ) [ E ( k ) ] t Th xprssio abov idicats th lctro is qually likly to b foud at vry atom, sic all th quatitis A k (t) ar qual (ad idp of ). Oly th phas is diffrt for diffrt atom locatio, chagig i th amout (kb) from atom to atom. b Ral [A k ] x Fig 9.1 Ral part of th amplitud probability A for fidig th atom locatio of th xtra lctro i th crystal, at a giv tim. Th rgy E as a fuctio of k is show i th figur blow. Oly rgis i th rag from E - W to E +W ar allowd. If a lctro i a crystal is i a statioary stat, it ca hav o rgy othr tha valus i this bad. This is th workig pricipl of th rgy bads formatio i a crystal, which xplais th lctrical coductivity i mtals ad smicoductors.) Notic th wih of th bad scals with W. Th smallst rgis E ~ (E o -W) ar obtaid for th smallst valus of k 0. As k icrass, th rgy icrass as wll util it rachs a maximum at k = / b For k largr tha / b th rgy would start to dcras, but thr is o d to cosidr such valus, bcaus thy do ot giv w stats. 7

8 No d to cosidr ths valus of k. E E o +W E o E o -W No d to cosidr ths valus of k. -/ b / b Fig 9. Ergy of th statioary stats as a fuctio of k, accordig to xprssio (11). Notic, th formatio of a rgy bad whos wih is proportioal to W. Th diffrt statioary stats (o stat for ach valu of k withi th cotiuum itrval < /b, /b>) ar giv by xprssio (1), k = A k = ikx ( i/ )[ E( k) ] t I ffct, whil a wav-vctor k = k 1 +/b has a profil of highr spatial ad thus is idd diffrt tha that th profil corrspodig to th wav-vctor k 1, it turs out that both profils hav th sam valu at ach atom locatio (s Fig. 9.3 blow). [ This is tru bcaus wh multiplyig k by x (which is multipl of b) th addd phas is, which is quivalt to a zro chag i phas. That is k x givs th sam quivalt phas as k 1 x ]. Hc, as far as th valuatio of th probability-amplitud at th atoms sit is cocrd, both wav-vctors k 1 ad k dscrib th sam quatum stat. k 1 = 8b, k 1 1 = (8/9)b, 8 b k k 1 9 b 8 b Ral A(x ) b x Fig 9.3 Ral part of A(x ) = ik x ( i/ )[ E(k) ] t for two diffrt valus of k, with k 1 ad k sparatd by /b). Although both profils ar i gral diffrt, both hav th sam valu at th atoms sits. Thus, both dscrib th sam quatum stat (th valu of A at th atom s sit is what mattr). Formally, this is a cosquc of th fact that. i[ k (/b)] x ik x i[/b] x 8

9 sic x = b o obtais, ik = x i π i[ k ( /b)] x ik = x Accordigly, th valus of k withi th rag /b < k </b ar sufficit to dscrib th quatum stats. I this rag: Th highr k th highr th rgy of th statioary stat. For small valus of k, th followig approximatio holds, E(k) = E o - W Cos(kb) ~ E o - W [1 - (½) (k b) ]; E(k) ~ (E o - W) + W b k Ergy of statioary stats (13) with small valus of k 9.1.C Tim-volutio of Stats I a statioary stat (s xprssio (1) abov), a lctro is qually likly to b foud i ay atom i th crystal. I cotrast, how to rprst a situatio i which a lctro of a crtai rgy is locatd i a crtai rgio? W ar rfrrig to a situatio i which a lctro is mor likly to b foud i o rgio tha at som othr plac. Elctro wav-packt A altrativ to addrss th qustio abov is to mak a wav-packt costructd usig a propr liar combiatio of th statioary stats k (t) foud i th prvious sctio (s xprssio (1)); ach statioary stat iifid by th valu of k. ( t ) = dk F(k) k (t) Wavpackt (14) = dk F(k) A k (t) = dk F(k) ik x = dk F(k) ik x ( i/ )[ E( k) ] t ( i/ ) [ E( k) ] t A(x, t) For xampl, w ca slct a wavpackt of wih X that has a maximum at aroud th atom locatd at x o. That is, th xpasio, 9

10 A(x, t) = dk F(k) ik x ( i/ )[ E( k) ] t has a maximum at aroud th o -th atom (at a giv tim t), as schmatically show i Fig Ral A(x, t) x o X Fig. 9.4 Profil (dashd li) of th ral part of th wavpackt at a giv tim t. b For a puls A(x, t) localizd (i spac) with a wih X, thr will b a associatd Fourir trasform F(k) with a prdomiat wavumbr k o, ad all th othr wavumbrs k withi a rag K; that is, th itgral i (13) xpads oly ovr a wih K, 1 k x - E k / t i { [ ( ) ] } A( x, t) ( dk) F( k) π K E( k) Usig ( k), 1 A( x, t) F k π ( ) K i [ k x - ( k) t ] dk Wavpackt (14) Group vlocity i{ k x - ( k) t } W lard i Chaptr-5 that, v though th k-compot travls at diffrt phas vlocitis, still w ca dfi a vlocity for th wav-packt calld th group vlocity, v wavfucti group-vllocity o's v g d ) 1 ) k dk ko d k dk k o Particular cas: Wav-packt with small valus of k I th particular cas that k o is i th low-valu rag, w ca us th xprssio drivd abov E(k) ~ (E o - W) + W b k, which lads to, 10

11 W b v v g k o (for small valus of k wavfucti group-vllocity o's o ) (15) Vlocity of a low-rgy wav-packt havig a prdomiat wavumbr k=k o, Effctiv mass (cas of low rgy lctros) Sic E(k) = (E o - W) + W b k Wb ad v g k (for small valus of k,) w raliz that low-rgy lctros mov with rgy proportioal to th squar of its vlocity, lik a classical particl. This rsmblac with a classical particl ca b mad mor traspart through th dfiitio of a ffctiv mass. For that purpos, i xprssio E(k) = (E o - W) + W b k lt s first mov our zro rgy rfrc to (E o -W); that is, E(k) = W b k Ergy of statioary stats (16) with small valus of k E 4W W 0 -/ b / b k Stats of low rgy ( k<</b ) Fig. 9.5 Compard to Fig. 9. th miimum rgy has b movd to th zro rgy lvl. Accordigly, xprssio (11) bcoms E(k) = W - W Cos( k b). A wav-packt of prdomiat wav-vctor k=k o will hav a rgy qual to th rgy of th compot of wav-vctor k=k o ; that is E= W b k o. I trms of th associatd group vlocity, giv i (15) abov, th rgy of th wavpackt ca b xprssd as, v g W b ko 11

12 E = W b v g ( ) = Wb 1 4Wb (v g ). Thus, for a wav-packt whos prdomiat wav-vctor k is rlativly small compard to /b, o obtais, E 1 b W v g This xprssio ca b writt as 1 E m ff v g (17) which maks th motio of th quatum packt to rsmbl th classical dscriptio of th lctro s motio. That is, amplitud-probability wavs rsmbl th motio of a classical fr particl. Th costat m is calld th ffctiv mass. m ff ff (for small valus of k ) (18) W b m ff has othig to do with th lctro s mass. It has b arbitrarily dfid just to mak th amplitud probability motio to rsmbl th motio of a classical particl. I a ral crystal m ff turs out to hav a magitud of about to 0 tims th fr-spac mass of a lctro. Notic, howvr, th xprssio for m ff is appalig to itrprt it as a ral mass. For xampl, xprssio (18) idicats that: th smallr W th gratr th m ff ; that is, a smallr amplitud probability (W) for a lctro to mov from o positio to aothr ca b itrprtd as a quivalt havir mass irtia. Notic also, from (15) ad (18), m ff v k (19) g o Th rprstativ d Brogli momtum of th wavpackt, xprssd as a classical momtum mass vlocity. ko = h/, ds up big Summary: W udrstad ow how a lctro ca rid through th crystal, flowig prfctly fr, uprturbd, with a vlocity qual to v, dspit havig all th atom to hit agaist. It dos by havig its amplitud probability to fly from o atom to th xt, with W providig th quatum probability to mov from o atom to th xt. That is how a solid coducts lctricity. g 1

13 Th pictur abov would lad to a ifiit coductivity, sic it prdicts that a lctro, oc acquirig a vlocity v it would drift forvr. I fact, it is currtly g accptd that th fiit coductivity of a crystal is ot du to th collisio of lctros with th atoms; istad th fiit coductivity is causd by th prsc of dfcts i th lattic, which caus scattrig of th lctros motio. 9. Hamiltoia quatios i th limit wh th lattic spac tds to zro I th prvious sctio w ralizd that a amplitud-probability (wavpackt) propagat i a crystal as if it wr a particl. For xampl, it was foud that its rgy is qual to ½ m v g, ad that its d Brogli momtum is qual to m ff v g. That is, th Hamiltoia dscriptio of ko amplitud-probability wavs i a lattic-crystal rsmbls th mchaistic dscriptio of a particl (udr propr iificatio of quatitis). O would xpct th that, th Hamiltoia dscriptio of amplitud probability wavs i a discrt lattic would ihrtly hav i it (or b othig but a mirror rflctio of) a mor gral quatum mchaical dscriptio for th motio of a particl. Mayb by takig th limitig cas of th lattic distac b qual to zro, th rsultig cotiuum quatio would rsmbl th mor formal Schrodigr quatio. Th lattr is prcisly th objctiv that w will pursu i this sctio: To obtai th Schrodigr quatio as a limit cas of th Hamiltoia quatios (dscribig th motio of a particl alog a array th atoms) wh th lattic costat b tds to zro. Icially, this chaptr offrs also a opportuity to illustrat th trasitio from a) th, util ow, somwhat loos gral dscriptio of a quatum stat I, to b) a much mor aild dscriptio of th wavfuctio, i.. givig a accout o how th amplitud-probability s (x) dpds o th positio coordiats. ff 9..A From a discrt basis to a cotiuum basis Lt s first formaliz th dfiitio of a cotiuum spatial basis Lt s tak a liar lattic of atoms, ad imagi that th lattic spacig b wr to b tak smallr ad smallr. I th limit, w would hav th cas i which th lctro could b aywhr alog th li. I othr words, assum th possibility of lablig th spac with ifiity of poits (i a cotiuous fashio.) Discrt cas x 0 Cotiuum cas x (0) b 13

14 Th xprssio for a wavfuctio = x A(x ) = may also b prfrably writt as = x A (x ) = Th last otatio uss A istad of simply A as to mak mor xplicit that th amplitud probabilitis corrspods to th stat. I (1), multiplyig o th lft by x k, o obtais: x stads for a stat i which a particl is locatd at th spatial coordiat x. x A (x) dx x A ( x ) dx (1) x k = A ( x k ) x A ( x ) () (Notic th xprssio o th lft sid costituts a bttr otatio, mphasizig that thr is o d to iclud th charactr A.) x k givs th amplitud probability that th stat b foud at xk. Altrativly th followig otatio is also usd, x ( x ) k k b 0 x (x) (3) amplitud probability to fid th stat at th stat x. Not: Wh writig x ( x ), th sam symbol is big usd, for i) lablig a particular physical stat for a lctro; ad for ii) dfiig a mathmatical fuctio of x(x). I both cass th xprssio givs th amplitud probability that a lctro i th stat b foud at th stat x. 14

15 If i = x A (x ) w watd to mphasiz th tim dpdc, w would writ, t = x A (x, t ) t = b 0 whr th amplituds ar xprssd i diffrt altrativ forms A ( x k, t ) = x k (t) = ( k x A (x, t ) dx (4) x, t ) A ( x, t ) = x t = ( x, t ) (5) 9..B From a gral dscriptio of th stat I to a mor aild accout of its dpdc with th positio: How dos x look lik? Startig from th solutio corrspodig to th cas of a lctro movig across a discrt lattic, w wat to work out th quatios that rlat th valu of th amplitud-probability at o poit x to th amplitud-probability at poits arbitrarily clos to it. This will b do by makig b 0. That way w would obtai a quatum mchaical dscriptio of a lctro s motio i th cotiuum spac. Lt s start with th most gral solutio of th Hamiltoia dscribig th motio of a lctro i a discrt lattic, t = x A ( x, t ) (6) th amplituds A ( x, t ) hav to satisfy th Hamiltoia quatios, da x i ( ) - W (hr w ar usig th otatio). A (x -1 ) + E o A (x ) - W A (x +1 ) (7) A (x ) istad of A ( x, t ) just to simplify Lt s aalyz ow th cotiuum cas, by takig th limit b 0. First, lt s choos th origi of th rgy scal at th miimum valu i Fig. 9.. That maks E 0 = W. Thus w rwrit (7) as, da (x ) i - W Rarragig trms, A (x -1 ) + W A (x ) - W A (x +1 ) (8) 15

16 Wb m ff d A (x ) i W [ A (x ) - A (x -1 )] - W [ A (x +1 ) - A (x ) ] A W b ( x ) A (x-1) x x-1 - W b A (x 1) A (x ) x 1 x Now w tak b 0 whr w hav usd b = x +1 - x da d A i ( x ) W b (x ) dx A = - W b d A (x 1) dx d i ( x ) - W b d A ( x b) dx d A - W b (x 1 ) dx (9) da If w blidly took b 0, th xprssio (9) would giv i ( x ) 0 ; such a rsults would rval o usful iformatio. Rcall, howvr, that W givs th liklihood th lctro jumps to th ighbor atom ad, thus, it is plausibly to thik that as th spacig b dcrass th valu of W simultaously icrass. Thus, w could covitly cosidr that as b dcrass, th product Wb rmais costat. Usig xprssio (18) w ca tak this factor to b qual to Wb / mff. So w will cosidr that as b 0, th valu of W icrass such that th product W b rmais costat ad qual to /( ). m ff Wb b 0 m ff (30) Cosqutly, as b 0, Eq. 9 bcoms, da ( x,t) i d A (x,t) (31) m ff dx 16

17 I summary, Wh th crystal lattic b td to zro, th Hamiltoia quatios tak th form: i (x,t) t (x,t) m ff x. (3) whr m is dfid through th xprssio ff Wb m ff ad with th rquirmts that th quatity Wb rmais costat as b 0. Th solutios of this quatio rmi th amplitud probabilitis (x,t) that dfi th stat (t) = x A x,t ) dx whr w hav xtrapolatd th discrt xpasio giv i xprssio (6) abov. Giv th quivalt otatios A ( x, t ) = x t = ( x, t ), as giv i (5) abov, th quatio abov is also writt as, (x,t) (x,t) i t m x ff. Motio of a lctro alog a cotiuum li spac (33) = x x d x = x (x,t) d x Fig. 9.6 Elctro i a discrt liar lattic. I th limit, wh th spacig b tds to zro, th problm rsmbls o i which th lctro movs alog a cotiuum li spac. b x 9..C Particl i a pottial V=V( x,t ) What would b th limitig cas b 0 of th Hamiltoia quatios if th atoms wr subjctd to a xtral pottial? 17

18 For compariso, lt s tak a look to th Hamiltoia quatios corrspodig to th cas V(x)=0, which was giv i th xprssio (6) ad rproducd blow, da-1 i -W A - + E o A -1 - W A da i - W A -1 + E o A - W A +1 (34) da 1 i -W A + E o A +1 - W A + First, lt s rcall that E o is th rgy of th lctro wh locatd i a isolatd atom.) Wh o xtral voltag is applid, all th atoms ar quivalt; accordigly E o =H ( =1,,3, ) Howvr wh a xtral voltag is applid, th lctro pottial rgy varis from atom to atom (for xampl if a uiform lctric fild wr applid, th a pottial V = - x would b stablishd.) Thus, i th cas of a gral voltag V = V(x) wr applid, th rgy E o of a lctro will b shiftd to E o +V whr V V( x ) da i -W A -1 + ( E o +V )A - W A +1 (35) This is practically th sam Eq. (7); oly th lmts alog th diagoal bcom modifid. Th applicatio of takig th limitig procss b 0 bcoms, th, vry similar. W should obtai, A (x,t) A (x,t) i V ( x,t) A (x, t) t mff x Or, quivaltly ( x,t) ( x,t) V ( x,t) ( x,t) (36) t m x i ff = x x dx = x (x,t) dx 9.3 Th Postulatd Schrodigr Equatio (No-rlativistic particl motio) Th corrct quatum mchaical quatio for th motio of a lctro i fr spac was first discovrd by Schrodigr. 18

19 i (x,t) t (x,t) m x Schrodigr quatio (fr particl) (37) Notic that th Schrodigr quatio is vry similar to th quatio (8) abov. W do ot itd to hav you thik w hav drivd th Schrodigr quatio but oly wish to show you o way of thikig about it. Wh Schrodigr first wrot it dow, h gav a kid of drivatio basd o som huristic argumts ad som brilliat ituitiv gusss. Som of th argumts h usd wr v fals, but that dos ot mattr; th oly importat thig is that th ultimat quatio givs a corrct dscriptio of atur. Th purpos of th discussio i th prvious sctios was to show that: th corrct fudamtal quatum mchaics quatio (37), has th sam form as, th quatio (35 ) obtaid wh cosidrig th limitig cas of a lctro movig alog a li of atoms spacd at discrt stps. This mas that w ca thik of Eq. (37) as dscribig th diffusio of a probability amplitud (x,t) from o poit to th xt alog th li. That is, if a lctro has a crtai amplitud probability to b at o poit, it will, a littl tim latr, hav som amplitud to b at ighborig poits. Tak from Th Fyma Lctur o Physics, Vol III, pag Th diffusio argumt giv abov sprigs from th fact that th Schrodigr quatio (37) looks vry similar to th diffusio quatio D, (th lattr govrig, for t x xampl, th tim volutio of a gas spradig alog a tub.) But, thr is a substatial diffrc btw ths two quatios du to th complx umbr i frot of th tim drivativ of th Schrodigr quatio i : t m x whil th diffusio quatio lads to xpotially dcayig solutios (i.. th dsity tds to a uiform distributio of molculs), i cotrast, i th Schrodigr quatio th imagiary cofficit i frot of th tim drivativ maks th bhavior compltly diffrt. Its solutios ar complx-variabl propagatig wavs. For a particl movig isid a pottial V(x,t) th Schrodigr is, 19

20 i t V ( x,t) m x Schrodigr Equatio (38) This quatio markd a historic momt costitutig th birth of th quatum mchaical dscriptio of mattr. Th grat historical momt markig th birth of th quatum mchaical dscriptio of mattr occurrd wh Schrodigr first wrot dow his quatio i 196. For may yars th itral atomic structur of th mattr had b a grat mystry. No o had b abl to udrstad what hld mattr togthr, why thr was chmical bidig, ad spcially how it could b that atoms could b stabl. (Although Bohr had b abl to giv a dscriptio of th itral motio of a lctro i a hydrog atom which smd to xplai th obsrvd spctrum of light mittd by this atom, th raso that lctros movd this way rmaid a mystry.) Schrodigr s discovry of th propr quatios of motio for lctros o a atomic scal providd a thory from which atomic phoma could b calculatd quatitativly, accuratly ad i ail. Fyma s Lcturs, Vol III, pag Although th rsult (38) is a postulat, w do hav ow som clus about how to itrprt it, basd o th kowldg w acquird by solvig th th particular cas of th dyamics of a lctro i a crystal lattic o APPENDIX 0

21 [Not: It turs out that dscribd by th wavfuctio k ik x = x k givs th liar momtum of th lctro ( i/ )[ E(k) ] t rgardlss of th valu of k. Th abov statmt ca b provd via P ~ k = P ~ { x [ A k ( x, t ) ] } = P ~ ik x { x ( i/ )[ E(k) ] t } = = = i x i x ik x { x ik x { x ik ( i/ )[ E(k) ] t } ( i/ )[ E(k) ] t } = i x = k ik x { x ik ( i/ )[ E(k) ] t } ik x { x ik ( i/ )[ E(k) ] t = k k } Thus, P ~ k = k k Howvr, k ik x = x ( i/ )[ E(k) ] t is ot th most gral wavfuctio dscribig a lctro. Wh a priodic pottial is cosidrd, th wavfuctio ar giv by th Bloch fuctios u(x ) k ik x Bloch= x ( i/ )[ E(k) ] t i k x Notic P ~ k Bloch = k k Bloch + i x So k k Bloch is ot a igstat of P ~ ] u( x ) ( i/ )[ E(k) t x 1

22 k is still th crystal momtum of th lctro, wh this is i th But, statioary stat k Bloch 1 Th Fyma Lcturs, Vol III, Chaptr 13