Coupled Pendulums. Two normal modes.

Transcription

1 Tim Dpndnt Two Stat Problm Coupld Pndulums Wak spring Two normal mods. No friction. No air rsistanc. Prfct Spring

2 Start Swinging Som tim latr - swings with full amplitud. stationary

3 M +n L M +m Elctron Transfr Elctron movs btwn mtal cntrs. v = v = * v = 3 v = v = Vibrational Transfr Vibration movs btwn two mods of a molcul. v = 0 v = 0 h * Elctronic Excitation Transfr Elctronic xcitd stat movs btwn two molculs.

4 Considr two molculs and thir lowst two nrgy lvls E E Tak molculs to b idntical, so latr will st E E E 0 Stats of Systm Molcul xcitd unxcitd Molcul xcitd unxcitd and Normalizd & Orthogonal

5 Initially, tak thr to b NO intraction btwn thm No spring H E HE H is tim indpndnt. Thrfor iet / iet / Tim dpndnt Part of wavfunction Tim indpndnt kts Spatial wavfunction

6 If molculs rasonably clos togthr Intrmolcular Intractions Coupls stats & (Lik spring in pndulum problm) Thn nrgy of molcul is influncd by. Enrgy of dtrmind by both & Will no longr b ignkt of H Thn: iet / iet / H H H ut & ar coupld H E Coupling strngth Enrgy of intraction

7 Thus: Coupling strngth / ie t H E / ie t H E For molculs that ar idntical E E E 0 Tim dpndnt phas factors Vry Important Pick nrgy scal so: Thrfor H H E0 0

8 Hav two kts & dscribing stats of th systm. Most gnral stat is a suprposition t C C Normalizd May b tim dpndnt Kts & hav tim dpndnt parts iet / For cas of idntical molculs bing considrd: E E E 0 0 Thn: & ny tim dpndnc must b in C & C.

9 Substitut t C C into tim dpndnt Schrödingr Equation: i t H t HC C t Tak drivativ. ic C C C C t C C C t & t indpndnt Lft multiply by ic C normalizd & orthogonal Lft multiply by ic C Thn: ic C ic C Eq. of motion of cofficints & tim indpndnt. ll tim dpndnc in C & C

10 Solving Equations of Motion: hav: ic C Tak d dt C ic C i C but: C i C thn: C C Scond drivativ of function quals ngativ constant tims function solutions, sin and cos. C Qsin( t/ ) Rcos( t/ )

11 C Qsin( t/ ) Rcos( t/ ) nd: C i C C i Qcos t/ Rsin t/

12 t normalizd * * tt C C C C CC CC * * CC * * CC Sum of probabilitis quals. This yilds R Q To go furthr, nd initial condition Tak for t = 0 C C 0 Mans: Molcul xcitd at t = 0, not xcitd.

13 C Qsin( t/ ) Rcos( t/ ) C i Qcos t/ Rsin t/ For t = 0 C C 0 Mans: Molcul xcitd at t = 0, not xcitd. R = & Q = 0 For ths initial conditions: C cos( t/ ) C isin( t/ ) probability amplituds

14 Proction Oprator: t C C cos( t/ ) isin( t/ ) Tim dpndnt cofficints Considr Proction Oprator t C Givs pic of t that is In gnral: S Ci i i k k S Ck k Cofficint mplitud (for normalizd kts)

15 Considr: Closd rackts Numbr S k k S C C C * k k k bsolut valu squard of amplitud of particular kt k in suprposition S. Ck Probability of finding systm in stat k givn that it is in suprposition of stats S

16 Proction Ops. Probability of finding systm in givn it is in t or t C C cos( t/ ) isin( t/ ) cos / * P t t C C t * P t t C C t sin / Total probability is always sinc cos + sin = nrgy nrgy-sc rad sc

17 * P t t C C t cos / * P t t C C t sin / t t = 0 P = ( xcitd) P = 0 ( not xcitd) Whn t/ / t h/4 P = 0 P = ( not xcitd) ( xcitd) Excitation has transfrrd from to in tim t h/4 t t h/ 4 ( xcitd again) ( not xcitd) In btwn tims Probability intrmdiat

18 Stationary Stats Considr two suprpositions of & H H H H Eignstat, Eignvalu H H Similarly H Eignstat, Eignvalu Obsrvabls of Enrgy Oprator

19 E 0 Rcall E 0 = 0 If E 0 not 0, splitting still symmtric about E 0 with splitting. Dimr splitting E 0 E = 0 Dlocalizd Stats Probability of finding ithr molcul xcitd is qual

20 Us proction oprators to find probability of bing in ignstat, givn that th systm is in t t t C C C C * *.. * * CC ( C ) C complx conugat of prvious xprssion t t lso t t [ * * * * CC CC CC ] CC [cos ( / ) sin t ( t / ) i cos( t / )sin( t / ) i sin( t / )cos( t / )] Mak nrgy masurmnt qual probability of finding or - t is not an ignstat

21 Expctation Valu Half of masurmnts yild +; half On masurmnt on many systms Expctation Valu should b 0. * * tht C C H C C CC H CC H CC H CC H * * * * Using H H CC CC * * isin( t )cos( t ) isin( t )cos( t ) tht 0 Expctation Valu - Tim indpndnt If E 0 0, gt E 0

22 Non-Dgnrat Cas E E E * E E 4 P CC cos t E 4 E 4 * E 4 P CC cos t E 4 s E incrass Oscillations Fastr Lss Probability Transfrrd cos cos x x Thrmal Fluctuations chang E &

23 Crystals Dimr splitting Thr lvls n lvls Using loch Thorm of Solid Stat Physics can solv problm of n molculs or atoms whr n is vry larg,.g., 0 0, a crystal lattic.

24 Excitation of a On Dimnsional Lattic - + lattic spacing ground stat of th th molcul in lattic (normalizd, orthogonal) xcitd stat of th molcul Ground stat of crystal with n molculs g n 0 Tak ground stat to b zro of nrgy. Excitd stat of lattic, th molcul xcitd, all othr molculs in ground stats 0 n nrgy of singl molcul in xcitd stat, E st of n-fold dgnrat ignstats in th absnc of intrmolcular intractions bcaus any of th n molculs can b xcitd.

25 loch Thorm of Solid Stat Physics Priodic Lattic Eignstats Lattic spacing - Translating a lattic by any numbr of lattic spacings,, lattic looks idntical. caus lattic is idntical, following translation Potntial is unchangd by translation Hamiltonian unchangd by translation Eignvctors unchangd by translation loch Thorm from group thory and symmtry proprtis of lattics ( ) ip / L x ( x) p p ik ( x) p Th xponntial is th translation oprator. It movs function on lattic spacing. p is intgr ranging from 0 to n-. L = n, siz of lattic k = p/l

26 ny numbr of lattic translations producs an quivalnt function, rsult is a suprposition of th kts with ach of th n possibl translations. ( k) n n 0 ik loch Thorm ignstats of lattic Kt with xcitd stat on th molcul (singl sit function) Sum ovr all possibl positions (translations) of xcitd stat. Normalization so thr is only a total of on xcitd stat on ntir lattic. k is a wav vctor. Diffrnt valus of k giv diffrnt wavlngths. Diffrnt numbr of half wavlngths on lattic. In two stat problm, thr wr two molculs and two ignstats. For a lattic, thr ar n molculs, and n ignstats. Thr ar n diffrnt orthonormal ( k) arising from th n diffrnt valus of th intgr p, which giv n diffrnt valus of k. k = p/l

27 On dimnsional lattic problm with narst nighbor intractions only M, H H H Molcular Hamiltonian in absnc of intrmolcular intractions. Intrmolcular coupling btwn adacnt molculs. Coupls a molcul to molculs on ithr sid. Lik coupling in two stat (two molcul) problm. n H H H H H Sum of singl molcul Hamiltonians M M M M M H M E Th th trm givs E, th othr trms giv zro bcaus th ground stat nrgy is zro. H k H n ik M ( ) M n 0 n ik H M n 0 E ( k) In th absnc of intrmolcular intractions, th nrgy of an xcitation in th lattic is ust th nrgy of th molcular xcitd stat.

28 Inclusion of intrmolcular intractions braks th xcitd stat dgnracy., H stat with th molcul xcitd coupling strngth H, Oprat on loch stats ignstats. H k H n n ik, ( ), 0 n ik H 0, n n ik ik 0 n

29 H k n ik ik, ( ) n 0 Each of th trms in th squar brackts can b multiplid by ik ik n ik ik ik ik ik ik H, ( k) n 0 combining combining n ik ik( ) ik ik( ) n 0 + In spit of diffrnc in indics, th sum ovr is sum ovr all lattic sits bcaus of cyclic boundary condition. Thrfor, th xp. tims kt, summd ovr all sits ( ) ( k)

30 Rplacing xp. tims kt, summd ovr all sits ( ) with ( k) ik ik H, ( k) ( k) ( k) ik ik ( k) factor out ( k) cos( k) ( k) dding this rsult to H ( k) E ( k) M Givs th nrgy for th full Hamiltonian. E( k) E cos( k ) Th narst nighbor intraction with strngth braks th dgnracy.

31 Rsult (on dimnsion, narst nighbor intraction only, ) E( k) E cos( k ) k wav vctor labls lvls lattic spacing Exciton and E( k) E cos( k ) Each stat dlocalizd ovr ntir crystal..5 E(k) - E (units of ) k rillouin zon Quasi-continuous Rang of nrgis from to -

32 Exciton Transport Exciton wav packt mor or lss localizd lik fr particl wav packt Disprsion Rlation: ( k) E( k) E cos( k) Group Vlocity: V g d( k) sin( k ) dk Exciton packt movs with wll dfind vlocity. Cohrnt Transport. Thrmal fluctuations (phonon scattring) localization, incohrnt transport, hopping