Lecure #11: Wavepacke Dynamics for Harmonic Oscillaor and PIB Las ime: Time Dependen Schrödinger Equaion Ψ HHΨ = iħ Express Ψ in complee basis se of eigenfuncions of ime independen H H {ψ n (x), E n } For 2-sae Ψ s, we saw ha Ψ(x, ) = j ie c j /n j e ψ j (x) 1 Ψ * (x, )Ψ(x, ) moves only if Ψ conains a leas 2 differen E j s; 2 dxψ * Ψ = 1 for all Ψ(x, ) Conservaion of probabiliy 3 (xˆ) and (pˆ) obey Newon s laws Moion of cener of wavepacke Ehrenfes s Theorem 2 4 Survival probabiliy P () = dxψ * (x, )Ψ(x, = 0) How fas does Ψ(x, ) move away from is iniial preparaion Ψ(x, 0) Dephasing, parial recurrence, grand recurrence 5 Recurrences occur when all ΔE ij are ineger muliples of common facor TODAY: Some examples of wavepackes in a Harmonic Oscillaor or PIB poenial well Mosly picorial We sar wih he iniial condiion, Ψ(x, = 0), which I call he pluck I is quie analogous o wha musicians undersand abou a wave on a sring ha is ied down a boh ends Ψ(x, 0) = c j ψ j j
561 Fall 2013 Lecure #11 Page 2 If we have a complee se of ψ j (x), hen we can expand any Ψ(x, 0) as a linear combinaion of ψ j (x) Like a Fourier series Once we have Ψ(x, 0) i is rivial o pu in he dependence Ψ(x, ) = ie c j /n j e ψ j (x) because for each known ψ j here is a known E j We usually like o creae a wavepacke localized near a urning poin The more ψ j (x) wavefuncions we use in describing Ψ(x, 0), he sharper we can make he = 0 wavepacke There are several experimenally or picorially simple schemes for creaing a wavepacke, which is a superposiion of eigensaes of H ha have differen values of E j (needed in order o have any moion a all) Creae a non-eigensae a = 0 Half Harmonic Oscillaor, barrier a x = 0 Remove barrier a = 0 To make such a = 0 wavepacke, we can use any of he ψ 2v+1 (odd) eigensaes ha have a node a x = 0 Bu in order o have ime dependen (xˆ) and (pˆ) we also need some ψ 2v (even) eigensaes in pairs, c 2 ψ 2 (0) = c 0 ψ 0 (0), so ha c 2 ψ 2 (0) + c 0 ψ 0 (0) = 0 Usually, in order o make life simple, we choose only 3 ψ v o creae a Ψ(x, = 0) wih approximaely he correc shape Ψ(x, 0) = c 0 ψ 0 (x)+ c 1 ψ 1 (x)+ c 2 ψ 2 (x) This will have a node a x = 0 and larger probabiliy for x < 0 han for x > 0 (xˆ) =2c 0 c 1 x 01 cos ω +2c 1 c 2 x 12 cos ω Noe ha x 00, x 11, x 22, and x 02, are all zero because of he Harmonic Oscillaor Δv = ±1 selecion rule for ˆx Noe ha probabiliy and (xˆ) sloshes back and forh beween he x < 0 and x > 0 regions a angular frequency ω Wha is H H?Is i dependen? H H = c 0 2 E 0 + c 1 2 E 1 + c 2 2 E 2
561 Fall 2013 Lecure #11 Page 3 H because he ψ v are eigenfuncions of H, herefore orhogonaliy ensures ha here are no ie v /n +ie v /n c i c j cross erms, and he pairs of e and e facors combine o yield 1 Of course, E has o be conserved Creae a non eigensae wavepacke by causing a verical elecronic ransiion a = 0 The excied sae poenial energy curve is displaced from ha of he elecronic ground sae R " (E) = R "" e v "» 0 Noaion: R " e " for upper sae "" for lower sae v =0 R e "" ħ ω "" /2 The v "" = 0 wavepacke is ransferred o he excied sae The Franck Condon principle says ha, since elecrons move much faser han nuclei, he elecronic ransiion is insananeous as far as he nuclei are concerned This means ha x and p do no change in an elecronic ransiion So we sar ou wih a wavepacke on he excied sae where R H = R "", (pˆ) =[2μ ħω "" /2] 1/2 I is clear ha he iniially formed wavepacke will be 0 e 0 localized near he inner uring poin of he excied sae and will be experiencing a large " " force in he +x direcion If we approximae Ψ(x, 0) as a mixure of v = 10 and v =11 saes Ψ(x, 0) = c 10 ψ v l =10(x)+ c 11 ψ 11 (x) Ψ * (x, )Ψ(x, ) = c 10 2 ψ 10 2 + c 11 2 ψ 11 2 +2c 10 c 11 ψ 10 ψ 11 cos ω T (allowing c j and ψ j o be real)
561 Fall 2013 Lecure #11 Page 4 P () = (Ψ * (x, )Ψ(x, 0)) 2 i105nω/n i115nω/n 2 = c 10 2 e + c 11 2 e 4 4 2 2 = c 10 + c 11 +2c 10 c 11 cos ω A = 0 P () is a is maximum value Bu here are a series of perfec rephasings a = n 2 π π and minimum values a = (2n +1) ω ω Why does he wavepacke behave in his way? HR R e "" π/2ω πω (p) π/ω HR = c 10 c 11 R 10,11 cos ω (c 10 c 11 < 0) (pˆ) = c 10 c 11 p 10,11 sin ω (he R 10,11 harmonic oscillaor inegral is posiive and he P 10,11 inegral is imaginary) The iniial wavepacke moves away from iself faser in momenum space han in coordinae space, so he iniial decay of P () is predominanly a momenum effec Dephasing and Rephasing of a Wavepacke A favorie kind of wavepacke is one ha is localized near a urning poin a = 0 Iisa paricle like sae ha we expec will ac in a classical mechanical paricle-like manner For a Harmonic Oscillaor, all E v l E v are ineger muliples of ħω Thus, if he ime dependen par of Ψ * (x, )Ψ(x, ) (he coherence erm) is phased up a = 0, hen i will be phased 1 1 h down a = τ = because he signs of all he Δv = ±1 coherence erms will be 2 2 nω reversed We expec
561 Fall 2013 Lecure #11 Page 5 Ψ * Ψ (E) (E) 0 0 x 1 2π = 0 = 2 ω phased up phased down A in beween imes, Ψ * Ψ is likely o look very un paricle like Dephased The wavepacke undergoes simple harmonic moion, and appears in all of is simple glory a alernaing urning poins Is expecaion values (xˆ) and (pˆ) move according o Newon s laws, bu he picure of Ψ * (x, )Ψ(x, ) can be more complicaed Speculae abou wha you migh expec for a wavepacke composed of eigensaes of an anharmonic oscillaor, wih energy levels G(v) = ω e (v + 1/2) ω e x e (v + 1/2) 2, where ω ex e 002 ω e Is he periodic rephasing perfec? Is each successive rephasing only parial? Does he wavepacke evenually lose is paricle like localizaion? Once his happens, does he localized wavepacke ever re emerge as a fully rephased eniy? There is no variaion of ω wih E for Harmonic Oscillaor All of he coherence erms in HO give (x) A cos ω (p) B sin ω
561 Fall 2013 Lecure #11 Page 6 Does his look familiar? Jus like classical HO d (x) = 1 (p x ) d m v = p/m here, d (p x ) = ( V (x)) Ehrenfes s Theorem d ma = F v is velociy, no vibraional quanum number Cener of wavepacke moves according o Newon s equaions! Tunneling For a hin barrier, all ψ v wih node in middle (odd v) hardly feel he barrier They are shifed o higher E only very slighly The ψ v ha have a local maximum a x = 0 (he even v saes) all feel he barrier very srongly They are shifed up almos o he energy of nex higher level, especially if he energy of he HO ψ v lies below he op of he barrier Why do I say ha he barrier causes all HO energy levels o be shifed up? [We will reurn o his problem once we have discovered non-degenerae perurbaion heory] We see some evidence for his difference in energy shifs for odd vs even-v levels by hinking abou he 1 HO 2
561 Fall 2013 Lecure #11 Page 7 This half-ho oscillaor only has levels a E 1, E 3 of he full oscillaor so v = 0 of he 1 oscillaor is a he energy of v = 1 of he full oscillaor 2 So a barrier causes even-v levels o shif up a lo and become near-degenerae wih he nex higher odd-v level [Can change energy order because he energy levels are in order of # of nodes] Energy Levels of Ordinary HO Energy Levels of HO wih finie heigh barrier in he middle 5 5 4 4 almos back o normal 3 3 2 } medium 2 1 1 0 0 Suppose we make a ψ 1, ψ 0 wo-sae superposiion Δ 0,1 T } small 2 ψ 2 2 ψ 2 Ψ * (x, )Ψ(x, ) = c 0 0 + c 1 1 +2c 1 c 2 ψ 0 ψ 1 cos Δ 01 Δ 0,1 = E 1 E 0 ħ (Δ 0,1 is small) Wha does he ψ v = 0 eigensae of he well wih barrier in he middle look like?
561 Fall 2013 Lecure #11 Page 8 v = 0 v = 1 shifed slighly up in E bu ψ is hardly disored v = 0 has zero nodes (wavefuncion ried bu barely failed o have one node) I resembles he v = 1 sae of he no-barrier oscillaor Ψ 1,0 (x, 0) = 2 1/2 [ψ 1 (x)+ ψ 0 (x)] looks like his a = 0 x 0 x + 1 1 Ψ 1,0(x, * )Ψ ψ 2 ψ 2 1,0 (x, ) = 0 + 1 + ψ 1 ψ 0 cos Δ 0,1 2 2 We ge oscillaion of nearly perfecly localized wavepacke righ lef righ ad infinium * Δ 0,1 is small so period of oscillaion is long (i is he energy difference beween he v = 0 and v = 1 eigensaes of he harmonic plus barrier poenial) Similarly for 3,2 wavepacke * lef/righ localizaion is less perfec * oscillaion is faser because Δ 2,3 is larger
561 Fall 2013 Lecure #11 Page 9 MESSAGE: As you approach op of barrier, unneling ges faser Tunneling is slow (small spliings of consecuive pairs of levels) for high barrier, hick barrier, or a E far below op of barrier Can use paern of energy levels (Δ 0,1 and Δ 2,3 ) observed in a specrum (frequencydomain) o learn abou ime-domain phenomena (unneling) Dynamics in he frequency-domain
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