( ) = b n ( t) n " (2.111) or a system with many states to be considered, solving these equations isn t. = k U I ( t,t 0 )! ( t 0 ) (2.

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Download "( ) = b n ( t) n " (2.111) or a system with many states to be considered, solving these equations isn t. = k U I ( t,t 0 )! ( t 0 ) (2."

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1 Andrei Tokmakoff, MIT Deparmen of Chemisry, 3/14/ PERTURBATION THEORY Given a Hamilonian H = H 0 + V where we know he eigenkes for H 0 : H 0 n = E n n, we can calculae he evoluion of he wavefuncion ha resuls fromv :! I = b n n.111 using he coupled differenial equaions for he ampliudes of n. For a complex imedependence or a sysem wih many saes o be considered, solving hese equaions isn pracical. Alernaively, we can choose o work direcly wihu I, 0 n, calculae as: = k U I, 0! 0.11 where U I, 0 %! = exp +!i 0 V I d.113 Now we can runcae he expansion afer a few erms. This is perurbaion heory, where he dynamics under H 0 are reaed exacly, bu he influence of V on b n is runcaed. This works well for small changes in ampliude of he quanum saes wih small coupling marix elemens relaive o he energy spliings involved! 0 ; V! E k E n As we ll see, he resuls we obain from perurbaion heory are widely used for specroscopy, condensed phase dynamics, and relaxaion. Transiion Probabiliy Le s ake he specific case where we have a sysem prepared in!, and we wan o know he probabiliy of observing he sysem in k a ime, due ov. = = k U I, 0!.114

2 -7 = k exp +! i d! V I % = k!! i d k V I 0 +!i %! d d 1 k V I V I 1 0 0! using k V I! = k U 0 V U 0! = e!i!k V k!.117 So, =! k! i d 1 e i!k % 1 V k! 1 firs order m!i%! e!i* m V m d d 1 e!i* mk V km 0 0 second order The firs-order erm allows only direc ransiions beween! and k, as allowed by he marix elemen in V, whereas he second-order erm accouns for ransiions occuring hrough all possible inermediae saes m. For perurbaion heory, he ime ordered inegral is runcaed a he appropriae order. Including only he firs inegral is firs-order perurbaion heory. The order of perurbaion heory ha one would exend a calculaion should be evaluaed iniially by which allowed pahways beween! and k you need o accoun for and which ones are allowed by he marix elemens. For firs order perurbaion heory, he expression in eq..118 is he soluion o he differenial equaion ha you ge for direc coupling beween! and k :!! b = i k! ei k V k b 0.10

3 -8 This indicaes ha he soluion doesn allow for he feedback beween! and k ha accouns for changing populaions. This is he reason we say ha validiy dicaes! 0. If! 0 is no an eigensae, we only need o express i as a superposiion of eigensaes, n! 0 = b n 0 and n = b n 0 k U I n!..11 Now here may be inerference effecs beween he pahways iniiaing from differen saes: n = c k = =! k b n n n.1 Also noe ha if he sysem is iniially prepared in a sae!, and a ime-dependen perurbaion is urned on and hen urned off over he ime inerval =! o +!, hen he complex ampliude in he arge sae k is jus he Fourier ransform of V evaluaed a he energy gap!!k. =! i + d e!i k V! % k.13! If he Fourier ransform is defined as F! V! V! % = 1 + * d V exp i!,.14 hen! = V! k!..15

4 -9 Example: Firs-order Perurbaion Theory Vibraional exciaion on compression of harmonic oscillaor. Le s subjec a harmonic oscillaor o a Gaussian compression pulse, which increases is force consan. Firs wrie he Hamilonian: Now pariion i according o H = H 0 + V : H = T + V = p m + 1 k x.16 k = k 0 +!k k 0 = m!!k =!k 0 exp 0 p H = m + 1 k 0 x + 1!k 0 x exp 0! %! H 0 V % H 0 n = E n n H 0 =!! a a + 1 % E =!! n + 1 % n.19

5 -30 If he sysem is in 0 a 0 =!, wha is he probabiliy of finding i in n a =!? For n! 0 : b n =!i! Using! n0 = E n E 0! = n : b n =!i! k n x + 0 d 0! d V n0 e i n % e in e!.131 So, b n =!i! k n x 0 e!n % / 0.13 Here we used: Wha abou he marix elemen? x =! m! a + a =! m! aa + a a + aa + a a.133 Firs-order perurbaion heory won allow ransiions o n = 1, only n = 0 and n =. Generally his wouldn be realisic, because you would cerainly expec exciaion o v=1 would dominae over exciaion o v=. A real sysem would also be anharmonic, in which case, he leading erm in he expansion of he poenial Vx, ha is linear in x, would no vanish as i does for a harmonic oscillaor, and his would lead o marix elemens ha raise and lower he exciaion by one quanum. However for he presen case, x 0 =! m!.134 So, b =!i k 0 m% e! %.135

6 -31!k and P = b = 0 e %4! = m k 0 k 0 + e %4.136 * From he exponenial argumen, significan ransfer of ampliude occurs when he compression pulse is shor compared o he vibraional period.! << Validiy: Firs order perurbaion heory doesn allow for b n o change much from is iniial value. For P << 1 %! k 0 k 0 * << Generally, he perurbaion δk mus be small compared o k 0, i.e. H 0 >> V, bu i should also work well for he impulsive shock limi σω<<1.

7 -3 FIRST-ORDER PERTURBATION THEORY A number of imporan relaionships in quanum mechanics ha describe rae processes come from 1 s order perurbaion heory. For ha, here are a couple of model problems ha we wan o work hrough: Consan Perurbaion Sep-Funcion Perurbaion! =!. A consan perurbaion of ampliude V is applied o 0. Wha is? V =! 0 V = % 0 < 0 V 0 To firs order, we have: k U 0 V U 0! = V e i! k! 0 =! k! i d e i% k! 0 Vk! 0.1 Here V k! is independen of ime. Now, assuming k!! and seing 0 = 0 we have =! i! V d k e i k =! V k! E k! E!! 1 exp i k! %.141 =! iv k! ei k! / sin E k! E k! /.14! Where I usede i! 1 = ie i! sin!. Now

8 -33 = = 4 V k! Wriing his as we did in Lecure 1: E k! E! sin 1 k!.143 = V! sin! /!.144 where! = E k E!. Compare his wih he exac resul we have for he wo-level problem: = Clearly he perurbaion heory resul works for V << Δ. We can also wrie he firs-order resul as V V +! sin! + V /!.145 = V where sinc x = sin x x. Since lim sinc x = 1, x!0! sinc! /!.146 lim P!0 k = V!.147 The probabiliy of ransfer from! o k as a funcion of he energy level spliing E k! E! : Area scales linearly wih ime. Since he energy spread of saes o which ransfer is efficien scales approximaely as E k! E! <, his observaion is someimes referred o as an uncerainy relaion

9 -34 wih!e!!. However, remember ha his is really jus an observaion of he principles of Fourier ransforms, ha frequency can only be deermined by he lengh of he ime period over which you observe oscillaions. Since ime is no an operaor, i is no a rue uncerainly relaion like!p!x!. Now urning o he ime-dependence: The quadraic growh for Δ=0 is cerainly unrealisic a leas for long imes, bu he expression shouldn hold for wha is a srong coupling case Δ=0. However, le s coninue looking a his behavior. In he long ime limi, he sinc x funcion narrows rapidly wih ime giving a dela funcion: lim! sin ax lim =! ax x.148 = V k! E k % E!.149 The dela funcion enforces energy conservaion, saying ha he energies of he iniial and arge sae mus be he same in he long ime limi. Wha is ineresing in eq..149 is ha we see a probabiliy growing linearly in ime. This suggess a ransfer rae ha is independen of ime, as expeced for simple firs order kineics:

10 -35 w k =!P k! = V k! E k E!.150 This is one saemen of Fermi s Golden Rule he sae-o-sae form which describes relaxaion raes from firs order perurbaion heory. We will show ha his rae properly describes long ime exponenial relaxaion raes ha you would expec from he soluion o dp d =!wp.

11 -36 Slowly Applied Adiabaic Perurbaion Our perurbaion was applied suddenly a > 0 sep funcion V =! 0 V This leads o unphysical consequences you generally can urn on a perurbaion fas enough o appear insananeous. Since firs-order P.T. says ha he ransiion ampliude is relaed o he Fourier Transform of he perurbaion, his leads o addiional Fourier componens in he specral dependence of he perurbaion even for a monochromaic perurbaion! So, le s apply a perurbaion slowly... V = V e! here η is a small posiive number.! 1 is he effecive urn-on ime of he perurbaion. The sysem is prepared in sae! a =!. Find. = k U I! =!i d e i k! % k V! e! =!iv k! exp + i k! * + i k! exp + i E k! E! / = V * k! E k! E! + i V = = k! exp * + k! = V k! exp * E k! E! + This is a Lorenzian lineshape in! k! wih widh!!.

12 -37 Gradually Applied Perurbaion Sep Response Perurbaion The gradually urned on perurbaion has a widh dependen on he urn-on rae, and is independen of ime. The ampliude grows exponenially in ime. Noice, here are no nodes in. Now, le s calculae he ransiion rae: w kl =!! = V k! e + k! Look a he adiabaic limi;!0. Seing e! 1 and using lim! 0! = %! + k! k! w k! =! V k! k! =! V k! Ek E! We ge Fermi s Golden Rule independen of how perurbaion is inroduced!

13 -38 Harmonic Perurbaion Ineracion of a sysem wih an oscillaing perurbaion urned on a ime 0 = 0. This describes how a ligh field monochromaic induces ransiions in a sysem hrough dipole ineracions. Again, we are looking o calculae he ransiion probabiliy beween saes! and k: V = V cos! = µe 0 cos!.151 To firs order, we have: V k! = k! I = V k! cos! = V k! ei! + e i! % = i! d V k 0 e i% k.15 = iv k! d e i % k +% e i % k % 0 seing 0! = V k! e i % k +% 1 % k + % + ei % k % 1 * + * % k % + Now, using e i! 1 = ie i! sin! as before: =!iv k! / sin k!! / % k!! e i k!! / sin k! + / % % k! + + ei k! Noice ha hese erms are only significan when!! k!. As we learned before, resonance is required o gain significan ransfer of ampliude.

14 -39 Firs Term Second Term max a :! = +! k!! =! k! E k > E! E k < E! E k = E! +! E k = E!! Absorpion resonan erm Simulaed Emission ani-resonan erm For he case where only absorpion conribues, E k > E!, we have:! = = V k! or sin 1! k!! E 0 µ k! sin 1! k!!!! k! %!! k! %.155 We can compare his wih he exac expression: V! = = k!! k!! 1 sin % + V k! V k! +! k!!.156 which poins ou ha his is valid for couplings V k! ha are small relaive o he deuning! = k!. The maximum probabiliy for ransfer is on resonance! k! =!

15 -40 Limiaions of his formula: By expanding sin x = x! x3 3! +, we see ha on resonance! = k! 0 lim! 0 = V k! This clearly will no describe long-ime behavior. This is a resul of 1 s order perurbaion heory no reaing he depleion of!. However, i will hold for small, so we require <<! V k.158 A he same ime, we can observe he sysem on oo shor a ime scale. We need he field o make several oscillaions for i o be a harmonic perurbaion. These relaionships imply ha > 1! 1! k!.159 V k! <<! k!.160

16 -41 Adiabaic Harmonic Perurbaion Wha happens if we slowly urn on he harmonic ineracion? V = V e! cos = i e d! V % k e i + e i * i k +!, + = V k e i k +! e! k + + i! + e i k k + i! Again, we have a resonan and ani-resonan erm, which are now broadened by!. If we only consider absorpion: V = = k! 1 e! 4 k! +! which is he Lorenzian lineshape cenered a! k! =! wih widh! =. Again, we can calculae he adiabaic limi, seing! 0. We will calculae he rae of ransiions! k! = /. Bu le s resric ourselves o long enough imes ha he harmonic perurbaion has cycled a few imes his allows us o neglec cross erms! resonances sharpen. w k! =! V k! + k! + % k! *, +,

ψ ( t) = c n ( t) t n ( )ψ( ) t ku t,t 0 ψ I V kn

ψ ( t) = c n ( t) t n ( )ψ( ) t ku t,t 0 ψ I V kn MIT Deparmen of Chemisry 5.74, Spring 4: Inroducory Quanum Mechanics II p. 33 Insrucor: Prof. Andrei Tokmakoff PERTURBATION THEORY Given a Hamilonian H ( ) = H + V ( ) where we know he eigenkes for H H