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 n = E n n we ofen wan o calculae changes in he ampliudes of n induced by V( ): where ψ ( ) = c n ( ) n n ()= kψ ( ) = c k ku, ( )ψ( ) In he ineracion picure, we defined b = e +iω kr k ( ) = k c k ( ) ψ I which conains all he relevan dynamics. The changes in ampliude can be calculaed by solving he coupled differenial equaions: i bk = i e ω nk V kn n ()b n () For a complex sysem or a sysem wih many saes o be considered, solving hese equaions isn pracical. Alernaively, we can choose o work direcly wih U I (, ), and we can calculae b k ()as: b k = ku I (, )ψ ( ) where U, ) = exp I ( + i V ( )d τ τ I Now we can runcae he expansion afer a few erms. This is perurbaion heory, where he dynamics under H 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. b k () b k () ; V E k En
p. 34 Transiion Probabiliy Le s ake he specific case where we have a sysem prepared in probabiliy of observing he sysem in k a ime, due o V( )., and we wan o know he P k ()= b b k k ( ) k ( )= U I (, ) i b k exp + d k ()= τ τ V I ( ) i = k dτ k V I ( τ ) i τ V I ( )V (τ 1 ) + dτ dτ 1 k + τ I using iω k kv I () = ku V ()U = e V k () b k ()= δ k i iω k τ ( ) dτ 1 e 1 V k τ 1 firs order + i τ iω dτ V km τ m τ dτ1 e iω mkτ 1 ( )e V m ( τ 1 ) + second order m This expression is usually runcaed a he appropriae order. Including only he firs inegral is firs-order perurbaion heory. If ψ is no an eigensae, we only need o express i as a superposiion of eigensaes, bu remember o conver o c k ()= e ω k b k (). 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 ω k.
p. 35 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 he frequency of he h.o. H = p m + k ()x A = δ k =A / πσ k() = k k() +δ δ k() = A exp ( ) σ k = mω p x A x H= H + V () = + k + exp m σ H () V ( ) H n = E n n H + 1 = Ω a a E n 1 = Ω n + If he sysem is in a =, wha is he probabiliy of finding i in n a =? n ()= i for n : b dτ V n ( τ )e iω n τ ω n = nω i + n = A n x d e e τ iω τ τ σ + Ωτ τ / σ n ()= i b A n x dτ e in + π 1 b exp(ax + bx + c dx = ) exp (c 4 a ) a
p. 36 e n ()== i n σ b A n x Ω / 4 Wha abou marix elemen? + x = (a a ) = (aa+ a a+ aa mω mω + a a ) Firs-order perurbaion heory won allow ransiions o n = 1, only n = 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 V(x), 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 = mω So, b = i A e σ Ω mω A 4σ Ω P = b = e A =δk πσ m Ω Significan ransfer of ampliude occurs when he compression pulse is shor compared o he vibraional period. 1 <<Ω σ Validiy: Firs order doesn allow for feedback and b n can change much from is iniial value. for P A << mω
p. 37 Firs-Order Perurbaion Theory A number of imporan relaionships in quanum mechanics ha describe rae processes come from 1 s order P.T. For ha, here are a couple of model problems ha we wan o work hrough: (1) Consan Perurbaion ψ() =. A consan perurbaion of ampliude V is applied o. Wha is P k? V() V ( ) = θ( )V < = V To firs order, we have: b k = δ k i dτ e iω k (τ ) Vk V k independen of ime ku V U = Ve iω k ( ) iω k (τ ) = δ k + i V k dτ e = δ k + V k [exp(iω k ( )) 1] E k E using e i 1 = ie i sin = δ k + iv k e iω k For k we have b k 4V k ( )/ E k E sin(ω k ( )/) P k = = sin 1 ω k ( ) E k E or seing = and wriing his as we did in lecure 1:
p. 38 P k = V sin ( / ) where = E k E l or P k = V sinc ( / ) Compare his wih he exac resul: V P k = sin ( + V + V / ) Clearly he P.T. resul works for V <<. ( no for degenerae sysems) The probabiliy of ransfer from o k as a funcion of he energy level spliing (E k E ) : V kl / ( ) π / Area scales linearly wih ime. 4π π π 4π E k E l Time-dependence: Ek=El (exac soluion for Ek=El ) Pk() Ek-El Vkl Ek-E l >> Vkl π /V kl
p. 39 Time dependence on resonance ( =): expand sin x = x 3 x 3! + V 3 3 P k = + 6 3 V = This is unrealisic, bu he expression shouldn hold for =. Long ime limi: The sinc (x) funcion narrows rapidly wih ime giving a dela funcion: sin (ax ) π lim = δ( x) ax π V lim P k k ()= δ (E k E )( ) A probabiliy ha is linear in ime suggess a ransfer rae ha is independen of ime! This suggess ha he expression may be useful o long imes: V k k ()= P π w = δ(e k E ) k ( ) This is one saemen of Fermi s Golden Rule, which describes relaxaion raes from firs order perurbaion heory. We will show ha his will give long ime exponenial relaxaion raes.
p. 4 () Harmonic Perurbaion Ineracion of a sysem wih an oscillaing perurbaion urned on a ime =. This describes how a ligh field (monochromaic) induces ransiions in a sysem hrough dipole ineracions. V ( ) = V cosω = µe cosω V() observe τ V k () = V k cosω = V k [ e iω + e iω ] To firs order, we have: b k = i = k ψ I () dτ V k ( τ )e iω τ k iv k d e i(ω k +ω ) τ e i(ω k ω ) = τ τ i(ω k +ω) e i(ω k +ω) e i(ω k ω) e i(ω k ω) V k e = + ω k +ω ωk ω Seing and using e iθ 1 =ie iθ sin θ b k = k k iv k e i(ω ω) / sin (ω ω) / e i(ω +ω) / sin (ω +ω) / k k + ω ω ω +ω k k Noice ha hese erms are only significan when ω ω k : resonance!
p. 41 Firs Term max a: ω =+ω k E k > E E k = E + ω Second Term ω = ω k E k < E E k = E ω Absorpion k Simulaed Emission l (resonan erm) l (ani-resonan erm) k For he case where only absorpion conribues, E k > E, we have: P k V k 1 b k = = sin (ω ω) (ω k ω) E µ k k or sin 1 (ω ω) (ω ω) k k The maximum probabiliy for ransfer is on resonance ω k = ω V kl / 4 π / - -1 1 ω ω kl /π
p. 4 We can compare his wih he exac expression: V k 1 + ω (ω ω) + k V k P = = sin ω) k b k V k ( k which poins ou ha his is valid for couplings ω = (ω ω). k V k ha are small relaive o he deuning Limiaions of his formula: By expanding sin x = x 3 x 3! +, we see ha on resonance ω=ω k ω lim V k P ω k ()= 4 This clearly will no describe long-ime behavior, bu he expression is no valid for ω =. Nonheless, i will hold for small P k, so << V k (depleion of 1 negleced in firs order P.T.) 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. 1 1 > π / <<ω kl ω ω k These relaionships imply ha V k << ω k