The time evolution of the state of a quantum system is described by the time-dependent Schrödinger equation (TDSE): ( ) ( ) 2m "2 + V ( r,t) (1.

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1 Adrei Tokmakoff, MIT Departmet of Chemistry, 2/13/ TIME-DEPENDENT QUANTUM MECHANICS 1 INTRODUCTION 11 Time-evolutio for time-idepedet Hamiltoias The time evolutio of the state of a quatum system is described by the time-depedet Schrödiger equatio (TDSE): i t " ( r, t ) = Ĥ" ( r, t ) (11) Ĥ is the Hamiltoia operator which describes all iteractios betwee particles ad fields, ad is the sum of the kietic ad potetial eergy For oe particle Ĥ = 2 2m "2 + V ( r,t) (12) The state of the system is expressed through the wavefuctio ( r, t) The wavefuctio is complex ad caot be observed itself, but through it we obtai the probability desity P = " * ( r, t) ( r, t)dr = ( r, t) ( r, t) (13) which characterizes the probability that the particle described by Ĥ is betwee r ad dr at time t Most of what you have previously covered is time-idepedet quatum mechaics, where we mea that the Hamiltoia Ĥ is assumed to be idepedet of time: Ĥ = Ĥ r We the assume a solutio with a form i which the spatial ad temporal variables i the wavefuctio are separable: ( r, t) = " ( r )T ( t) (14) i 1 T ( t) t T ( t ) = Ĥ ( r )" r " r (15)

2 1-2 Here the left-had side is a fuctio oly of time (t), ad the right-had side is a fuctio of space oly ( r, or rather positio ad mometum) Equatio (15) ca oly be satisfied if both sides are equal to the same costat, E Takig the right had side we have Ĥ ( r ) r r = E " Ĥ ( r ) ( r ) = E ( r ) (16) This is our beloved Time-Idepedet Schrödiger Equatio (TISE) The TISE is a eigevalue equatio, for which r are the eigestates ad E is the eigevalue Here we ote that Ĥ = Ĥ = E, so Ĥ is the operator correspodig to E ad drawig o classical mechaics we associate Ĥ with the expectatio value of the eergy of the system Now takig the left had side of (15): i 1 T ( t) T t = E " t + ie ' ( T t = 0 (17) T ( t) = T 0 exp (iet / ) = T 0 exp (i"t ) (18) So, i the case of a boud potetial we will have a discrete set of eigefuctios ( r ) with correspodig eergy eigevalues E from the TISE, ad there are a set of correspodig solutios to the TDSE ( r, t) = c " ( r ) exp (i t) (19) where = E / ad c is a (complex) amplitude The eigefuctios form a orthoormal set, so that c 2 = 1 Sice the oly time-depedece is a phase factor, the probability desity (13) is idepedet of time for the eigefuctios ( r, t) Therefore, the eigestates ( r ) do ot chage with time ad are called statioary states However, more geerally, a system may exist as a liear combiatio of eigestates:

3 = c r, t " r, t = c e i t ( r ) 1-3 " (110) where c are (complex) amplitudes For such a case, the probability desity will oscillate with time As a example, cosider two eigestates ( r, t) = = c 1 " 1 e i 1 t + c 2 " 2 e i 2 t (111) For this state the probability desity oscillates i time ascos ( 2 " 1 )t : P( t) = 2 2 = = Re " * = c c2 2 + * c1 c 2 * 1 2 e 'i(( 2 '( 1 t) + c * 2 c 1 * 2 1 e +i ( 2 '( 1 t (112) 2 2 = cos (( 2 ' ( 1 )t We refer to this as a coherece, a coheret superpositio state If we iclude mometum (a wavevector) of particle associated with this state, we ofte describe this as a wavepacket Time Evolutio Operator More geerally, we wat to uderstad how the wavefuctio evolves with time The TDSE is liear i time Sice the TDSE is determiistic, we will defie a operator that describes the dyamics of the system: t = ˆ U ( t,t 0 ) t 0 (113) U is a propagator that evolves the quatum system as a fuctio of time For the timeidepedet Hamiltoia: t " ( r, t ) + iĥ " ( r, t ) = 0 (114) To solve this, we will defie a operatort ˆ = exp (iĥt / ), which is a fuctio of a operator A fuctio of a operator is defied through its expasio i a Taylor series:

4 1-4 T ˆ = exp (iĥt ) = 1 iĥt = f ( Ĥ ) + 1 " 2 iĥt ' 2 " (115) Note: about fuctios of a operator  : Give a set of eigevalues ad eigevectors of Â, ie,  = a, you ca show by expadig the fuctio as a polyomial that f ( Â) = f ( a ) You ca also cofirm from the expasio thatt ˆ 1 = exp( iĥt ), otig that Ĥ is Hermetia ad Ĥ commutes with ˆ T Multiplyig eq (114) from the left by ˆ T 1, we ca write ad itegratigt 0 t, we get ) " exp iĥt t ' ( r, t, + ) * + - = 0, (116) exp iĥt " ' r, t r, t So, we see that the time-propagator is ad therefore " ( exp iĥt 0 = exp "Ĥ ( t " t 0 ) ˆ U t, t 0 ' r, t 0 " = exp Ĥ ( t t 0 ) = 0 (117) ( r, t 0 ' (118) ', (119) ( r, t) = e " E ( t"t 0 )/ ( r, t 0 ) (120)

5 1-5 I which I have used the defiitio of the expoetial operator fore iĥt / " = e ie t " Alteratively, if we substitute the projectio operator (or idetity relatioship) ito eq (119), we see " = 1 (121) Uˆ ( t, t 0 ) = e iĥ tt 0 / " " ( ) " " = = e i tt 0 E (122) This form is useful whe are characterized So ow we ca write our time-developig wavefuctio as = " e i t t 0 r, t " r, t 0 = e i ( t t 0 ) c = c t " (123) As writte i eq (113), we see that the time-propagator Uˆ ( t, t 0 ) acts to the right (o kets) to evolve the system i time The evolutio of the cojugate wavefuctios (bras) is uder the Hermetia cojugate of ˆ U t, t 0 actig to the left: ( t) = ( t 0 ) U ˆ ( t,t 0 ) (124) From its defiitio as a expasio ad recogizig " = exp iĥ ( t t 0 ) ˆ U t,t 0 Ĥ as Hermetia, you ca see that ' ' (125)

6 Time-evolutio of a coupled two-level system (2LS) Let s use this propagator usig a example that we will refer to ofte It is commo to reduce or map quatum problems oto a 2LS We will pick the most importat states the oes we care about ad the discard the remaiig degrees of freedom, or icorporate them as a collectio or cotiuum of other degrees of freedom termed a bath, Ĥ = Ĥ 0 + Ĥ bath We will discuss the time-evolutio of a 2LS with a time-idepedet Hamiltoia Cosider a 2LS with two (uperturbed) states a ad b with eergies a ad b, which are the coupled through a iteractiov ab We will ask: If we prepare the system i state a, what is the time-depedet probability of observig it i b? Ĥ = a " a a + b " b b + a V ab b + b V ba a = " V a ab ( ' V ba " b (126) The states a ad b are i the ucoupled or oiteractig basis, ad whe we talk about spectroscopy these might refer to uclear or electroic states i what I refer to as a site basis or local basis The couplig V mixes these states, givig two eigestates of Ĥ, + ad ", with correspodig eergy eigevalues + ad " We start by searchig for the eigevalues ad eigefuctios of the Hamiltoia Sice * the Hamiltoia is Hermetia, ( H ij = H ji ), we write V ab = V ba * = V e i" (127)

7 1-7 Ĥ = a V e "i ' ) (128) V e +i ( Now we defie variables for the mea eergy ad eergy splittig betwee the ucoupled states b E = a + b 2 (129) = " a " b 2 (130) The we obtai the eigevalues of the coupled system by solvig the secular equatio givig det( H "I ) = 0, (131) ± = E ± " 2 + V 2 (132) Because the expressios get messy, we do t use this expressio to fid the eigevectors for the coupled system, ± Rather, we use a substitutio where we defie: ta2 = V " (133) with 0 θ π/2 Now, Ĥ = E I + 1 ' ta 2"e +i 1 ta 2"ei ( ) * (134) ad we ca express the eigevalues as ± = E ± "sec2 (135) We wat to fid the eigestates of the Hamiltoia, ± from Ĥ ± = " ± ± where eg + = c a a + c b b This gives + = cos" e i / 2 a + si" e i / 2 b = si" e i / 2 a + cos" e i / 2 b (136)

8 1-8 Note that this basis is orthoormal (complete ad orthogoal): " " = 1 Now, let s examie the expressios for the eigestates i two limits: (a) Weak couplig (V/Δ << 1) Here θ 0, ad + correspods to a weakly perturbed by the V ab iteractio " correspods to b I aother way, as " 0, we fid + " a ad " b (b) Strog couplig (V/Δ >> 1) I this limit θ = π/4, ad the a/b basis states are idistiguishable The eigestates are symmetric ad atisymmetric combiatios: ± = 1 ( 2 ± b a ) (137) Note from eq (136) that the sig of V dictates whether + or " correspods to the symmetric or atisymmetric combiatio For egative V >> Δ, θ = π/4, ad the correspodece chages We ca schematically represet the eergies of these states with the followig diagram Here we explore the rage of ± available give a fixed value of the couplig V ad a varyig splittig Δ

9 1-9 This diagram illustrates a avoided crossig effect The strog couplig limit is equivalet to a degeeracy poit (Δ~0) betwee the states a ad b The eigestates completely mix the uperturbed states, yet remai split by the stregth of iteractio 2V Such a avoided crossig is observed where two weakly iteractig potetial eergy surfaces cross with oe aother at a particular uclear displacemet The time-evolutio of this system is give by our time-evolutio operator U ( t, t 0 ) = + e "i + ( t "t 0 ) + + " e "i " ( t "t 0 ) " (138) where ± = " ± Sice a ad b are ot the eigestates, preparig the system i state a will lead to time-evolutio Let s prepare the system so that it is iitially i a What is the probability that it is foud i state b at time t? ( t 0 = 0) ( 0) = " a (139) P ba ( t) = b " t 2 2 = b U ( t, t 0 ) a (140)

10 1-10 To evaluate this, you eed to kow the trasformatio from the a b basis to the ± basis, give i eq (136) This leads to P ba V ( t) 2 = V si2 " R t (141) R = 1 "2 + V 2 (142) R, the Rabi Frequecy, represets the frequecy at which probability amplitude oscillates betwee a ad b states Notice for V 0, ± " a,b (the statioary states), ad there is o time-depedece For V >>, the R = V ad P = 1 aftert = 2" R = 2V

11 1-11 Quatities we ofte calculate Correlatio amplitude: Measures the resemblace betwee the state of your system at time t ad a target state : C( t) = " ( t) (143) The probability amplitude P( t) = C( t) 2 If you express the iitial state of your wavefuctio i your eigebasis 0 = c " (144) C( t) = U ( t,t 0 ) " ( 0) = c m * m,, j * = c m c e i t m j e i j t j c (145) Here c m are the coefficiets that project your target wavefuctio oto your eigebasis Note m = " m Expectatio values give the time-depedet average value of a operator Physical, ad observables correspod to the expectatio values of Hermetia operators  =  therefore must be real Expectatio values of operators are give by For a iitial state 0 = ( t)  ( t ) = ( 0) U ˆ ( t,0)  Uˆ ( t,0) ( 0)  t = c " (146)

12 1-12 = Uˆ ( t,0) ( 0) = e "i t c t ( t) = ( 0) U ˆ ( t,0) = e i m t * c m = c t m * = c m m m ( t) m (147) = c c m * e i" m t m  m,  t * = c ( t)c m ( t) A m (148) m, m = E " E m = " m (149) Note that for a Hermetia operator eq (148) is real Readigs The material i this sectio draws from the followig: 1 Cohe-Taoudji, C; Diu, B; Lalöe, F Quatum Mechaics (Wiley-Itersciece, Paris, 1977) pp Mukamel, S Priciples of Noliear Optical Spectroscopy (Oxford Uiversity Press: New York, 1995) Ch2 3 Liboff, R L Itroductory Quatum Mechaics (Addiso-Wesley, Readig, MA, 1980) p 77 4 Sakurai, J J Moder Quatum Mechaics, Revised Editio (Addiso-Wesley, Readig, MA, 1994)