2.3 SCHRÖDINGER AND HEISENBERG REPRESENTATIONS

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1 Andrei Tokmakoff, MIT Deparmen of Chemisry, 2/22/ SCHRÖDINGER AND HEISENBERG REPRESENTATIONS The mahemaical formulaion of he dynamics of a quanum sysem is no unique. So far we have described he dynamics by propagaing he wavefuncion, which encodes probabiliy densiies. This is known as he Schrödinger represenaion of quanum mechanics. Ulimaely, since we can measure a wavefuncion, we are ineresed in observables probabiliy ampliudes associaed wih Hermeian operaors). Looking a a ime-evolving expecaion value suggess an alernae inerpreaion of he quanum observable: = Â ) = 0) U ÂU 0) = 0) U ) Â U 0 ) ) = 0) U ÂU Â ) The las wo expressions here sugges alernae ransformaion ha can describe he dynamics. These have differen physical inerpreaions: 1) Transform he eigenvecors: U. Leave operaors unchanged. 2) Transform he operaors: Â ) U ÂU. Leave eigenvecors unchanged. 1) Schrödinger Picure: Everyhing we have done so far. Operaors are saionary.. Eigenvecors evolve underu, 0 2) Heisenberg Picure: Use uniary propery of U o ransform operaors so hey evolve in ime. The wavefuncion is saionary. This is a physically appealing picure, because paricles move here is a ime-dependence o posiion and momenum. Le s look a ime-evoluion in hese wo picures: Schrödinger Picure We have alked abou he ime-developmen of, which is governed by i = H 2.66)

2 2-18 in differenial form, or alernaively = U, 0 ) 0 ) in an inegral form. In he Schrödinger picure, for operaors ypically A values of operaors Â) =  ) : = 0. Wha abou observables? For expecaion i Â) = i %  % +  +  = ) H  2.67) = Â, H = Â, H Alernaively, wrien for he densiy marix: i Â) = i Tr  = itr  % = Tr Â)* H, +, = Tr ) * Â, H +, 2.68) If  is independen of ime as we expec in he Schrödinger picure) and if i commues wih H, i is referred o as a consan of moion. Heisenberg Picure From eq. 2.65) we can disinguish he Schrödinger picure from Heisenberg operaors: Â) =  ) = 0 S U ÂU 0 ) =  S H 2.69) where he operaor is defined as 0 = U, 0 = ÂS ÂS U, 0 ) 2.70)

3 2-19 Also, since he wavefuncion should be ime-independen H = 0, we can relae he Schrödinger and Heisenberg wavefuncions as S = U, 0 ) H 2.71) So, H = U, 0 ) S = S 0 ) 2.72) In eiher picure he eigenvalues are preserved:  i S = a i i S U ÂUU i S = a i U i S 2.73) i H = a i i H The ime-evoluion of he operaors in he Heisenberg picure is: ÂH = U  S U ) = U  S U + U  S U = i U H ÂS U i U  S H U +  % H + U ÂS U 2.74) = i H H i  H H H = i ) * Â, H +, H The resul: i = Â, H % 2.75) H is known as he Heisenberg equaion of moion. Here I have wrien H H = U H U. Generally speaking, for a ime-independen Hamilonian U = e ih /, U and H commue, and H H = H. For a ime-dependen Hamilonian, U and H need no commue. Paricle in a poenial

4 2-20 Ofen we wan o describe he equaions of moion for paricles wih an arbirary poenial: For which he Heisenberg equaion gives: H = p2 + V x) 2.76) 2m p = V x. 2.77) x = p m 2.78) Here, I ve made use of ˆx n, ˆp = inˆxn%1 2.79) ˆx, ˆp n = inˆpn%1 2.80) These equaions indicae ha he posiion and momenum operaors follow equaions of moion idenical o he classical variables in Hamilon s equaions. These do no involve facors of. Noe here ha if we inegrae eq. 2.78) over a ime period we find: x = p m + x 0 ) 2.81) implying ha he expecaion value for he posiion of he paricle follows he classical moion. These equaions also hold for he expecaion values for he posiion and momenum operaors Ehrenfes s Theorem) and indicae he naure of he classical correspondence. In correspondence o Newon s equaion, we see m 2 x 2 = V 2.82)

5 2-21 THE INTERACTION PICTURE The ineracion picure is a hybrid represenaion ha is useful in solving problems wih imedependen Hamilonians in which we can pariion he Hamilonian as H = H 0 + V 2.83) H 0 is a Hamilonian for he degrees of freedom we are ineresed in, which we rea exacly, and can be alhough for us generally won be) a funcion of ime. V is a ime-dependen poenial which can be complicaed. In he ineracion picure we will rea each par of he Hamilonian in a differen represenaion. We will use he eigensaes of H 0 as a basis se o describe he dynamics induced by V, assuming ha V is small enough ha eigensaes of H 0 are a useful basis o describe H. If H 0 is no a funcion of ime, hen here is simple ime-dependence o his par of he Hamilonian, ha we may be able o accoun for easily. Seing V o zero, we can see ha he ime evoluion of he exac par of he Hamilonian H 0 is described by where, mos generally, U 0, 0 U, 0 0 ) = i H 0 U 0, 0 ) 2.84) i = exp + % 0 d H ) bu for a ime-independen H 0 U 0, 0 ) = e ih 0 0 ) 2.86) We define a wavefuncion in he ineracion picure I hrough: S U 0, 0 ) I 2.87) or I = U 0 S 2.88) Effecively he ineracion represenaion defines wavefuncions in such a way ha he phase accumulaed under e ih 0 is removed. For small V, hese are ypically high frequency oscillaions relaive o he slower ampliude changes in coherences induced by V.

6 2-22 We are afer an equaion of moion ha describes he ime-evoluion of he ineracion picure wave-funcions. We begin by subsiuing eq. 2.87) ino he TDSE: i S = H S 2.89) U, 0 0 ) I = i H ) U 0, 0 ) I U 0 I + U 0 I ) U 0, 0` ) I = i H + V ) i H U + U I 0 0 I 0 ) U 0 I = i H + V 0 i I = V I I 2.91) where V I = U 0, 0 ) V U 0, 0 ) 2.92) I saisfies he Schrödinger equaion wih a new Hamilonian: he ineracion picure Hamilonian, V I, which is he U 0 uniary ransformaion ofv. Noe: Marix elemens in V I = k V I l = e i lk V kl where k and l are eigensaes of H 0. We can now define a ime-evoluion operaor in he ineracion picure: where U I, 0 I = U I, 0 ) I 0 ) 2.93) % = exp + i d V I 0 ). 2.94) Now we see ha S = U 0, 0 ) I = U 0, 0 )U I, 0 ) I 0 ) = U 0, 0 )U I, 0 ) S 0 ) 2.95)

7 2-23 U, 0 ) = U 0, 0 )U I, 0 ) 2.96) Using he ime ordered exponenial in eq. 2.94), U can be wrien as U, 0 ) = U 0, 0 ) + ) n=1 i % U 0 * 2,* 1 n + 0 V * 1 * n * 2 d* n d* n1 d* 1 U 0,* n )V * n )U 0 * n,* n1 ) + 0 U 0 * 1, ) where we have used he composiion propery ofu, 0 ). The same posiive ime-ordering applies. Noe ha he ineracions Vτ i ) are no in he ineracion represenaion here. Raher we used he definiion in eq. 2.92) and colleced erms. For ransiions beween wo eigensaes of H 0 l and k): The sysem evolves in eigensaes of H 0 during he differen ime periods, wih he ime-dependen ineracions V driving he ransiions beween hese saes. The ime-ordered exponenial accouns for all possible inermediae pahways. Also, he ime evoluion of conjugae wavefuncion in he ineracion picure is expressed as % +i U, 0 ) = U I, 0 ) U 0, 0 ) = exp d V I 0 ) exp % or U 0 = e ih 0 ) when H 0 is independen of ime. The expecaion value of an operaor is: +i d H 0 0 ) 2.98) = Â ) = 0 ) U, 0 ) ÂU, 0 ) 0 ) = 0 ) U I U 0 ÂU 0 U I 0 ) = I ÂI I Â 2.99)

8 2-24 where A I U 0 A S U ) Differeniaing A I gives: Â = i I H, 0 ÂI % 2.101) also, = i I V I I 2.102) Noice ha he ineracion represenaion is a pariion beween he Schrödinger and Heisenberg represenaions. Wavefuncions evolve underv I, while operaors evolve under H 0. For H 0 = 0, V = H Â = 0; = i S H Schrödinger S For H 0 = H, V = 0 Â % = 0 Heisenberg = i H, Â ; 2.103)

9 2-25 The relaionship beween U I, 0 ) and b n ) For problems in which we pariion a ime-dependen Hamilonian, H = H 0 + V 2.104) H 0 is he ime-independen exac zero-order Hamilonian and V is a ime-dependen poenial. We know he eigenkes and eigenvalues of H 0 : H 0 n = E n n 2.105) and we can describe he sae of he sysem as a superposiion of hese eigensaes: = c n n The expansion coefficiens c k are given by c k = k 2.106) n = k U, 0 ) 0 ) 2.107) Alernaively we can express he expansion coefficiens in erms of he ineracion picure wavefuncions b k = k I = k U I 0 ) 2.108) This noaion follows Cohen-Tannoudji.) Noice c k = k U 0 U I 0 = e i k k U I 0 = e i k b k 2.109) This is he same ideniy we used earlier o derive he coupled differenial equaions ha describe he change in he ime-evolving ampliude of he eigensaes: i b k e i nk b n = V kn 2.110) n So, b k is he expansion coefficien of he ineracion picure wavefuncions. Remember b k 2 = c k 2 and b k 0) = c k 0). If necessary we can calculae b k and hen add in he exra oscillaory erm a he end.