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lecture07_2011 - 19Jan2011 Chemistry 21b Spectroscopy...

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19Jan2011 Chemistry 21b – Spectroscopy Lecture # 7 – Time-Dependent Perturbation Theory & Light-Matter Interactions In Ch 21a you considered the effects of small, time independent perturbations to various systems (see also the supplemental notes from Week #1). Spectroscopy, of course, is concerned with the interaction of matter with electromagnetic radiation, which is inherently a time-dependent process. The general procedure we will follow is similar, but here the question becomes “What is the probability that, some time t after the perturbation is applied, the system has undergone a transition from an initial state i to a final state f ?” The problem we wish to solve has the total Hamiltonian ˆ H ( r ,t ) = ˆ H 0 ( r ) + λ ˆ H ( r ,t ) , where λ again is a parameter of smallness (perturbation theory notes). Note that we have assumed the zeroth-order Hamiltonian ˆ H 0 is time-independent. In that case, we know that the time-dependent eigenfunctions of ˆ H 0 may be written as ψ n ( r ,t ) = ϕ n ( r ) e n t ˆ H 0 ϕ n = E (0) n ϕ n ¯ n ϕ n . Let us suppose that at time t> 0 the system is in a state characterized by ψ ( r ,t ) = summationdisplay n c n ( t ) ψ n ( r ,t ) . (7 . 1) The superposition principle tells us that the probability that the system is in the state ψ n at time t is simply given by | c n ( t ) | 2 . Thus, to answer the question above we must find the { c n ( t ) } coefficients. To obtain them we proceed much as in the time-independent case. Clearly, the total wavefunction ψ ( r ,t ) must satisfy ı ¯ h ∂ψ ∂t = ( ˆ H 0 + λ ˆ H ) ψ . If we insert Eq. (7.1) into this expression, and as before operate from the left with f ( r ,t ) | , we are left with ı ¯ h dc f dt = λ summationdisplay n <f | H | n>c n . (7 . 2) In general, this is an infinite series of coupled equations for the coefficients { c f ( t ) } . As λ 0, the time-dependent perturbation becomes negligible and the c f coefficients remain constant. Thus, as for the time-dependent case let us seek a solution in a power series of corrections to the initial constants, or c f ( t ) = c (0) f + λc (1) f ( t ) + λ 2 c (2) f ( t ) + ... 57
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Inserting this series into Eq. (7.2) and collecting like terms in λ results in the following corrections to the time-dependent coefficients [where the dot denotes a time derivative and the H nf are the matrix elements of ˆ H ( r ,t )]: ı ¯ h ˙ c (0) f = 0 ı ¯ h ˙ c (1) f = summationdisplay n H fn c (0) n ı ¯ h ˙ c (2) f = summationdisplay n H fn c (1) n . . . ı ¯ h ˙ c ( s +1) f = summationdisplay n H fn c ( s ) n (7 . 3) So, to get the corrections to order ( s + 1), we must know all of the coefficients to order s . The first equation simply states that the initial coefficients are constant in time until the perturbation is applied. These are complex, long equations, and so at this point in the analysis it is common to make a simplifying assumption, namely that the system starts in a definite eigenstate of ˆ H 0 , call it ψ i ( r ,t ). If we use the initial time as t = −∞ , we have ψ ( r ,t ) ψ i ( r ,t ) = summationdisplay n δ ni ψ n ( r ,t ) c (0) n = δ ni Substituting these expressions into the first-order part of Eq. (7.3) gives
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