lecture07_2011

Lecture07_2011 - 19Jan2011 Chemistry 21b – Spectroscopy Lecture 7 – Time-Dependent Perturbation Theory& Light-Matter Interactions In Ch 21a

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Unformatted text preview: 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 ( 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 is time-independent. In that case, we know that the time-dependent eigenfunctions of ˆ H may be written as ψ n ( r , t ) = ϕ n ( r ) e − iω n t ˆ H ϕ n = E (0) n ϕ n ≡ ¯ hω 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 + λ ˆ H ′ ) ψ . If we insert Eq. (7.1) into this expression, and as before operate from the left with < ψ 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 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 ....
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This note was uploaded on 01/03/2012 for the course CH 21b taught by Professor List during the Fall '10 term at Caltech.

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Lecture07_2011 - 19Jan2011 Chemistry 21b – Spectroscopy Lecture 7 – Time-Dependent Perturbation Theory& Light-Matter Interactions In Ch 21a

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