This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 19Jan2011 Chemistry 21b – Spectroscopy Lecture # 7 – TimeDependent Perturbation Theory & LightMatter 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 timedependent 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 zerothorder Hamiltonian ˆ H is timeindependent. In that case, we know that the timedependent 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 timeindependent 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 timedependent perturbation becomes negligible and the c f coefficients remain constant. Thus, as for the timedependent 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 timedependent 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 ....
View
Full
Document
This note was uploaded on 01/03/2012 for the course CH 21b taught by Professor List during the Fall '10 term at Caltech.
 Fall '10
 list
 Chemistry

Click to edit the document details