5 - MIT Department of Chemistry 5.74, Spring 2004:...

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MIT Department of Chemistry 5.74, Spring 2004: Introductory Quantum Mechanics II p. 33 Instructor: Prof. Andrei Tokmakoff PERTURBATION THEORY Given a Hamiltonian Ht ( ) ( ) = H 0 + Vt where we know the eigenkets for H 0 n n = E n H 0 we often want to calculate changes in the amplitudes of n induced by ( ) : ψ t ( ) = c n n ( ) t where n () = k = ( ) t kU t , t 0 ( ) ( ) t 0 c k t In the interaction picture, we defined = e + i ω k r c k t ( ) = k ( ) b k t I which contains all the relevant dynamics. The changes in amplitude can be calculated by solving the coupled differential equations: i b k = i e −ω nk t V k n b n t t t = n For a complex system or a system with many states to be considered, solving these equations isn’t practical. Alternatively, we can choose to work directly with U I t , t 0 as: ( ) , and we can calculate b k t b k = kU I t , t 0 ( ) t 0 ( ) where Ut , t ) = e x p − i t V d I ( 0 + = I ττ t 0 Now we can truncate the expansion after a few terms. This is perturbation theory, where the dynamics under H 0 are treated exactly, but the influence of ) on b n is truncated. This works ( well for small changes in amplitude of the quantum states with small coupling matrix elements relative to the energy splittings involved. ; V ± E k E b k ( t ) b k ( 0 ) n
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p. 34 Transition Probability Let’s take the specific case where we have a system prepared in probability of observing the system in A , and we want to know the k at time t , due to Vt ) . ( 2 P k () = t bt k U I ( t , t 0 ) A k () k t i bt k τ τ e x p + d V I A k t 0 = i t = k A d τ k τ A V I = t 0  − i 2 t τ 2 + t 0 d τ 2 t 0 d τ 1 k A + V I V 1 ) τ 2 I = using i ω A k t V k A t kV I t A = kU 0 U 0 A = e i A k τ b k t “first order” = δ k A i t t 0 d 1 e 1 V k A 1 = t + i 2 t 0 d 2 t 0 V km 2 i A m 1 + “second order” 2 d 1 e i mk 2 e V m A 1 = m This expression is usually truncated at the appropriate order. Including only the first integral is first-order perturbation theory. If is not an eigenstate, we only need to express it as a superposition of eigenstates, but ψ 0 remember to convert to c k t = e k t b k ( t ) .
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5 - MIT Department of Chemistry 5.74, Spring 2004:...

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