4 - 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. 28 Instructor: Prof. Andrei Tokmakoff THE RELATIONSHIP BETWEEN U(t,t 0 ) AND c n (t) For a time-dependent Hamiltonian, we can often partition H = H 0 + Vt ( ) H 0 : time-independent; () : time-dependent potential. We know the eigenkets and eigenvalues of H 0 : n n We describe the initial state of the system ( t = t 0 ) as a superposition of these eigenstates: ψ( t 0 ) = E n H 0 = c n n n t ψ For longer times , we would like to describe the evolution of in terms of an expansion in these kets: ( t ) = c n ( t n ) n The expansion coefficients c k are given by t c k = t k t ( ) = kU t , t 0 ( ) t 0 ( ) Alternatively we can express the expansion coefficients in terms of the interaction picture wavefunctions ( ) = k ( ) b k t I t (This notation follows Cohen-Tannoudji.) Notice ct k t ) = U U I t 0 ) k () = 0 it = e −ω k kU I ψ t 0 = e k bt k b k 2 = t t so that c k 2 . Also, b k 0 0 = c k ( ) . It is easy to calculate b k ( t ) and then add in the extra oscillatory term at the end.
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p. 29 Now, starting with ψ I i = = V I I t we can derive an equation of motion for b k b k ψ I () I = b n n i = = k V U I I t 0 t 0 t n = 1 = kV I n n inserting nn U I ψ I ( t 0 ) n n I n b ( t ) = n n i t −ω
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4 - MIT Department of Chemistry 5.74, Spring 2004:...

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