34 - Physics 6210/Spring 2007/Lecture 34 Lecture 34...

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Unformatted text preview: Physics 6210/Spring 2007/Lecture 34 Lecture 34 Relevant sections in text: 5.6 Time-dependent perturbation theory (cont.) We are constructing an approximation scheme for solving i h d dt c n ( t ) = X m e i h ( E n- E m ) t V nm ( t ) c m ( t ) , V nm = h n | V ( t ) | m i . For simplicity we shall suppose (as is often the case) that the initial state is an eigen- state of H . Setting | (0) i = | i i , i.e., taking the initial state to be one of the unperturbed energy eigenvectors, we get as our zeroth-order approximation: c n ( t ) c (0) n = ni . We get our first-order approximation c (1) n ( t ) by improving our approximation of the right-hand side of the differential equations to be accurate to first order. This we do by using the zeroth order approximation of the solution, c (0) n in the right hand side of the equation. So, we now approximate the differential equation as i h d dt c n ( t ) e i h ( E n- E i ) t V ni ( t ) , with solution c n ( t ) c (0) n + c (1) n ( t ) = ni + 1 i h Z t dt e i h ( E n- E i ) t V ni ( t ) . Successive approximations are obtained by iterating this procedure. We shall only deal with the first non-trivial approximation, which defines first-order time-dependent perturbation theory . We assumed that the system started off in the (formerly stationary) state | i i defined by H . Of course, generally the perturbation will be such that | i i is not a stationary state for H , so that at times t > 0 the state vector will change. We can still ask what is the probability for finding the system in an eigenstate...
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34 - Physics 6210/Spring 2007/Lecture 34 Lecture 34...

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