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Unformatted text preview: Homework Solutions # 4 (Liboff Chapter 6) 6.7 Note that ( x, 0) is an eigenstate of H . (a) ( x, t ) = ( x, 0) e it where = E/ h = hk 2 / 2 m . (b) ( x, t ) = 1 2 i e ik x e ik x e it in terms of p eigenfunctions. Therefore, hk can be observed, with equal probability ( 1 2 ). (c) at t = 3 s, = e ik x . So ( x, t ) = e ik x e i ( t 3) (The 3 s part is irrelevant; just an overall phase.) 6.13 At t = 0, the state is  i , with A  i = a  i . For t > 0, using the time evolution operator U = exp( it h H ),  i = e it h H  i Then, since U = exp( it h H ) = U 1 , h A i = h  e it h H Ae it h H  i We have to move A to the other side of U , so we need the commutator [ U, A ]. Start with using H A = A H + c , and finding [ H n , A ] = H n A A H n = H n 1 A H + c H n 1 A H n = = nc H n 1 after interchanging...
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 Spring '09
 Mucciolo
 mechanics, Work

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