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pset3 - University of Central Florida Department of Physics...

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University of Central Florida Department of Physics Fall 2009 PHY 4604 Problem Set # 3 Due: October 2, 2009 There are still some problems from Chapter 3. The book gives the answer to the most interesting problems from Chap. 4. Study them carefully. The list below contains other problems which are to be handed in for grading. 1. Liboff , 3.15 2. Liboff , 3.23 3. Liboff , 3.18 in a slightly modified (more logical) form: Assuming that ˆ H ( t ) ˆ H ( t ) = ˆ H ( t ) ˆ H ( t ) for any two times t and t , show that ψ ( r ,t ) = exp bracketleftbigg i ¯ h integraldisplay t 0 dt ˆ H ( t ) bracketrightbigg ψ ( r , 0) . (1) 4. When the Hamiltonian is time dependent but ˆ H ( t ) ˆ H ( t ) negationslash = ˆ H ( t ) ˆ H ( t ), the dynamics cannot be solved as in Eq. (1). Another strategy is the following. Assume that ψ ( r ,t ) = summationdisplay n a n ( t ) e iE n t/ ¯ h φ n ( r ) , where { E n } and { φ n ( r ) } are the eigenvalues and eigenfunctions of the Hamiltonian at t = 0, respectively. (a) Find a set of differential equations which the time-dependent coefficients { a n ( t ) } need to satisfy. Hint : Recall that if ˆ H (0) is Hermitian, than its set of eigenfunctions
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