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# p137bp3 - Physics 137B Fall 2007 Problem set 3 more...

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Physics 137B, Fall 2007 Problem set 3 : more perturbation theory; review time dependence and identical particles Assigned Friday, 14 September. Due Friday, 21 September. 1. Consider a spin s = 1 / 2 that starts off at t = 0 in the eigenstate of S z with eigenvalue ¯ h/ 2. In our usual basis, this eigenstate can represented by the spinor (1 , 0). (a) Suppose that the Hamiltonian is H = BS z (1) where B is some constant with units of (energy/angular momentum). How does this state evolve in time? (b) Now suppose that the Hamiltonian is H = BS x (2) where again B is some constant. Is the initial state (1 , 0) an eigenstate of this Hamiltonian? How does this initial state evolve in time? Give the value S z as a function of time. You may wish to use the matrix representation of S from the last problem set. 2. Consider the nonrelativistic infinite square-well potential, with V = 0 between x = 0 and x = L , and V = elsewhere. Calculate the first- and second-order energy shifts for all eigenstates from the potential shift H = ( x - L/ 2) . (3) Some energy shifts may be zero.
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