Physics 137B, Fall 2007Problem set 3: more perturbation theory; review time dependence and identical particlesAssigned Friday, 14 September. Due Friday, 21 September.1. Consider a spins= 1/2 that starts off att= 0 in the eigenstate ofSzwith eigenvalue ¯h/2. In our usualbasis, this eigenstate can represented by the spinor (1,0).(a) Suppose that the Hamiltonian isH=BSz(1)whereBis some constant with units of (energy/angular momentum). How does this state evolve in time?(b) Now suppose that the Hamiltonian isH=BSx(2)where againBis some constant. Is the initial state (1,0) an eigenstate of this Hamiltonian? How does thisinitial state evolve in time?Give the valueSzas a function of time.You may wish to use the matrixrepresentation ofSfrom the last problem set.2. Consider the nonrelativistic infinite square-well potential, withV= 0 betweenx= 0 andx=L, andV=∞elsewhere. Calculate the first- and second-order energy shifts for all eigenstates from the potentialshiftH=Aδ(x-L/2).(3)Some energy shifts may be zero.
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