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Unformatted text preview: Physics 137B, Fall 2007 Quantum Mechanics II Midterm II solutions 1. Consider a spin-half system with H = H + H , H = (2 B / h ) S z , H = 2 CS x , where C is a nonzero constant and much less than B / h . Suppose that at t = 0 the system is in the spin-down state, i.e., the ground state of H . It may help to use the representations of spin-half operators S z = h 2 1- 1 , S x = h 2 0 1 1 0 . (1) (a) Write an expression for the transition probability from spin-up to spin-down as a function of time. Hint: you should find some sort of oscillatory behavior. We start from the formula P ba ( t ) = 2 | H ba ( t ) | 2 h 2 F ( t, ba ) . (2) Here ba is ( E - E ) / h = 2 B / h . All that remains is to compute H ba . Using the supplied matrices, H is found to be C h . (b) Find the exact eigenvalues of the full two-dimensional Hamiltonian for this two-state system by solving a quadratic equation. The determinant equation for the eigenvalues of H is ( - B )( + B ) + C 2 h 2 = 0 2 = C 2 h 2 + 2 B 2 . (3) Hence the energy eigenvalues are = q C 2 h 2 + 2 B 2 . (4) (c) Is the initial state | i an eigenstate of the full Hamiltonian? A simple yes or no is fine; you do not need to find the actual eigenstates. No, since acting on it with the Hamiltonian gives a vector that has a nonzero up component....
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This note was uploaded on 08/01/2008 for the course PHYSICS 137B taught by Professor Moore during the Fall '07 term at University of California, Berkeley.
- Fall '07