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p137bnewmt2sol

p137bnewmt2sol - Physics 137B Fall 2007 Quantum Mechanics...

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Physics 137B, Fall 2007 Quantum Mechanics II Midterm II solutions 1. Consider a spin-half system with H = H 0 + H , H 0 = (2 μB 0 / ¯ h ) S z , H = 2 CS x , where C is a nonzero constant and much less than μB 0 / ¯ h . Suppose that at t = 0 the system is in the spin-down state, i.e., the ground state of H 0 . It may help to use the representations of spin-half operators S z = ¯ h 2 1 0 0 - 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, ω ba ) . (2) Here ω ba is ( E - E ) / ¯ h = 2 μB 0 / ¯ 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 ( λ - μB 0 )( λ + μB 0 ) + C 2 ¯ h 2 = 0 λ 2 = C 2 ¯ h 2 + μ 2 B 0 2 . (3) Hence the energy eigenvalues are λ = ± C 2 ¯ h 2 + μ 2 B 0 2 . (4) (c) Is the initial state | ↓ 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. (d) Show that the oscillation frequency expected from the exact solution is consistent with the oscillation frequency from perturbation theory in the limit where perturbation theory is valid.

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p137bnewmt2sol - Physics 137B Fall 2007 Quantum Mechanics...

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