852_2010hw6

# 852_2010hw6 - PHYS852 Quantum Mechanics II, Spring 2010...

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PHYS852 Quantum Mechanics II, Spring 2010 HOMEWORK ASSIGNMENT 6 Topics covered: Time-independent perturbation theory. 1. [30] Two-Level System : Consider the system described by H = δS z + Ω S x , with δ > 0, where S x and S z are components of the spin vector of an s = 1 / 2 particle. Treat the S z term as the bare Hamiltonian. (a) [15] Use perturbation theory to compute the eigenvalues and eigenvectors of H . Compute all terms up to fourth-order in Ω. (b) [5] Expand the exact eigenvalues and eigenvectors around Ω = 0 and compare to the perturbation theory results. (c) [10] Verify that the states computed in (a) are normalized to unity and orthogonal up to fourth- order. 2. [20] Resonance-frequency shifts : Consider a system with a 3-dimensional Hilbert space spanned by states | a i , | b i , and | c i . In the basis {| a i , | b i , | c i} , let the bare Hamiltonian of the system be H 0 = ~ Δ 1 - 1 1 - 1 1 - 1 1 - 1 3 . (1) For the case where the system is perturbed by the operator
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