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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.
Ask a homework question - tutors are online