Introduction to Quantum Mechanics · Problem 7.12

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · degenerate_perturbation_theory

Problem

Consider a quantum system with just three linearly independent states. Suppose the Hamiltonian, in matrix form, is H=V0((1−ϵ)0001ϵ0ϵ2),\mathsf{H} = V_0 \begin{pmatrix} (1 - \epsilon) & 0 & 0 \\ 0 & 1 & \epsilon \\ 0 & \epsilon & 2 \end{pmatrix}, where V0V_0 is a constant, and ϵ\epsilon is some small number (ϵ≪1\epsilon \ll 1). (a) Write down the eigenvectors and eigenvalues of the *unperturbed* Hamiltonian (ϵ=0\epsilon = 0). (b) Solve for the *exact* eigenvalues of H\mathsf{H}. Expand each of them as a power series in ϵ\epsilon, up to second order. (c) Use first- and second-order *non*-degenerate perturbation theory to find the approximate eigenvalue for the state that grows out of the nondegenerate eigenvector of H0\mathsf{H}^0. Compare the exact result, from (b). (d) Use degenerate perturbation theory to find the first-order correction to the two initially degenerate eigenvalues. Compare the exact results.

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