Introduction to Quantum Mechanics · Problem 7.35

Time-Independent Perturbation Theory — David J. Griffiths

Physics — Quantum Mechanics · degenerate_perturbation_application

Problem

Here is an application of the technique developed in Problem 7.34. Consider the Hamiltonian

\mathsf{H}^0 = \begin{pmatrix} \epsilon & 0 & 0 \\ 0 & \epsilon & 0 \\ 0 & 0 & \epsilon' \end{pmatrix}, \quad \mathsf{H}' = \begin{pmatrix} 0 & a & b \\ a^* & 0 & c \\ b^* & c^* & 0 \end{pmatrix}.

(a) Find the projection operator

P_D

(it's a

3 \times 3

matrix) that projects onto the subspace spanned by

|\psi^0_a\rangle = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \quad \text{and} \quad |\psi^0_b\rangle = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}.

Then construct the matrices

\tilde{\mathsf{H}}^0

and

\tilde{\mathsf{H}}'

. (b) Solve for the eigenstates of

\tilde{\mathsf{H}}^0

and verify... i. that its spectrum is nondegenerate, ii. that the nondegenerate eigenstate of

\mathsf{H}^0

|\psi^0_c\rangle = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}

is also an eigenstate of

\tilde{\mathsf{H}}^0

with the same eigenvalue. (c) What are the "good" states, and what are their energies, to first order in the perturbation?