Time-Dependent Degenerate Pertubation Theory for 3x3 matrix

[jcharles513]

Homework Statement

H₀ = [2,0,0;0,2,0;0,0,4]
H₁ = [0,1,0;1,0,1;0,1,0]

Find energy eigenvalues to 2nd order.

Homework Equations

The Attempt at a Solution

I know that I need to diagonalize the perturbation in the 2x2 subspace (for my 2 degenerate eignevalues of 2 but I'm not sure how to diagonalize my perturbation in this subspace. As far as I can tell H1 is not block diagonal so I can't separate it. What am I missing? From there I think I can do the rest of the problem. Just stuck here.

 

[TSny]

Let $|u_1\rangle$ and $|u_2\rangle$ be the eigenvectors of H₀ that correspond to the degenerate eigenvalue.

Form the 2x2 matrix $V$ with elements $V_{11} = \langle u_1|H_1|u_1\rangle$, $V_{12} = \langle u_1|H_1|u_2\rangle$, etc.

That's the 2x2 matrix that you need to diagonalize.