Diagonalizing a matrix using perturbation theory.

[PsychoDash]

Homework Statement

Consider the following Hamiltonian.

H=$\begin{pmatrix} 20 & 1 & 0 \\1 & 20 & 2 \\0 & 2 & 30 \end{pmatrix}$
Diagonalize this matrix using perturbation theory. Obtain eigenvectors (to first order) and eigenvalues (to second order).

Ho=$\begin{pmatrix} 20 & 0 & 0 \\0 & 20 & 0 \\0 & 0 & 30 \end{pmatrix}$
H'=$\begin{pmatrix} 0 & 1 & 0 \\1 & 0 & 2 \\0 & 2 & 0 \end{pmatrix}$

Homework Equations

The Attempt at a Solution

In general, diagonalizing a matrix involves finding its eigenvalues and then writing the eigenvalues on the diagonal with zeros elsewhere. Despite that, I'm just not sure how to approach this question.

 

[vela]

How do you calculate corrections in perturbation theory? What equations do you have? Do you have degenerate states?

It's really just a matter of figuring out which formulas you need to use and plugging everything in. You can get the needed matrix elements by inspection.

 

[PsychoDash]

That's what's throwing me off. There is no "physical system" as I have come to understand it. All I'm given is what I wrote above. I understand that to calculate perturbations in general, you use <Psi|H'|Psi>, but that gets me back to needing a wavefunction to operate on. All I have is this set of matrices.

 

[vela]

You don't need a wave function for the system. You need the eigenstates of the unperturbed Hamiltonian. Let's call those $\vert 1 \rangle$, $\vert 2 \rangle$, and $\vert 3 \rangle$. The matrix you've been given is the representation of $\hat{H}$ relative to that basis. In other words,
$$\begin{pmatrix} \langle 1 \lvert \hat{H} \rvert 1 \rangle & \langle 1 \lvert \hat{H} \rvert 2 \rangle & \langle 1 \lvert \hat{H} \rvert 3 \rangle \\ \langle 2 \lvert \hat{H} \rvert 1 \rangle & \langle 2 \lvert \hat{H} \rvert 2 \rangle & \langle 2 \lvert \hat{H} \rvert 3 \rangle \\ \langle 3 \lvert \hat{H} \rvert 1 \rangle & \langle 3 \lvert \hat{H} \rvert 2 \rangle & \langle 3 \lvert \hat{H} \rvert 3 \rangle \end{pmatrix} = \begin{pmatrix} 20 & 1 & 0 \\1 & 20 & 2 \\0 & 2 & 30 \end{pmatrix}$$
Does that clear things up?

 

[PsychoDash]

How does that help diagonalize H?