Diagonalizing a matrix using perturbation theory.

[PsychoDash]

Homework Statement

Consider the following Hamiltonian.

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

Ho=\begin{pmatrix} 20 &amp; 0 &amp; 0 \\0 &amp; 20 &amp; 0 \\0 &amp; 0 &amp; 30 \end{pmatrix}<br />
H'=\begin{pmatrix} 0 &amp; 1 &amp; 0 \\1 &amp; 0 &amp; 2 \\0 &amp; 2 &amp; 0 \end{pmatrix}<br />

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}<br /> \langle 1 \lvert \hat{H} \rvert 1 \rangle &amp; \langle 1 \lvert \hat{H} \rvert 2 \rangle &amp; \langle 1 \lvert \hat{H} \rvert 3 \rangle \\<br /> \langle 2 \lvert \hat{H} \rvert 1 \rangle &amp; \langle 2 \lvert \hat{H} \rvert 2 \rangle &amp; \langle 2 \lvert \hat{H} \rvert 3 \rangle \\<br /> \langle 3 \lvert \hat{H} \rvert 1 \rangle &amp; \langle 3 \lvert \hat{H} \rvert 2 \rangle &amp; \langle 3 \lvert \hat{H} \rvert 3 \rangle<br /> \end{pmatrix} = \begin{pmatrix} 20 &amp; 1 &amp; 0 \\1 &amp; 20 &amp; 2 \\0 &amp; 2 &amp; 30 \end{pmatrix}<br />
Does that clear things up?

 

[PsychoDash]

How does that help diagonalize H?