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How to Find the Eigenvalues and Eigenvectors for Degenerate Hamiltonians

Using quantum physics, you can determine the f eigenvalues and matching eigenvectors for systems in which the energies are degenerate. Take a look at this unperturbed Hamiltonian:

image0.png

In other words, several states have the same energy. Say the energy states are f-fold degenerate, like this:

image1.png

How does this affect the perturbation picture? The complete Hamiltonian, H, is made up of the original, unperturbed Hamiltonian, H0, and the perturbation Hamiltonian,

image2.png

In zeroth-order approximation, you can write the eigenfunction

image3.png

as a combination of the degenerate states

image4.png

Note that in what follows, you assume that

image5.png

if m is not equal to n. Also, you assume that the

image6.png

are normalized — that is,

image7.png

Plugging this zeroth-order equation into the complete Hamiltonian equation, you get

image8.png

Now multiplying that equation by

image9.png

gives you

image10.png

Using the fact that

image11.png

if m is not equal to n gives you

image12.png

Physicists often write that equation as

image13.png

where

image14.png

And people also write that equation as

image15.png

where E(1)n = En – E(0)n. That's a system of linear equations, and the solution exists only when the determinant to this array is nonvanishing:

image16.png

The determinant of this array is an fth degree equation in E(1)n, and it has f different roots,

image17.png

Those f different roots are the first-order corrections to the Hamiltonian. Usually, those roots are different because of the applied perturbation. In other words, the perturbation typically gets rid of the degeneracy.

So here's the way you find the eigenvalues to the first order — you set up an f-by-f matrix of the perturbation Hamiltonian,

image18.png

Then diagonalize this matrix and determine the f eigenvalues

image19.png

and the matching eigenvectors:

image20.png

Then you get the energy eigenvalues to first order this way:

image21.png

And the eigenvectors are

image22.png
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