Quantum Physics For Dummies
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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:

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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,

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In zeroth-order approximation, you can write the eigenfunction

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as a combination of the degenerate states

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Note that in what follows, you assume that

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

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are normalized — that is,

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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

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if m is not equal to n gives you

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Physicists often write that equation as

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where

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And people also write that equation as

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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:

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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

About This Article

This article is from the book:

About the book author:

Steven Holzner is an award-winning author of technical and science books (like Physics For Dummies and Differential Equations For Dummies). He graduated from MIT and did his PhD in physics at Cornell University, where he was on the teaching faculty for 10 years. He’s also been on the faculty of MIT. Steve also teaches corporate groups around the country.

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