Perturbation Theory - Perturbation Theory of Degenerate States

Perturbation Theory of Degenerate States

One may notice that the problem occurs in the first order perturbation theory when two or more eigenfunctions of the unperturbed system correspond to one eigenvalue i.e. when the eigenvalue equation becomes

.

and the index labels many states with the same eigenvalue . Expression for the eigenfunctions having the energy differences in the denominators becomes infinite. In that case the degenerate perturbation theory must be applied. The degeneracy must be removed first for higher order perturbation theory. The function is first assumed to be the linear combination of eigenfunctions with the same eigenvalue only

which again from the orthogonality of leads to the following equation

c_{n,i}\lambda^{(0)}_{n,i} + \epsilon \sum_k c_{n,k} \int f^{(0)}_{n,i}(x) D^{(1)} f^{(0)}_{n,k} (x)\,dx = \lambda c_{n,i}

for each . As for the majority of low quantum numbers the changes over small range of integers the later equation can be usually solved analytically as at most 4x4 matrix equation. Once the degeneracy is removed the first and any order of the perturbation theory may be further used with respect to the new functions.

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