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.

Read more about this topic:  Perturbation Theory

Famous quotes containing the words theory, degenerate and/or states:

    “We commonly say that the rich man can speak the truth, can afford honesty, can afford independence of opinion and action;—and that is the theory of nobility. But it is the rich man in a true sense, that is to say, not the man of large income and large expenditure, but solely the man whose outlay is less than his income and is steadily kept so.”
    —Ralph Waldo Emerson (1803–1882)

    “James Bond in his Sean Connery days ... was the first well-known bachelor on the American scene who was not a drifter or a degenerate and did not eat out of cans.”
    —Barbara Ehrenreich (b. 1941)

    “[Urging the national government] to eradicate local prejudices and mistaken rivalships to consolidate the affairs of the states into one harmonious interest.”
    —James Madison (1751–1836)