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
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:
“It is not enough for theory to describe and analyse, it must itself be an event in the universe it describes. In order to do this theory must partake of and become the acceleration of this logic. It must tear itself from all referents and take pride only in the future. Theory must operate on time at the cost of a deliberate distortion of present reality.”
—Jean Baudrillard (b. 1929)
“In this distribution of functions, the scholar is the delegated intellect. In the right state, he is, Man Thinking. In the degenerate state, when the victim of society, he tends to become a mere thinker, or, still worse, the parrot of other mens thinking.”
—Ralph Waldo Emerson (18031882)
“During the first World War women in the United States had a chance to try their capacities in wider fields of executive leadership in industry. Must we always wait for war to give us opportunity? And must the pendulum always swing back in the busy world of work and workers during times of peace?”
—Mary Barnett Gilson (1877?)