Understanding Degenerate States: Rules & Equations

[Void123]

This might seem like a foolish query, but I'm having a hard time understanding the rules of degenerate states. I know that it describes how different quantum numbers lead to equal energy values, but I'm not sure how you exactly determine that. Is it just trial and error, thinking about different possible combinations, mentally? Or is there an actual equation? For instance, I know there is a formula for the isotrophic harmonic oscillator $$g_{n}$$, but how does this vary for other phenomena? I've seen it in the forms $$n^{2} + 1$$, $$n^{2}$$, etc. Which one applies to which?

 

[tom.stoer]

It's an eigenvalue equation. The only difference is that in qm it is usually an equation in an infinite dimensional Hilbert space.

So you have to solve an equation like

(H-E)|E> = 0

If you find degenerate eigenvalues all you have to do is to determine the dimension of the eigenspace for a certain E and a set of vectors spanning exactly this subspace.

 

[diazona]

Is it just trial and error, thinking about different possible combinations, mentally?

In general, yes, it's just trial and error. Only for specific cases can you find an equation or formula to describe the degeneracy of an operator's spectrum.