Relationship between good quantum numbers and degenerate perturbation theory

I'm currently studying time-independent perturbation theory and I ran into a couple of doubts that for the life of me I can't seem to solve. A "good quantum number" is the eigenvalue of an eigenvector of an operator O that remains an eigenvector of O with the same eigenvalue as time evolves, according to Wikipedia: https://en.wikipedia.org/wiki/Good_quantum_number. There's a proof in the link above that shows that this is equivalent to having your operator O commute with the hamiltonian H of the problem.

However, I don't understand how these good quantum numbers are related to degenerate perturbation theory. When there is degeneracy, you don't know which states to use in the equations derived in the nondegenerate case, because any linear combination of the degenerate eigenfunctions is also an eigenfunction with the same eigenvalue. Plus, using the nondegenerate equations when there is degeneracy would involve division by zero. How are these two problems solved by the usage of good quantum numbers?

And what exactly all of this have to do with diagonalizing the perturbation matrix?

Thanks in advance for any help!

when there is a degeneracy, this means that we have several eigenvectors $$v_1, ... v_k$$ with same eigenvalue for the original hamiltonian (i.e. energy), let $$\mathbf{V}$$ be the subspace spanned by them, it can be proven that the pertuperation $$\delta H$$ has to be diagonal in this subspace, for that we need to find any set of basis that spans all of $$\mathbf{V}$$ such that $$\delta H$$ is diagnonal, this doesn't change the physics because the new basis are still eigenvectors of the original hamiltonian with the same eigenvalue, after that several cases can happen which I won't go in detail into instead I refer you to MIT's 8.06 notes starting from page 15 which goes into detail of first and second order pertuperation theory.
• I think the reference to time here is simply in the sense that some operator commutes with $H$. Commented Dec 10, 2021 at 2:28