- #1

goodphy

- 216

- 8

I read the textbook and found that common eigenfunctions are even possible for degenerate eigenvalues.

Let's say operators

*A*and

*B*commutes and eigenvalue

*a*of operator

*A*is N-fold degenerate, means that there are N linearly independent eigenfunctions having same eigenvalue

*a*. These eigenfunctions are not necessarily common eigenfunctions of operator

*B*, unlikely to non-degenerate eigenfunctions that are always common for commuting operators. However, the textbook said that we can always find a set of new N linearly independent eigenfunctions of eigenvalue

*a*that are common for operator B by linear combination of old eigenfunctions of

*a*. It is really fantastic theorem but I couldn't find its proof.

Could you please give me the proof of this? If I fully understand the proof, then I will be really confident that commuting operators have common eigenfunctions for all their eigenvalues, even some eigenvalues are degenerate.