- #1
foxjwill
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Homework Statement
Theorem 5 of a text I've been reading that I downloaded from online (for interested parties, the link (a pdf) is http://bohr.physics.berkeley.edu/classes/221/0708/notes/hilbert.pdf) says that
"If two observables A and B commute, [A, B] = 0, then any nondegenerate eigenket of A is also an eigenket of B. (It may be a degenerate eigenket of B.)"
What I don't understand is how the eigenket could be degenerate in B. I am assuming, I hope correctly, that by "degenerate eigenket" the author means "an eigenket in a degenerate eigenspace". My line of thought is as follows:
Since [A,B]=0, A and B have the same eigenbasis and therefore the same eigenspaces. But then because the degeneracy of an eigenspace is defined to be its dimension, any degenerate eigenspace of A must also be a degenerate eigenspace of B, and vice-versa.