I've been looking at some basic quantum mechanics all day in an attempt to better my understanding of the subject. While going over the proof that commuting operators are compatible, I started getting questions relating to complete sets of commuting observables (CSCO's).
I apologize for the fact that these questions might be trivial, it's been a long day.
Suppose $A$ and $B$ are two observables that commute.
- Do $A$ and $B$ have the same amount of eigenvalues?
- Do $A$ and $B$ have the same amount of eigenvectors?
- What do the previous answers mean for the cardinality of the sets (of eigenvectors of $A$, $B$ and $AB$)?
Basically, I'm wondering about dimensions and sizes of spaces and sets in the case where two observables commute.