hmm, I see... this also brought up another question. How do we know which observable is a given operator corresponds to mathematically if all we have is a given basis, and the operator wrote under this basis representation, and also some quantum states supplied for the operators to act on
That is, suppose I have self adjoint operators $A$ and $B$ where all its components under a given basis is known, acting on some known quanutm states, is there a way to tell what observables they are giving me?