Uniqueness of simultaneous eigenstates of two linear operators

Question

I was solving a homework problem where the question gives the representation of two operators in matrix form, in some arbitrary set of basis vectors. It then asks to find the simultaneous eigenstates of the two operators (they commute). Finally, it asks if specifying the eigenvalues uniquely specifies the eigenstates. I first found the eigenvalues of each operator by writing out the characteristic equation for each matrix - I assume the eigenvalues should not change when you change basis set. The spectrum of each operator has degeneracies. Then, I wrote down constraints on each each component of the eigenvector and guessed a set of simultaneous eigen states. From the constraints, I showed that there exists a unique choice of simultaneous eigen states for these two operators.

My question is are there situations in which the choice of simultaneous eigenstates is not unique. Or must they never be unique - in which case my conclusion was wrong?

Answer 1

Yes. A very simple examples would be $L^2$ and $L_z^2$. The $\vert \ell m\rangle$ states $\vert 1,1\rangle$ and $\vert 1,-1\rangle$ are both eigenvectors of the two operators, but so is the arbitrary combination $$ \vert\psi(\theta,\phi)\rangle =\cos\theta \vert 1,1\rangle + e^{i\phi}\sin\theta \vert 1,-1\rangle $$ given in terms of the continuous parameters $\theta$ and $\phi$.