I have a 2D wavefunction $\psi(\rho , \phi) = A \exp(-\rho^2/2)\cos ^2\phi$ from which I have to find the possible values of $L_z$. I have tried using the operator form of $L_z = \frac{\hbar}{i} \frac{\partial}{\partial \phi}$ but that is not very helpful.
The 2nd part is to find the probability of $L_z = -2 \hbar$ . For this is realize I need to do the inner-product of the m= -2 state with the original wavefunction but how do I find the $m=-2$ state?