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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?

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Your $\psi(\rho,\phi)$ is not an eigenstate of $L_z$ (as you correctly guessed), but the eigenstates of $L_z$ are of the form $e^{im\phi}$ so try to expand $\psi(\rho,\phi)$ in terms of eigenstates of $L_z$ as $$ \psi(\rho,\phi)=A\exp(-\rho^2/2)\sum_{m}a_m e^{im\phi}\, . $$ With this you should be able to identify which eigenstates of $L_z$ occur and so answer your questions.

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