While dealing with a circling particle in an spherical symetric potential our professor said that we can replace an operator of $z$ component of angular momentum $\hat{L}_z$ with the expectation value - he denoted it just $L_z$ - of the angular momentum if $L_z$ is constant. Why is that?
So we first had this equation:
\begin{align} \underbrace{\psi (r,\varphi,\vartheta)}_{\llap{ \text{wave function in spherical coordinates}}} &= \exp\left[\hat{L}_z \frac{i}{\hbar}\, \varphi\right] \underbrace{\psi (r,0,\vartheta)}_{\rlap{\text{wave function in spherical coordinates at $\varphi=0$}}} \end{align}
and we got this one (notice that there is no operator over an $L_z$):
\begin{align} \psi (r,\varphi,\vartheta) &= \exp\left[L_z \frac{i}{\hbar}\, \varphi\right] \psi (r,0,\vartheta) \end{align}
Anyway here is the spherical coordinate system we ve been using all the time (the blue spherical aure is supposed to be a spherical potential...):