In quantum mechanics, Ehrenfest's theorem states that $\langle p_x\rangle = m \frac{d}{dt}\langle x\rangle$. My question is, does there exist a similar relationship between $\langle L_z\rangle$, the expectation value of the z-component of the orbital angular momentum operator, and the time derivative of $\langle\theta\rangle$, the expectation value of the position operator $\hat{\theta}$ in spherical coordinates?
If not, is there any way to relate $\langle L_z\rangle$ to the time derivatives of expectation values of one or more operators? For instance if you knew what $\langle x\rangle$, $\langle y\rangle$, $\langle z\rangle$, $\langle p_x\rangle$, $\langle p_y\rangle$, and $\langle p_z\rangle$ are as a function of time, would that give you enough information to determine $\langle L_z\rangle$?
And are the answers to these questions affected at all by whether the particle has spin or not? By the way, this question was inspired by the comment section of this answer.