By Ehrenfest’s theorem one finds that the quantum operators $\hat{q}$ and $\hat{p}$ satisfy $$\frac{d \langle\hat{q}\rangle}{dt} = \frac{\langle\hat{p}\rangle}{m}$$ and $$\frac{d \langle\hat{p}\rangle}{dt} = \langle F(\hat{q})\rangle$$ Classically, by applying Liouvilles theorem the canonical coordinates $p$ and $q$ satisfy $$\frac{d \langle q \rangle}{dt} = \frac{\langle p \rangle}{m}$$ and $$\frac{d \langle p \rangle}{dt} = \langle F(q)\rangle$$
It seems to me that the relations yield identical predictions for the values of $q$ and $p$ if the potential is the harmonic oscillator, does that seem correct? Is this always the case?
