I have the Ehrenfest relation $\frac{d}{dt}\langle\mathbf{p}\rangle=-\langle\nabla V(\mathbf{r})\rangle$. I need to write this in the case of the harmonic oscillator potential $V(\mathbf{r})=\frac{1}{2}m\omega^2\mathbf{r}^2$. From this, how would I solve this using a coupled system of differential equations for $\langle\mathbf{r}\rangle$ and $\langle\mathbf{p}\rangle$ as functions of time?
I tried using the original Ehrenfest relation $\frac{d}{dt}\langle\mathbf{r}\rangle= \frac{1}{m}\langle\mathbf{p}\rangle$ but got nowhere.
The two equations you are looking for are simply: \begin{align*} \frac{d}{dt}\langle\mathbf{p}\rangle = -\langle \nabla V(\mathbf r)\rangle = -\langle \nabla \frac{1}{2}m\omega^2 r^2\rangle = \frac{-1}{2}m\omega^2 \langle \nabla (x^2+y^2+z^2)\rangle = -m\omega^2\langle \mathbf{r} \rangle \end{align*} and \begin{align*} \frac{d}{dt}\langle\mathbf{r}\rangle= \frac{1}{m}\langle\mathbf{p}\rangle \end{align*}