Given a system with Hamiltonian $ \hat{H} = \frac {\hat{p} ^2}{2m} + \hat{V}(r)$ in a certain state $|\psi \rangle$, I want to find if $\langle r \rangle$ varies with time.
From
$$ i \hbar\frac {d \langle r \rangle} {dt} = \langle [\hat{r},\hat{H}] \rangle$$
Since $[\hat{r}, \hat{V}(r)] = 0$, we have
$$[\hat{r},\hat{H}] = -\frac {\hbar ^2}{2m}[\hat{r},\nabla^2]$$
- What is the most efficient way to compute $[\hat{r},\nabla^2]$ ?
My approach:
Write $\nabla ^2$ in spherical coordinates, and since $\hat{r}$ commutes with the angular part what remains is (omitting the hats)
$$ \left[r, \frac {1}{r^2} \frac{\partial}{\partial r}\left( r^2\frac{\partial}{\partial r}\right)\right]$$
to be computed with the help of a test function.