I am trying to calculate how $\langle r\rangle$ in the hydrogen atom evolves with time I am working on the Hydrogen atom and I was trying to calculate $\frac{d<r>}{dt}$ using $$\frac{d\langle r\rangle}{dt} = \frac{i}{\hbar} \langle [\hat{H} , \hat{r}]\rangle$$ Here $r = \sqrt{x^2 + y^2 + z^2}$ and $H = \frac{p^2}{2m} + V$ where $p^2 = -\hbar^2 \nabla^2 $. Now according to Ehrenfest's theorem  should behave classically and give me some equivalent of velocity, and indeed I do get something but it does't resemble velocity: $\frac{-\hbar^2}{2m} (2\nabla r \nabla f + f \nabla^2 r)$ 
where $f$ is a test function.
Steps:
$$\begin{align}
[H ,r]f &= \left[\frac{p^2}{2m} + V , r\right]f \\
 &= \frac{p^2(rf)}{2m} + Vrf - \frac{rp^2(f)}{2m} - rVf \\
 &= \frac{1}{2m}[p^2 ,r]f \\
 &= \frac{1}{2m}[-\hbar^2\nabla^2 , r]f \\
 &= \frac{-\hbar^2}{2m}[\nabla^2 , r]f \\
 &= \frac{-\hbar^2}{2m} \left( \nabla^2(rf) - r\nabla^2(f) \right) \\
 &= \frac{-\hbar^2}{2m} \left( \nabla r\nabla f + r\nabla^2f + \nabla f \nabla r + f\nabla^2 r - r\nabla^2f \right) \\
 &= \frac{-\hbar^2}{2m} \left( 2\nabla r \nabla f + f \nabla^2 r \right)
\end{align}$$
Am I doing something wrong?
 A: By using an arbitrary test function you haven't included any information about the state of the electron. This means you are considering all possible states of an electron in a coulomb potential, so that formula you derived is true for all bound states of the hydrogen atom and all the unbound states of an electron scattering off the proton. It is true in the classical limit, where it should give you back the Kepler orbits, but it also true for states which are behaving distinctly quantum mechanically, notably the eigenstates for the hydrogen atom, which are stationary.
Given that the expression is so general, its not that surprising that you get a result that doesn't immediately resemble the classical result. Normally to take the classical limit of a system you have to think a bit about what is physically happening as you take that limit. In this case you would probably need to consider a superposition of a large number of very high $n$ states, i.e. states with a large energy and a large uncertainty in the energy. (You would probably need similar conditions on $l$ as well) 
