View Single Post

Recognitions:
$$\frac{\mathrm{d}}{\mathrm{d} t} \langle A \rangle =\frac{1}{\mathrm{i} \hbar}\langle [\hat{A},\hat{H}] \rangle,$$
where $A$ is a not explicitly time dependent observable. For momentum you find
$$\frac{\mathrm{d}}{\mathrm{d} t} \langle \vec{p} \rangle =-\langle \vec{\nabla} V(\hat{\vec{x}}) \rangle.$$
Except for a constant force or a force that is linear in $\vec{x}$ the expectation value on the right-hand side is not the same as $-\vec{\nabla} V(\langle x \rangle)$!