Are newton's laws also an approximation?
Ehrenfest's theorem is not saying that the average values obey Newton's laws, which is wrong! This is only the case for the motion in a harmonicoscillator potential or in a constant force field. Ehrenfest's theorem says
[tex]\frac{\mathrm{d}}{\mathrm{d} t} \langle A \rangle =\frac{1}{\mathrm{i} \hbar}\langle [\hat{A},\hat{H}] \rangle,[/tex]
where [itex]A[/itex] is a not explicitly time dependent observable. For momentum you find
[tex]\frac{\mathrm{d}}{\mathrm{d} t} \langle \vec{p} \rangle =\langle \vec{\nabla} V(\hat{\vec{x}}) \rangle.[/tex]
Except for a constant force or a force that is linear in [itex]\vec{x}[/itex] the expectation value on the righthand side is not the same as [itex]\vec{\nabla} V(\langle x \rangle)[/itex]!
