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Feb18-13, 02:27 AM
Sci Advisor
P: 2,552
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 harmonic-oscillator 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 right-hand side is not the same as [itex]-\vec{\nabla} V(\langle x \rangle)[/itex]!