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Ehrenfest's theorem, as usually stated, is the following statement about quantum-mechanical operators: the derivative with respect to time of the expectation of an operator is 2*PI*i/h times the expectation of the commutator of the Hamiltonian and the operator, plus the expectation of the partial derivative of the operator with respect to time. It's a bit cumbersome when written out in words like that, but I can't really express the relevant mathematical notation in html. For this reason, I have omitted the proof of the theorem.

Now, although that is the usual statement of the theorem, the results that Ehrenfest originally published are, I belive, the special cases for the position and momentum operators. That is,

d<x>/dt = <p>/m and d<p>/dt = <-grad(V)>

where V is the potential. These correspond to the classical equations of motion

dx/dt = p/m andn dp/dt = -grad(V)

So the wave packet moves like a classical particle whenever the expectation values of the operators give a good representation of the classical variables; in particular, we recover the classical motion in the macroscopic limit in which the size and internal structure of the wavepacket may be neglected.

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