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.