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.