The evolution of the mean value of an observable
In the Heisenberg picture we have an equation for the evolution of operators.
dAH/dt = (∂AS/∂t)H + (iħ)-1[AH,HH].
The corresponding equation in the Schroedinger picture describes the evolution of the mean value of an observable.
(d/dt)<Ψs(t)|As|Ψs(t)> = <ΨS(t)|∂AS/∂t|ΨS(t)> + (iħ)-1<ΨS(t)|[AS,Hs]|ΨS(t)>.
(d/dt)<As> = <∂AS/∂t> + (iħ)-1<[AS,Hs]>.
If A does not explicitly depend on time and [A,H] = 0, then
the mean value <A> is constant.
Let H = P2/(2m) + U(R).
Then (d/dt)<R> = (iħ)-1<[R,Hs]>
= (iħ)-1<[R,P2/(2m)]> = <P>/m.
Then (d/dt)<P> = (iħ)-1<[P,Hs]>
= (iħ)-1<[P,U(R)]> = -<∇U(R)>.
These two equations express Ehrenfests theorem.
<R(t)> denotes the position of the center of the wave packet at time t.
Is the trajectory followed by the center of the wave packet the same as predicted by the
laws of classical mechanics?
md2<R>/dt2 = d<P>/dt = -<∇U(R)>.
The classical force at the position <R(t)> of the center of the
wave packet is Fcl = -∇U(r)|r=R.
In general ∇U(r)|r=R ≠
<∇U(R)>, which is the
average force over the whole wave packet.
Therefore the trajectory of the center of the wave packet is in general not the
one predicted by classical mechanics. However if the dimensions of the
wave packet
are much smaller than the dimensions over which U(r) or changes appreciably,
we reach the classical limit.