Sudden Approximation and Adiabatic theorem

[Yiping]

I am reading quantum mechanics (Messiah) now. And I get confused about the condition for the validity of the sudden approximation in CH. XVII. The author use perturbation theory to derive the result
$$T<<\hbar/\delta \overline{H}$$
,when the Hamiltonian change over time T. The condition tells me that I have to vary the Hamiltonian fast enough such that the wave function is not able to response the sudden change. But uncertainty relation between energy and time tells me that
$$\delta t \delta H\geq \hbar$$
Does the two condition violate each other? If it is so, does it means that in principle we can never use the sudden approximation?

My second question is about the adiabatic theorem.
The author mentions that
$$\lim_{T\rightarrow \infty} U_T(s)P_j(0)=P_j(s)\lim_{T\rightarrow\infty}U_T(s)$$
in which the Hamiltonian is changing during time period T, s labels the portion the system has evolved. If s=1, means the system finishs the whole changing process. And $$P_j(s)$$ is the projection operator of the eigenstate j at s.
The asymptotic property is hard for me to understand the physical meaning of it.
Does it means that when the evolving time, T, is large enough, projecting the wave function and then evolve it to s is equivalent to evolving the wave function to s first and then project to the eigen state j at s are actually the same?

 

[tim_lou]

For the first question, you have to be careful about the energy time uncertainty principle. There are many different versions and they all have different (but precise meanings).

In your case, I suppose you are talking about the uncertainty relation described by Griffith Ch3 pg. 113, where it explicitly says Δt is the time needed for an observable to change by ~1 standard deviation, and that observable must have NO explicit time dependence, which is not your case here.

As for the second question, apply the equation to a state |n(t0)>, the instantaneous eigenstate for the time dependent Hamiltonian with continuously varying eigenvalue. This gives precisely the statement of adiabatic theorem,

U(t, t0)|n(t0)> = P_n U(t, t0)|n(t0)> so that U(t, t0)|n(t0)> is just |n(t)> (neglecting some additional phase factor, this comes from the fact that any eigenvector of the projection must either be 0 or in the space P_n projects to)

You might argue, isn't it kinda obvious |n(t0)> will evolve into |n(t)>? but keep in mind that n(t) DOES NOT solve the Schrödinger's eq in general, it is merely an eigenvector of H, the adiabatic theorem says that if H varies slowly enough, n(t) is actually a solution of the Schrödinger eq. (neglecting time dependent phase factors)