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 Schrodinger'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 schrodinger eq. (neglecting time dependent phase factors)

 

[Entropia]

The sudden approximation and adiabatic theorem are two important concepts in quantum mechanics that allow us to simplify complex systems and make predictions about their behavior. The sudden approximation is a mathematical technique used to simplify the calculation of a system's wave function when there is a sudden change in the Hamiltonian. This approximation is valid when the time scale of the change is much smaller than the characteristic time scale of the system, which is given by T<<\hbar/\delta \overline{H}. This condition ensures that the wave function is not able to respond to the sudden change and can be approximated as a constant.

The uncertainty relation between energy and time does not necessarily violate this condition. The uncertainty relation states that there is a fundamental limit to the precision with which we can measure the energy and time of a system. However, the sudden approximation does not rely on precise measurements, but rather on the time scale of the change in the Hamiltonian. Therefore, the two conditions do not contradict each other and the sudden approximation can still be used in principle.

The adiabatic theorem, on the other hand, is a fundamental principle in quantum mechanics that states that if a system is changing slowly enough, the system will remain in its initial state. This means that the probability of finding the system in a particular state will remain constant over time. The equation mentioned in the question, \lim_{T\rightarrow \infty} U_T(s)P_j(0)=P_j(s)\lim_{T\rightarrow\infty}U_T(s), shows that the wave function will evolve to the same state, regardless of whether we project it first and then evolve it, or vice versa. This asymptotic property is a powerful tool in simplifying calculations and understanding the behavior of quantum systems.

In summary, both the sudden approximation and adiabatic theorem are important tools in quantum mechanics that allow us to make predictions and simplify complex systems. While they may seem contradictory at first, they are actually complementary and can both be used to understand the behavior of quantum systems.