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(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

T<<\hbar/\delta \overline{H}

[/tex]

,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

[tex]

\delta t \delta H\geq \hbar

[/tex]

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

[tex]

\lim_{T\rightarrow \infty} U_T(s)P_j(0)=P_j(s)\lim_{T\rightarrow\infty}U_T(s)

[/tex]

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 [tex]P_j(s)[/tex] 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?

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# Sudden Approximation and Adiabatic theorem

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