What does energy scale (compared to time scale) means?

Question

I am reading about the sudden approximation in Sakurai. He writes the Schrodinger equation for the time evolution operator in terms of the variable t=sT: $$ i\frac{\partial}{\partial s}U(t,t_0)=\frac{H}{\hbar/T}U(t,t_0)=\frac{H}{\hbar\Omega}U(t,_t0)$$ with $\Omega=1/T$.

He then says that in the time scale $T\rightarrow 0$, $\hbar\Omega$ will be much larger than the energy scale represented by $H$. What is the energy scale represented by $H$? Is it its eigenvalues?

It also says that $T$ should be small compared to $2\pi/\omega_{ab}$ where $E{ab}=\hbar \omega_{ab}$ is the difference in energy between two relevant eigenvalues. But how do I know which are the relevant eigenvalues? And why is this condition required?

Answer 1

Let's say you had a superposition \begin{equation} |\psi(t)\rangle = \sum_{n=0}^N c_n | n \rangle e^{i \hbar \omega_n t} \end{equation} where the energy levels are $E_n = \hbar \omega_n$, and $N$ is some finite number.

Then a sudden transition is one that happens over a timescale $T \gg 1/\omega_N$ (note the use of capital $N$, meaning the maximum value of $\omega_n$ in the above sum). In words, the transition happens over a timescale very short compared to the periods of all the modes in the superposition. If you look at differences between eigenvalues, you also want to look at the set of differences $\omega_m - \omega_n$ for $m, n \leq N$.

You might note that this definition depends on the state of the system. This is true; eigenstates with very high energies (aka frequencies) $\omega_n \gg 1/T$ will not see the transition as instantaneous. You can also generalize the above statements to work even if there are terms with $N\rightarrow \infty$, so long as $c_n \approx 0$ for all $n>N$ and some finite $N$.

Typically there is a "natural" cutoff $N$ where you don't expect to excite states with energy above $\hbar \omega_N$. For example, if you are driving a harmonic oscillator with an external force with frequency $\Omega$, you don't expect to excite modes with $\omega \gg \Omega$.