First order time dependent perturbation theory tells us that under the influence of a perturbation $Ve^{i\omega t}$, a system that started in the state $|n\rangle$ at time $t=0$ has probability $$P_k(t)=4|\langle k|V|n\rangle|^2\frac{\sin^2[\frac{(E_n-E_k-\hbar\omega)t}{2\hbar}]}{(E_n-E_k-\hbar\omega)^2}$$ of transitioning to the state $|k\rangle$.
We can imagine wanting to know the probability of transitioning to a set of energies at long times. To find this, we multiply the above probability by the density of states, integrate over all energies and use the large time limit (which incidentally, tells us all transitions have to be down by $\hbar\omega$) to find that the probability of making the transition down by an energy $\hbar\omega$ per unit time is$$\frac{2\pi t}{\hbar}g(E_{out})|\langle out|V|n\rangle|^2$$ where $|out\rangle$ refers to the allowed final states, with energy $\hbar\omega$ less.
I'm having trouble understanding exactly what this means. I said I wanted to find 'the probability of transitioning to a set of energies' but my final expression has one final state, namely $|out\rangle$. So have I actually found the probability of transitioning to a set of energies, or just the state $|out\rangle$ (which all the sources I'm reading suggest is the case)? If the latter, why has this happened (the maths I mentioned seems to be geared towards the former), and why couldn't I have just taken the large $t$ limit in the expression $P_k(t)$?
