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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 $P_k(t)$?

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Your second equation is strange. If it is 'per unit time', then you should not have the 't' in the numerator. If it is the total transition probability at time $t$, then you should have the 't'. Anyway, your second equation is the result of a summation over the final states. Written in the way you wrote, $|out\rangle $ should be understood as a representative state------you have many many states around the energy $E_{out}$, and you can pick anyone of them. This is justified because the matrix element $\langle out |V|n \rangle$ is almost independent of the specific out state chosen.

It is somehow common to get confused with Fermi golden rule. But we have a paper (see below) with a crystal-clear derivation of it. You cannot lose sight of the physics there.

https://www.scienceopen.com/document/vid/cdf576df-bf02-47a2-b17e-21283b806b11?0

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