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I am currently reading through Sakurai (1st ed) and he states that Fermi's golden rule can sometimes be written in terms of a Dirac delta function, with the assumption that $$ \rho (E_n) \equiv \delta (E_n - E_i). $$ What is the explicit working out or method that was used here to jump from 1 equation to the next? I don't quite understand what is meant by "it must be understood that this expression is integrated with $\int d E_f \rho (E_n)$".

This is used later on in the chapter for all the different applications of the approximation methods used but never really explained which we can make this "jump". Any help appreciated.

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I find that there is much confusion on this. And sometimes I even think a lot that you can read on FGR is somewhat semi-correct or even wrong. I hope I am not building on this pile with my following answer.

To get the total rate you must sum over all possible final states.

$$ w_{i \rightarrow [n]} = \sum_{n_i} w_{i\rightarrow n_i} $$

If these states are in a continuum you may exchange the sum by an intergation over a density of states:

$$ \sum_{n_i} \longrightarrow \int \mathrm{d}E_n \rho(E_n) \\ \Rightarrow \quad w_{i \rightarrow [n]} = \int \mathrm{d}E_n \rho(E_n) \delta(E_i -E_n) \left( \frac{2 \pi}{\hbar} \right) |V_{in}|^2 \\ = \left\{ \rho(E_n) \left( \frac{2 \pi}{\hbar} \right) |V_{in}|^2 \right\}_{E_n = E_i} $$

You can retrieve the sum over discrete final states by introducing a discrete density distribution $ \rho(E) = \sum_{n_i} \delta( E_{n_i} - E)$.

I hope this helps.

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