I am trying to understand an aspect of Fermi's golden rule in the case of a constant perturbation, $V$. The formula for the transition probability from an initial state $i$ to a final state $f$ is given by: $$P_{i \rightarrow f} = \frac{2\pi}{\hbar} \tau |V_{i,f}|^2 \delta(E_f - E_i)$$ I understand that the Dirac delta $\delta(E_f - E_i)$ ensures energy conservation by imposing that the transition occurs only when $E_f=E_i$. However, I have a doubt: how can this expression make sense since the Dirac delta goes to infinity when $E_f=E_i$? Shouldn't the transition probability always be less than 1?
I know that the Dirac delta is often used to describe probability densities, but in this case, we are talking about the probability of transitioning from a specific state $i$ to a specific state $f$. Am I missing something about the correct interpretation of the Dirac delta in this context? Could someone clarify how this formula is properly interpreted and how a finite, meaningful transition probability is obtained?
