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There is a very clear derivation of Fermi's golden rule (actually Dirac's) here. Everything runs smoothly until, somehow, the equivalence

$$ \Gamma_{a \rightarrow b} = \frac{P_{a \rightarrow b}}{t} $$

is introduced and it is assumed that $\Gamma$ is some kind of rate (e.g. the $\beta$-decay rate of the neutron). However I don't see the equivalence with a rate (i.e. some kind of event counting per unit of time). If a process follows, say, Poissonian statistics, then the mean time until the next event is $1/\lambda$, so $\lambda$ events are happening per unit of time. But here the only thing I see is a probability divided by $t$.

Edit: here it is stated that "the transition probability is also called the decay probability and is related to the mean lifetime $t$ of the state by $P_{a \rightarrow b} = 1/t$"

Could someone help clarifying the concepts?

PS: I have checked previous posts with similar questions, to make sure there is no duplicate. I found this question which, I think, is essentially asking the same as me

Interpretation of "transition rate" in Fermi's golden rule

However, there is no good accepted answer (e.g, it is not very obvious what Qmechanic answer is actually answering).

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1 Answers1

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The explanation you are looking for is on the bottom of pg.1 in the notes you linked to:

The central result is that for weak perturbations and long times, the transition probability $P_b(t)$ raises linear in time as $P_b(t) = \Gamma_{a \rightarrow b} \times t$ for $b\neq a$.

Now the transition probability $P_b(t)$ is by definition the average number of transitions into state $b$ during time $t$, in the sense of the average number of systems that end up in state $b$ from among a given initial ensemble of such systems. If all transitions to $b$ come from state $a$ (say the ensemble is initially prepared in state $a$), and if $P_b(t) \sim t$, then the average number of transition events per unit time is genuinely $\Gamma_{a \rightarrow b} = P_b(t)/t$. Hence $\Gamma_{a \rightarrow b}$ is indeed a transition rate in this case.

udrv
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