= Fermi's golden rule
{c}
{title2=$\Gamma_{i\to f}=(2\pi/\hbar)|V_{fi}|^2\delta(E_f-E_i)$}
For a weak time-independent perturbation switched on at zero time, first-order <time-dependent perturbation theory> gives a transition <probability>
$$
P_{i\to f}(t)=\frac{|V_{fi}|^2}{\hbar^2}
\frac{4\sin^2[(\omega_f-\omega_i)t/2]}{(\omega_f-\omega_i)^2}.
$$
The frequency kernel has area $2\pi t$ and width of order $1/t$. Against a smoothly varying continuum of final states, it approaches $2\pi t\delta(\omega_f-\omega_i)$, giving the rate $2\pi|V_{fi}|^2\delta(\omega_f-\omega_i)/\hbar^2$. Using $E=\hbar\omega$ gives the energy form above. One integrates the <Dirac delta function> against the appropriate <density of states>; it is not an ordinary finite number for one isolated resonant level.
The time must be long enough to resolve the smooth continuum and short enough that the total transition <probability> remains small. For continuum variation scale $\Delta\omega$ and total rate $\Gamma$, the usual window is $(\Delta\omega)^{-1}\ll t\ll\Gamma^{-1}$. A discrete finite spectrum also limits this approximation through its recurrences.
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