Fermi's golden rule (${\Gamma }_{i\to f}=(2\pi /\hslash )|V_{fi}{|}^2\delta (E_f-E_i)$)

For a weak time-independent perturbation switched on at zero time, first-order time-dependent perturbation theory gives a transition probabilityThe 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 $2\pi |V_{fi}{|}^2\delta ({\omega }_f-{\omega }_i)/{\hslash }^2$. Using $E=\hslash \omega$ 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.

For box-normalized plane waves in volume $\mathcal{V}$, a potential has matrix element $\tilde{W}({\mathbf{k}}_f-{\mathbf{k}}_i)/\mathcal{V}$. The final-state density per energy and solid angle is $\mathcal{V}m{k}_f/[(2\pi {)}^3{\hslash }^2]$. Dividing the golden-rule rate by incident flux $\hslash k_i/(m\mathcal{V})$ gives $(m^2/4{\pi }^2{\hslash }^4)(k_f/k_i)|\tilde{W}{|}^2$. Elastic scattering has $k_f=k_i$; the arbitrary box volume cancels.