Fermi's golden rule
/ FUR-mee /
Many quantum processes are not a jump between two sharp levels but a decay into a whole continuum of possibilities — an excited atom emitting a photon into any of countless field modes, a particle scattering into any outgoing direction. For these, the useful quantity is not a single probability but a constant transition rate: how many transitions per second. Fermi's golden rule is the compact formula that gives that rate.
It states that the transition rate from an initial state i into final states f under a perturbation V is Gamma = (2 pi/hbar) |<f|V|i>|^2 rho(E_f), where |<f|V|i>|^2 is the squared coupling matrix element and rho(E_f) is the density of final states per unit energy, evaluated where energy is conserved. It follows from first-order time-dependent perturbation theory in the long-time limit: the resonance factor sin^2[(omega - omega_fi)t/2]/(omega - omega_fi)^2 narrows as time grows and, integrated against the continuum of final states, becomes a delta function enforcing energy conservation, leaving a probability that grows linearly in time — hence a constant rate. A constant rate applied to a decaying population produces the familiar exponential decay law.
The golden rule is everywhere: it gives spontaneous-emission and absorption rates, scattering cross-sections, nuclear and particle decay rates, and electron-transport rates in solids. Its honest preconditions: there must be a genuine (quasi-)continuum of final states for rho(E_f) to make sense, the coupling must be weak enough for first-order perturbation theory, and the elapsed time must be long compared with 1/(level spacing) yet short compared with the decay time. At very short times the probability grows quadratically, not linearly, so the rule does not apply there. (It was actually derived by Dirac; Fermi named it 'golden' for its usefulness.)
The spontaneous-emission rate of an excited atom follows from the golden rule with V the atom-field coupling and rho the density of photon modes in vacuum; the result is a constant rate A (the Einstein A coefficient), so an excited population decays as N(t) = N_0 e^(-A t) with lifetime 1/A — for hydrogen's 2p state about 1.6 ns.
A constant golden-rule rate is exactly what makes decay exponential.
The rule needs a continuum of final states and first-order validity; it does not describe coherent two-level dynamics (that is Rabi flopping) and it fails at very short times, where transition probability is quadratic rather than linear in t.