We consider the system to begin in an eigenstate of a given Hamiltonian . We consider the effect of a (possibly time-dependent) perturbing Hamiltonian . If is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If is oscillating as a function of time with an angular frequency , the transition is into states with energies that differ by from the energy of the initial state. In both cases, the one-to-many transition probability per unit of time from the state to a set of final states is given, to first order in the perturbation, by
Fermi's golden rule is valid when the initial state has not been significantly depleted by scattering into the final states.
The most common way to derive the equation is to start with time-dependent perturbation theory and to take the limit for absorption under the assumption that the time of the measurement is much larger than the time needed for the transition.
Although named after Fermi, most of the work leading to the Golden Rule was done by Dirac who formulated an almost identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference. It is given its name due to the fact that, being such a useful relation, Fermi himself called it "Golden Rule No. 2.