The Fermi (or Fermi–Dirac) golden rule (FGR) (3.27) allows for calculation of the transition probability rate between two eigenstates of a quantum system using the time-dependent perturbation theory. It is the probability of this scattering-caused transition per unit time. So, the probability is time dependent and is is derived from the time-dependent perturbation theory (the perturbation Hamiltonian, i.e., the scattering potential, is time dependent), under the assumption that the time of the measurement is much larger than the time needed for the transition.

It is the rate of gain of probability per unit time in the manifold of final eigenstate ∣ψκ′〉, which is equal to the rate of loss of probability per unit time from the initial eigenstate ∣ψκ〉.

A brief derivation of the FGR is given and more details can be found in [94, 294, 325]

Time-Dependent Perturbation

The general Hamiltonian of interest is of the form (5.89)

where H∘ is a time independent Hamiltonian with a known solution ψκ, which is related to Ψκ through (2.68), i.e.,

and E(κ) and ψκ are time-independent. Here H′ causes time-dependent transitions between the states ψκ. The time-dependent Schrödinger equation is

We now express Ψ as an expansion of the eigenfunctions ψκ exp[−i Ee(κ)t/ħ]of the unperturbed system, (2.67), i.e.,

The time-dependent problem is solved when the coefficients aκ (t) are known, and from this we will develop time-rate equation, i.e., which is the derivation of the probability of transition.