The Fermi (or Fermi Dirac) golden rule (3.27) allows for calculation of the transition probability rate between two eigenstates of a quantum system using the time-dependent perturbation theory. It can be 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 ? _{? ?} ?, which is equal to the rate of loss of probability per unit time from the initial eigenstate ? _? ?.

A brief derivation of the FGR is given and more details can be found in [74, 219, 241]

E.1 Time-Dependent Perturbation

The general Hamiltonian of interest is of the form

where H _o is a time independent Hamiltonian with a known solution ? _?, which is related to ? _? through (2.68), i.e.,

and E( ?) and ? _? are time-independent. Here H ? causes time-dependent transitions between the states ? _?. The time-dependent Schr dinger equation is

using time-dependent coefficients a _?( t).

The approximation will involve expressing ? as an expansion of the time-independent eigenfunctions ? _? exp[ ? iE _e( ?) t/?] of the unperturbed, time-independent system, i.e., (2.68)

The time-dependent problem is solved when the coefficients a _?( t)...