# Fermi's golden rule

In quantum physics, Fermi's golden rule is a formula that describes the transition rate (probability of transition per unit time) from one energy eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak perturbation. This transition rate is effectively constant and depends on the strength of the coupling between the initial and final states of the system (due to the perturbation) as well as the density of states in the continuum.

## General

Although named after Enrico Fermi, most of the work leading to the Golden Rule is due to Paul Dirac who formulated 20 years earlier a virtually identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference.[1][2] It was given this name because, on account of its importance, Fermi dubbed it "Golden Rule No. 2." [3]

## The rate and its derivation

Fermi's golden rule describes a system which begins in an eigenstate, ${\displaystyle |i\rangle }$ , of an unperturbed Hamiltonian, H0 and considers the effect of a perturbing Hamiltonian, H' applied the system. If H' is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If H' is oscillating sinusoidally as a function of time (i.e. it is a harmonic perturbation) with an angular frequency ω, the transition is into states with energies that differ by ħω from the energy of the initial state.

In both cases, the transition probability per unit of time from the initial state ${\displaystyle |i\rangle }$  to a set of final states ${\displaystyle |f\rangle }$  is essentially constant. It is given, to first order approximation, by

${\displaystyle \Gamma _{i\rightarrow f}={\frac {2\pi }{\hbar }}\left|\langle f|H'|i\rangle \right|^{2}\rho (E_{f})}$

where ${\displaystyle \langle f|H'|i\rangle }$  is the matrix element (in bra–ket notation) of the perturbation H' between the final and initial states and ${\displaystyle \rho (E_{f})}$  is the density of states (number of continuum states in an infinitesimally small energy interval ${\displaystyle E+dE}$  ) at the energy ${\displaystyle E_{f}}$  of the final states.

This transition probability is also called "decay probability" and is related to the inverse of the mean lifetime. Fermi's golden rule is valid when the initial state has not been significantly depleted by scattering into the final states.

The standard 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.[4][5]

Only the magnitude of the matrix element ${\displaystyle \langle f|H'|i\rangle }$  enters the Fermi's Golden Rule. The phase of this matrix element, however, contains separate information about the transition process. It appears in expressions that complement the Golden Rule in the semiclassical Boltzmann equation approach to electron transport.[7]