In quantum physics, Fermi's golden rule is a way to calculate the transition rate between two eigenstates of a quantum system using time-dependent perturbation theory, which means it's an approximation.

We consider the system to begin in an eigenstate $|\; i\backslash rangle$ of a given Hamiltonian $H\_0$. We consider the effect of a time-independent perturbing Hamiltonian $H\text{'}$.

The one-to-many transition probability per unit of time from the state $|\; i\backslash rangle$ to a set of states $|\; f\backslash rangle$ is given, to first order in the perturbation, by:

$T\_\{i\; \backslash rightarrow\; f\}=\; \backslash frac\{2\; \backslash pi\}\; \{\backslash hbar\}\; \backslash left\; |\; \backslash langle\; f|H\text{'}|i\; \backslash rangle\; \backslash right\; |^\{2\}\; \backslash rho$

where ρ is the density of final states, and < f | H' | i > is the matrix element (in bra-ket notation) of the perturbation, H', between the final and initial states.

Fermi's golden rule is valid when $H\text{'}$ is time-independent, $|\; i\backslash rangle$ is an eigenstate of the unperturbed Hamiltonian, the states $|\; f\backslash rangle$ form a continuum, and the initial state has not been significantly depleted (eg, 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.

External links

• More information on Fermi's golden rule
• Derivation using time-dependent perturbation theory

Perturbation theory

Fermis Goldene Regel | Золотое правило Ферми