Fermi's golden rule

In quantum physics, Fermi's golden rule is a simple formula for the constant transition rate (probability of transition per unit time) from one energy eigenstate of a quantum system into other energy eigenstates in a continuum, effected by a perturbation. This rate is effectively constant.

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

Consider the system to begin in an eigenstate, $| i\rangle$, of a given Hamiltonian, H0 . Consider the effect of a (possibly time-dependent) perturbing Hamiltonian, H' . 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 as a function of time 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 one-to-many transition probability per unit of time from the state $| i \rangle$ to a set of final states $| f\rangle$ is essentially constant. It is given, to first order in the perturbation, by

$\Gamma_{i \rightarrow f}= \frac{2 \pi} {\hbar} \left | \langle f|H'|i \rangle \right |^{2} \rho$

where ρ is the density of final states (number of continuum states per unit of energy) and $\langle f|H'|i \rangle$ is the matrix element (in bra–ket notation) of the perturbation H' between the final and initial 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 $\langle f|H'|i \rangle$ 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]