Fermi's golden rule

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In quantum physics, Fermi's golden rule is a way to calculate the transition rate (probability of transition per unit time) from one energy eigenstate of a quantum system into a continuum of energy eigenstates, due to a perturbation.

We consider the system to begin in an eigenstate <math>| i\rangle</math> of a given Hamiltonian <math> H_0 </math>. We consider the effect of a (possibly time-dependent) perturbing Hamiltonian <math>H'</math>. If <math>H'</math> is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If <math>H'</math> is oscillating as a function of time with an angular frequency <math>\omega\,</math>, the transition is into states with energy that differs by <math>\hbar\omega</math> from the energy of the initial state. In both cases, the one-to-many transition probability per unit of time from the state <math>| i \rangle</math> to a set of final states <math>| f\rangle</math> is given, to first order in the perturbation, by:

<math> T_{i \rightarrow f}= \frac{2 \pi} {\hbar} \left | \langle f|H'|i \rangle \right |^{2} \rho </math>

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 the initial state has not been significantly depleted 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. It is given its name due to that fact that, being such a useful relation, Fermi himself called it "Golden Rule No. 2."<ref>Fermi, E. (1950). Nuclear Physics. University of Chicago Press.</ref>