User:Anotherwikipedian/Fermi's golden rule

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.

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


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

where &rho; 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.

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.

Derivation
Fermi's Golden Rule is derived from the results of time-dependent perturbation theory.

Consider an oscillating perturbation of the form $$\hat H'(r)e^{(-i\omega t)}$$ for $$\omega>0$$ where $$\hat H'(r)$$. has no explicit time dependence.

Time-dependent perturbation theory gives, to first order,
 * $$ c_k(t) = \frac{1}{i\hbar} \int_{0}^{t} e^{i(\omega _k - \omega_0)t}\hat H_{k0}(t') dt'$$ where $$\hat H _{kj} = \langle\psi_k|\hat H'|\psi_j\rangle$$