Fermi’s golden rule
In quantum mechanics, Fermi’s golden rule expresses the transition rate between two states of a system, when a sinusoidal perturbation is applied at the resonance frequency of the energy gap . The main conclusion is that the rate is independent of time.
From time-dependent perturbation theory, we know that the transition probability for a particle in state to go to is as follows for a sinusoidal perturbation at frequency :
Where . If we assume that irreversibly absorbs an unlimited number of particles, then we can interpret as the “amount” of the current particle that has transitioned since the last period .
For generality, let be the center of a state continuum with width . In that case, must be modified as follows, where is the destination’s density of states:
If is not in a continuum, then . The integrand is a sharp sinc-function around . For large , it is so sharp that we can take out . In that case, we also simplify the integration limits. Then we substitute to get:
This definite integral turns out to be , so we find, because clearly :
The transition rate , i.e. the number of particles per unit time, then takes this form:
Note that the -dependence has disappeared, and all that remains is a constant factor involving , where is the resonance frequency.
- D.J. Griffiths, D.F. Schroeter, Introduction to quantum mechanics, 3rd edition, Cambridge.