# 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 another energy eigenstate, due to a perturbation.

We consider the system to begin in an eigenstate, $\scriptstyle | i\rangle$, of a given Hamiltonian, $\scriptstyle H_0$. We consider the effect of a (possibly time-dependent) perturbing Hamiltonian, $\scriptstyle H'$. If $\scriptstyle H'$ is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If $\scriptstyle H'$ is oscillating as a function of time with an angular frequency $\scriptstyle \omega$, the transition is into states with energies that differ by $\scriptstyle \hbar\omega$ from the energy of the initial state. In both cases, the one-to-many transition probability per unit of time from the state $\scriptstyle| i \rangle$ to a set of final states $\scriptstyle| f\rangle$ 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 $\scriptstyle \rho$ is the density of final states (number of states per unit of energy) and $\scriptstyle \langle f|H'|i \rangle$ is the matrix element (in bra–ket notation) of the perturbation $\scriptstyle H'$ between the final and initial states.This transition probability is also called decay probability and is related to mean lifetime.

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. In the special case of equally spaced levels in the continuum, some simple and beautiful mathematical equalities about the sinc function can be invoked, and the derivation is especially clear. [1]

Although named after Fermi, most of the work leading to the Golden Rule was done by Dirac[2] who formulated an almost identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference. It is given its name because, being such a useful relation, Fermi himself called it "Golden Rule No. 2."[3]

Only the magnitude of the matrix element $\scriptstyle \langle f|H'|i \rangle$ 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.[4]

## References

1. ^ Zhang, Jiang-Min; Haque, Masudul (2014). Nonsmooth and level-resolved dynamics illustrated with a periodically driven tight binding model. arXiv:1404.4280.
2. ^ Dirac, P.A.M. (1 March 1927). "The Quantum Theory of Emission and Absorption of Radiation". Proceedings of the Royal Society A 114 (767): 243–265. Bibcode:1927RSPSA.114..243D. doi:10.1098/rspa.1927.0039. JSTOR 94746. See equations (24) and (32).
3. ^ Fermi, E. (1950). Nuclear Physics. University of Chicago Press.
4. ^ N. A. Sinitsyn, Q. Niu and A. H. MacDonald (2006). "Coordinate Shift in Semiclassical Boltzmann Equation and Anomalous Hall Effect". Phys. Rev. B 73 (7): 075318. arXiv:cond-mat/0511310. Bibcode:2006PhRvB..73g5318S. doi:10.1103/PhysRevB.73.075318.