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. Fig. ... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... In quantum mechanics, perturbation theory is a set of approximation schemes directly related to mathematical perturbation for describing a complicated quantum system in terms of a simpler one. ...

We consider the system to begin in an eigenstate of a given Hamiltonian H₀. We 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 , the transition is into states with energy that differs by from the energy of the initial state. In both cases, the one-to-many transition probability per unit of time from the state to a set of final states is given, to first order in the perturbation, by: In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ... It has been suggested that this article or section be merged into Angular velocity. ...

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. Density of states (DOS) is a property in statistical and condensed matter physics that quantifies how closely packed energy levels are in some physical system. ... Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. ...

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^[1] 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 due to that fact that, being such a useful relation, Fermi himself called it "Golden Rule No. 2."^[2] Enrico Fermi (September 29, 1901 – November 28, 1954) was an Italian physicist most noted for his work on the development of the first nuclear reactor, and for his contributions to the development of quantum theory, particle physics and statistical mechanics. ... Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ...

Notes

1. ^ Dirac, P.A.M. (1927). "The Quantum Theory of Emission and Absorption of Radiation". Proc. Roy. Soc. (London) A 114: 243-265. Retrieved on 2007-05-12.  See equations (24) and (32).
2. ^ Fermi, E. (1950). Nuclear Physics. University of Chicago Press. 

Link title Paul Adrien Maurice Dirac, OM, FRS (IPA: [dɪræk]) (August 8, 1902 – October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Year 2007 (MMVII) is the current year, a common year starting on Monday of the Gregorian calendar and the AD/CE era. ... May 12 is the 132nd day of the year (133rd in leap years) in the Gregorian calendar. ...

External links

• More information on Fermi's golden rule
• Derivation using time-dependent perturbation theory