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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 theory (quantum mechanics)|perturbation]].
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| We consider the system to begin in an [[eigenstate]], <math>\scriptstyle | i\rangle</math>, of a given [[Hamiltonian (quantum mechanics)|Hamiltonian]], <math>\scriptstyle H_0 </math>. We consider the effect of a (possibly time-dependent) perturbing Hamiltonian, <math>\scriptstyle H'</math>. If <math>\scriptstyle 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>\scriptstyle H'</math> is oscillating as a function of time with an [[angular frequency]] <math>\scriptstyle \omega</math>, the transition is into states with energies that differ by <math>\scriptstyle \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>\scriptstyle| i \rangle</math> to a set of final states <math>\scriptstyle| f\rangle</math> is given, to first order in the perturbation, by
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| :<math> T_{i \rightarrow f}= \frac{2 \pi} {\hbar} \left | \langle f|H'|i \rangle \right |^{2} \rho,</math>
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| where <math>\scriptstyle \rho </math> is the [[density of states|density of final states]] (number of states per unit of energy) and <math>\scriptstyle \langle f|H'|i \rangle </math> is the matrix element (in [[bra-ket notation]]) of the perturbation <math>\scriptstyle H'</math> between the final and initial states.This transition probability is also called decay probability and is related to mean lifetime.
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| Fermi's golden rule is valid when the initial state has not been significantly depleted by scattering into the final states.
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| 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.
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| Although named after [[Enrico Fermi|Fermi]], most of the work leading to the Golden Rule was done by [[Paul Dirac|Dirac]]<ref>{{cite journal | last = Dirac | first = P.A.M. | authorlink = Paul Dirac | title = The Quantum Theory of Emission and Absorption of Radiation | journal = [[Proceedings of the Royal Society A]] | volume = 114 | pages = 243–265 | date=1 March 1927| issue = 767 | doi = 10.1098/rspa.1927.0039 | jstor=94746|bibcode = 1927RSPSA.114..243D }} See equations (24) and (32).</ref> 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."<ref>{{cite book | last = Fermi | first = E. | title = Nuclear Physics | publisher = University of Chicago Press | year = 1950 }}</ref>
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| Only the magnitude of the matrix element <math>\scriptstyle \langle f|H'|i \rangle</math> enters the Fermi's Golden Rule. The phase of this matrix element, however, contains separate information about the transition process.
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| It appears in expressions that complement the Golden Rule in the semiclassical Boltzmann equation approach to electron transport.<ref name='sinitsyn-08jpa'>{{cite journal|title=Coordinate Shift in Semiclassical Boltzmann Equation and Anomalous Hall Effect|author=N. A. Sinitsyn, Q. Niu and A. H. MacDonald|journal=Phys. Rev. B|volume=73|year=2006|pages=075318|arxiv=cond-mat/0511310|doi=10.1103/PhysRevB.73.075318|bibcode = 2006PhRvB..73g5318S|issue=7 }}</ref><!-- N.A. Sinitsyn 2006 --> | |
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| ==References==
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| {{Reflist}}
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| ==External links==
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| *[http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/fermi.html More information on Fermi's golden rule]
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| *[http://www.ph.utexas.edu/~schwitte/PHY362L/QMnote.pdf Derivation using time-dependent perturbation theory]
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| {{DEFAULTSORT:Fermi's Golden Rule}}
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| [[Category:Concepts in physics]]
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| [[Category:Perturbation theory]]
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| [[Category:Enrico Fermi|Golden Rule]]
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