In
quantum physics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, qua ...
, Fermi's golden rule is a formula that describes the transition rate (the probability of a transition per unit time) from one energy
eigenstate of a quantum system to a group of energy eigenstates in a continuum, as a result of a weak
perturbation
Perturbation or perturb may refer to:
* Perturbation theory, mathematical methods that give approximate solutions to problems that cannot be solved exactly
* Perturbation (geology), changes in the nature of alluvial deposits over time
* Perturbat ...
. This transition rate is effectively independent of time (so long as the strength of the perturbation is independent of time) and is proportional to the strength of the coupling between the initial and final states of the system (described by the square of the
matrix element of the perturbation) as well as the
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
. It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some
decoherence
Quantum decoherence is the loss of quantum coherence. In quantum mechanics, particles such as electrons are described by a wave function, a mathematical representation of the quantum state of a system; a probabilistic interpretation of the wa ...
in the process, like relaxation or collision of the atoms, or like noise in the perturbation, in which case the density of states is replaced by the reciprocal of the decoherence bandwidth.
General
Although the rule is named after
Enrico Fermi
Enrico Fermi (; 29 September 1901 – 28 November 1954) was an Italian (later naturalized American) physicist and the creator of the world's first nuclear reactor, the Chicago Pile-1. He has been called the "architect of the nuclear age" and ...
, most of the work leading to it is due to
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
, who twenty years earlier had formulated a virtually identical equation, including the three components of a constant, the matrix element of the perturbation and an energy difference. It was given this name because, on account of its importance, Fermi called it "golden rule No. 2".
Most uses of the term Fermi's golden rule are referring to "golden rule No. 2", but Fermi's "golden rule No. 1" is of a similar form and considers the probability of indirect transitions per unit time.
The rate and its derivation
Fermi's golden rule describes a system that begins in an
eigenstate of an unperturbed
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
and considers the effect of a perturbing Hamiltonian applied to the system. If is time-independent, the system goes only into those states in the continuum that have the same energy as the initial state. If is oscillating sinusoidally as a function of time (i.e. it is a harmonic perturbation) with an
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
, the transition is into states with energies that differ by from the energy of the initial state.
In both cases, the ''transition probability per unit of time'' from the initial state
to a set of final states
is essentially constant. It is given, to first-order approximation, by
where
is the
matrix element (in
bra–ket notation) of the perturbation between the final and initial states, and
is the
density of states
In solid state physics and condensed matter physics, the density of states (DOS) of a system describes the number of modes per unit frequency range. The density of states is defined as D(E) = N(E)/V , where N(E)\delta E is the number of states i ...
(number of continuum states divided by
in the infinitesimally small energy interval
to
) at the energy
of the final states. This transition probability is also called "decay probability" and is related to the inverse of the
mean lifetime
A quantity is subject to exponential decay if it decreases at a rate Proportionality (mathematics), proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and ...
. Thus, the probability of finding the system in state
is proportional to
.
The standard 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.
Only the magnitude of the matrix element
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
The Boltzmann equation or Boltzmann transport equation (BTE) describes the statistical behaviour of a thermodynamic system not in a state of equilibrium, devised by Ludwig Boltzmann in 1872.Encyclopaedia of Physics (2nd Edition), R. G. Lerne ...
approach to electron transport.
While the Golden rule is commonly stated and derived in the terms above, the final state (continuum) wave function is often rather vaguely described, and not normalized correctly (and the normalisation is used in the derivation). The problem is that in order to produce a continuum there can be no
spatial confinement (which would necessarily discretise the spectrum), and therefore the continuum wave functions must have infinite extent, and in turn this means that the normalisation
is infinite, not unity. If the interactions depend on the energy of the continuum state, but not any other quantum numbers, it is usual to normalise continuum wave-functions with energy
labelled
, by writing
where
is the
Dirac delta function
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
, and effectively a factor of the square-root of the density of states is included into
.
In this case, the continuum wave function has dimensions of
, and the Golden Rule is now
where
refers to the continuum state with the same energy as the discrete state
. For example, correctly normalized continuum wave functions for the case of a free electron in the vicinity of a hydrogen atom are available in Bethe and Salpeter.
Applications
Semiconductors
The Fermi golden rule can be used for calculating the transition probability rate for an electron that is excited by a photon from the valence band to the conduction band in a direct band-gap semiconductor, and also for when the electron recombines with the hole and emits a photon. Consider a photon of frequency
and wavevector
, where the light dispersion relation is
and
is the index of refraction.
Using the Coulomb gauge where
and
, the vector potential of the EM wave is given by
where the resulting electric field is
For a charged particle in the valence band, the Hamiltonian is
where
is the potential of the crystal. If our particle is an electron (
) and we consider process involving one photon and first order in
. The resulting Hamiltonian is