Fermi's golden rule
In quantum physics, 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.
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.
It is also applicable when the final state is discrete, i.e. it is not part of a continuum, if there is some decoherence 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.
Although the rule is named after Enrico Fermi, most of the work leading to it is due to Paul Dirac, 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.
1" is of a similar form and considers the probability of indirect transitions per unit time.
[4] Fermi's golden rule describes a system that begins in an eigenstate
If H' is oscillating sinusoidally as a function of time (i.e. it is a harmonic perturbation) with an angular frequency ω, the transition is into states with energies that differ by ħω from the energy of the initial state.
In both cases, the transition probability per unit of time from the initial state
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.
[5][6] The golden rule is a straightforward consequence of the Schrödinger equation, solved to lowest order in the perturbation H' of the Hamiltonian.
In the interaction picture, we can expand an arbitrary quantum state’s time evolution in terms of energy eigenstates of the unperturbed system
The coefficients an(t) are yet unknown functions of time yielding the probability amplitudes in the Dirac picture.
and it is clear from the previous section, major role is played by the energies
Note that the delta function in the expression above arises due to the following argument.
[8] As a constant, it underlies the exponential particle decay laws of radioactivity.
It appears in expressions that complement the golden rule in the semiclassical Boltzmann equation approach to electron transport.
[9] 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).
is the Dirac delta function, and effectively a factor of the square-root of the density of states is included into
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.
We assume that the interactions depend on the energy of the continuum state, but not any other quantum numbers.
The Fermi's 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.
From perturbation it is evident that the heart of the calculation lies in the matrix elements shown in the bracket.
Calculating using these wavefunctions, and focusing on emission (photoluminescence) rather than absorption, we are led to the transition rate
defined as the optical transition dipole moment is qualitatively the expectation value
Hence we need to sum over all possible initial and final states that can satisfy the energy conservation (i.e. an integral of the Brillouin zone in the k-space), and take into account spin degeneracy, which after calculation results in
We note that in a general way we can express the Fermi's golden rule for semiconductors as[13]
In the same manner, the stationary DC photocurrent with amplitude proportional to the square of the field of light is
are the difference of the group velocity and Fermi-Dirac distribution between possible the initial and final states.
and the Hamiltonian, we can also rewrite the transition dipole and photocurrent in terms of position operator matrix using
When considering energy level transitions between two discrete states, Fermi's golden rule is written as
