In quantum physics, Fermi's golden rule is a formula that describes the transition rate 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 and is proportional to the strength of the coupling between the initial and final states of the system 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.
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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.
Contents
Historical background
Although the rule is named after Enrico Fermi, the first to obtain the formula was Paul Dirac, as he had twenty years earlier 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
|
i
⟩
{\displaystyle |i\rangle }
of an unperturbed Hamiltonian H0 and considers the effect of a perturbing Hamiltonian H' applied to the system. 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 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
|
i
⟩
{\displaystyle |i\rangle }
to a set of final states
|
f
⟩
{\displaystyle |f\rangle }
is essentially constant. It is given, to first-order approximation, by
Γ
i
→
f
=
2
Applications
Semiconductors
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. Consider a photon of frequency
Fermi's golden rule predicts that the probability that an excited state will decay depends on the density of states. This can be seen experimentally by measuring the decay rate of a dipole near a mirror: as the presence of the mirror creates regions of higher and lower density of states, the measured decay rate depends on the distance between the mirror and the dipole.
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is the matrix element (in bra–ket notation) of the perturbation H' between the final and initial states, and
ρ
(
E
f
)
{\displaystyle \rho (E_{f})}
is the density of states (number of continuum states divided by
d
E
{\displaystyle dE}
in the infinitesimally small energy interval
E
{\displaystyle E}
to
E
+
d
E
{\displaystyle E+dE}
) at the energy
E
f
{\displaystyle E_{f}}
of the final states. This transition probability is also called "decay probability" and is related to the inverse of the mean lifetime. Thus, the probability of finding the system in state
|
i
⟩
{\displaystyle |i\rangle }
is proportional to
e
−
Γ
i
→
f
t
{\displaystyle e^{-\Gamma _{i\to f}t}}
.
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
⟨
f
|
H
′
|
i
⟩
{\displaystyle \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.
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
refers to the continuum state with the same energy as the discrete state
i
{\displaystyle i}
. 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.
operator does not act on the spin, the electron stays in the same spin state and hence we can write the Bloch wavefunction of the initial and final states as
Finally, we want to know the total transition rate
Γ
(
ω
)
{\displaystyle \Gamma (\omega )}
. 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
is the joint valence-conduction density of states (i.e. the density of pair of states; one occupied valence state, one empty conduction state). In 3D, this is
. This effect can only exist in systems with broken inversion symmetry and nonzero components of the photocurrent can be obtained by symmetry arguments.
is the density of photon states at a given energy,
ℏ
ω
{\displaystyle \hbar \omega }
is the photon energy, and
ω
{\displaystyle \omega }
is the angular frequency. This alternative expression relies on the fact that there is a continuum of final (photon) states, i.e. the range of allowed photon energies is continuous.
