Quantum Fermi's golden rule describes the transition rate from an initial quantum state to a set of final states when a weak perturbation acts on the system. It is one of the standard results of time-dependent perturbation theory and is especially important when the final states form a continuum, such as in atomic decay, ionization, scattering, and optical absorption.^[1]^[2]

Basic idea

If a Hamiltonian is written as $H = H_{0} + V$ , where $H_{0}$ is the unperturbed Hamiltonian and $V$ is a weak interaction, then the perturbation can induce transitions from an initial eigenstate $| i ⟩$ of $H_{0}$ to final states $| f ⟩$ . In the long-time limit, and for weak coupling, the transition rate is proportional to the square of the matrix element $⟨ f | V | i ⟩$ and to the density of final states at the allowed energy.^[1]^[3]

Standard formula

For transitions into a continuum of final states, the usual form is

w_{i \to f} = \frac{2 π}{ℏ} | ⟨ f | V | i ⟩ |^{2} ρ (E_{f})

where:

$w_{i \to f}$ is the transition rate,
$⟨ f | V | i ⟩$ is the transition matrix element,
$ρ (E_{f})$ is the density of final states at the final energy,
and energy conservation requires the final energy to match the allowed transition energy.^[4]^[3]

For a harmonic perturbation of angular frequency $ω$ , the final energy is typically

E_{f} = E_{i} + ℏ ω

for absorption, or

E_{f} = E_{i} - ℏ ω

for emission.^[3]

Derivation sketch

In first-order time-dependent perturbation theory, the transition amplitude from an initial state $| i ⟩$ to a final state $| f ⟩$ is given by

c_{f} (t) \propto \int_{0}^{t} e^{i ω_{f i} t^{'}} ⟨ f | V | i ⟩ d t^{'}

This leads to a transition probability containing a sharply peaked factor of the form

\frac{\sin^{2} (x)}{x^{2}}

in the long-time limit. As $t$ becomes large, this function approaches a delta function,

\frac{\sin^{2} (x)}{x^{2}} \to π δ (x)

in the distributional sense, which enforces energy conservation between initial and final states.

When the final states form a continuum, the discrete sum over states becomes an integral over the density of states, leading directly to Fermi's golden rule.^[3]

Physical meaning

The rule states that quantum transitions are controlled by two ingredients. The first is the strength of coupling between the initial and final states, represented by the matrix element. The second is the number of available final states, represented by the density of states. Strong perturbation cannot produce an efficient transition if there are no accessible final states at the required energy, a modest coupling can produce a large rate if many final states are available.^[1]^[5]

Relation to density of states

Fermi's golden rule is closely connected to the concept of density of states. When the final spectrum is continuous, the discrete sum over final states is replaced by an integral over energy:

\sum_{f} \to \int ρ (E_{f}) d E_{f}

This is why transition rates in solids, atoms, and molecules often depend strongly on the structure of the available spectrum. Peaks or gaps in the density of states can strongly enhance or suppress observable transition rates.^[4]^[5]

Example: atomic spontaneous emission

In atomic spontaneous emission, an excited atom interacts with the electromagnetic field and can transition to a lower-energy state while emitting a photon. The transition rate depends on the dipole matrix element between the atomic states and on the density of photon states at the emitted frequency. Fermi's golden rule explains why some transitions are strong while others are weak or forbidden.^[3]

Applications

Fermi's golden rule is widely used in:

spontaneous emission and absorption of radiation,
atomic and molecular spectroscopy,
radioactive decay and scattering theory,
phonon and electron relaxation in solids,
photoionization and semiconductor optics.^[4]^[3]

Limitations

The rule is an approximation. It works best when the perturbation is weak, the observation time is long enough for the rate description to apply, and the back-action of the final states on the initial state can be neglected. For strong driving, very short times, or strongly coherent dynamics between a few discrete states, more complete treatments such as full time-dependent evolution or Rabi oscillations are needed.^[4]^[3]

Physics:Quantum Fermi's golden rule

Basic idea

Standard formula

Derivation sketch

Physical meaning

Relation to density of states

Example: atomic spontaneous emission

Applications

Limitations

See also

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