| Fermi's golden rule | |
|---|---|
| Name | Fermi's golden rule |
| Introduced by | Enrico Fermi |
| Year | 1950s |
| Field | Quantum mechanics |
| Expression | \Gamma = \tfrac{2\pi}{\hbar} |
Fermi's golden rule
Fermi's golden rule is an approximate formula in Quantum mechanics that gives the transition rate from an initial quantum state to a continuum of final states under a weak perturbation. It is widely used to compute decay rates and scattering probabilities in contexts ranging from atomic spectroscopy to solid-state condensed matter physics and nuclear reactions. The rule connects matrix elements of a perturbing Hamiltonian with the density of states at the transition energy, providing practical predictions for lifetimes and cross sections.
Fermi's golden rule provides the transition probability per unit time, typically written as \Gamma = (2\pi/\hbar)\,|V_{fi}|^2\,\rho(E_f), where V_{fi} is the matrix element of the perturbation between initial state |i\rangle and final states |f\rangle, and \rho(E_f) is the density of final states at energy E_f. The rule emerges in time-dependent perturbation theory to lowest nontrivial order and assumes a continuum of final states such as those encountered in radiative decay, ionization, or conduction-band transitions in semiconductors. It is attributed to Enrico Fermi, who applied related reasoning in early treatments of beta decay and atomic transitions.
The derivation begins with the time-dependent Schrödinger equation for a Hamiltonian H = H_0 + V(t), where H_0 has known eigenstates and V(t) is treated as a weak, typically harmonic or switch-on perturbation. First-order time-dependent perturbation theory yields transition amplitudes proportional to the Fourier transform of V(t) and a resonance denominator. Taking the long-time limit converts squared sinc functions into delta functions via the identity \lim_{T\to\infty} ( \sin^2(\omega T/2) / (\omega^2/4) ) \propto 2\pi T \delta(\omega), producing a constant transition rate. The calculation invokes the Born approximation for weak scattering and often parallels methods used in the golden-rule context in scattering theory. The derivation also commonly employs the concept of the resolvent and connections to Fermi's approach to continua in early quantum electrodynamics.
Fermi's golden rule is applied across many subfields: - In atomic physics to compute spontaneous emission and stimulated transition rates using matrix elements of the electric dipole moment and the electromagnetic field modes. - In solid-state physics to estimate carrier scattering rates from phonons, impurities, and other electrons; relevant to semiconductor transport, mobility, and resistivity calculations. - In nuclear physics for decay widths and transition probabilities between nuclear states, including models of beta decay and gamma emission. - In quantum optics for calculations that connect to the Weisskopf–Wigner theory of spontaneous emission and to master-equation approaches in open quantum systems. - In chemical physics and molecular spectroscopy for unimolecular dissociation and nonradiative transition rates, including connections to Fermi resonance phenomena. These applications frequently interface with experimental observables such as lifetime measurements, scattering cross sections, and spectral line shapes.
Fermi's golden rule is valid under specific assumptions: weak, time-independent or slowly varying perturbations; Markovian behavior in the long-time limit; and a continuum of final states with smooth density of states. It breaks down for strong coupling, near-threshold discrete spectra, or when nonperturbative effects (e.g., Rabi oscillations or quantum Zeno effect) dominate. Extensions include higher-order perturbative corrections, use within the Kubo formalism for linear response and transport coefficients, and non-Markovian generalizations using the Nakajima–Zwanzig projection operator techniques. In mesoscopic and nanophysics systems with structured reservoirs, modifications employ an energy-dependent self-energy or full spectral-function approaches common in many-body theory and Green's function methods.
Central to the rule is the density of states \rho(E), a property of H_0 that counts available final states per energy interval; \rho(E) can be computed for free particles, bands in crystals, or quantized cavity modes. The product |V_{fi}|^2\rho(E) governs selection rules and scaling of rates: matrix elements encode symmetries (e.g., angular momentum and parity selection rules), while \rho(E) imparts dimensional and dispersion dependence (e.g., power-law energy dependence in 1D, 2D, 3D). In scattering theory, the golden rule relates to Fermi's formula for the transition rate used to derive cross sections via connection to the incoming flux, and it underlies the use of Fermi factors in transport calculations in mesoscopic physics and electronic structure theory.
Numerous experiments corroborate predictions of Fermi's golden rule. Measurements of atomic spontaneous emission rates in isolated atoms and trapped ions align with dipole matrix-element estimates and mode densities in cavities and optical cavities, consistent with Purcell-effect modifications. In solids, phonon-limited mobilities and impurity scattering in silicon and gallium arsenide devices match golden-rule based calculations. Nuclear lifetimes and gamma-decay widths measured in facilities such as CERN and national nuclear physics laboratories validate applications in nuclear spectroscopy. Modern tests in quantum dots, superconducting qubits, and cold atoms probe regimes where deviations from the golden rule signal onset of strong coupling or non-Markovian reservoir correlations, guiding development of more complete open system descriptions.