Fermi's golden rule

Suppose a system begins in an initial quantum state and a weak perturbation V can couple it to final states. For a continuum of final states, a common form of Fermi's golden rule gives the transition rate as 2 pi divided by h-bar, multiplied by the squared absolute value of the transition matrix element and by the density of available final states per unit energy, evaluated at the energy allowed by the transition. The transition matrix element is the matrix element of V between the final and initial states. When final states have different matrix elements, their contributions are summed or integrated. The rule therefore combines how strongly the interaction connects states with how many final states are available.

The rule follows from time-dependent perturbation theory, which describes how a system's state can change when a relatively small interaction is added to its Hamiltonian. For a single discrete final state, perturbation theory first gives a time-dependent transition probability. When many final states form a sufficiently dense continuum, the probability can grow approximately linearly with time over a suitable interval; its proportionality coefficient is interpreted as a rate. Replacing the sum over final states with a density of states yields the familiar golden-rule expression.

The matrix element expresses the effective coupling between the initial and final states. A larger matrix-element magnitude generally increases the rate quadratically. The density of final states measures how many possible destination states lie near the energy allowed by the process; a larger density can also increase the rate. Selection rules or symmetries can make a matrix element vanish, suppressing a transition even when final states are available. Energy conservation appears through the requirement that the final states contribute at the appropriate energy.

Fermi's golden rule is used across quantum physics, including calculations of atomic and molecular transitions, scattering, radiation processes and decay. Its usefulness comes from reducing a complicated time evolution to a rate under a specified interaction and set of states. It is an approximation, not a universal law: the perturbation is ordinarily assumed weak, the relevant final states are often treated as a continuum, and the rate description is useful over times for which the transition probability remains in the appropriate regime. Strong coupling, very small or structured sets of final states, or substantial changes to the system over time can require a different treatment or a more careful generalization.

The expression is associated with Enrico Fermi, who called it 'Golden Rule No. 2' in lectures on nuclear physics. The underlying derivation has a wider history in early quantum theory: later historical analyses trace an early derivation to Paul Dirac's work on the emission and absorption of radiation, before the terminology became established. The familiar name thus refers to an especially useful result within perturbation theory rather than to a fundamental law independent of assumptions.