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European Journal of Physics

Teaching Fermi's golden rule without a continuum

Benedict N Murdin, Carter Matthews

Published Oct 28, 2026• 0 citations

Abstract

Abstract Fermi's Golden Rule is one of the most important results in quantum mechanics, predicting transition rates in applications ranging from beta decay to optical transitions, autoionization and photoionization. Most textbooks derive the Golden Rule only for the canonical case of a discrete state coupled by a time-independent perturbation to a continuum of final states. Here we present a unified treatment of several physical situations, including light absorption in atoms and solids, that all lead to the same Golden Rule expression. The essential ingredients are the transition matrix element and a spectral weighting function, whose physical origin need not be the density of final states, nor necessarily anything to do with the matter.The conventional discrete-to-continuum derivation introduces pedagogical complications associated with continuum normalization and the interpretation of the density of states. Among the cases considered here, the derivation for discrete transitions driven by a noisy perturbation is no more difficult mathematically but conceptually clearer, while leading directly to practical calculations. This suggests that it provides a more natural first introduction to the Golden Rule, with conventional continuum derivations better viewed later as another realization of the same underlying mathematical structure, rather than the other way around.

DOI: 10.1088/1361-6404/aebac8

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Abstract Fermi’s golden rule is of great importance in quantum dynamics. However, in many textbooks on quantum mechanics, its contents and limitations are obscured by the approximations and arguments in the derivation, which are inevitable because of the generic setting considered. Here we propose to introduce it by an ideal model, in which the quasi-continuum band consists of equaldistant levels extending from − ∞ to + ∞ , and each of them couples to the discrete level with the same strength. For this model, the transition probability in the first order perturbation approximation can be calculated analytically by invoking the Poisson summation formula. It turns out to be a piecewise linear function of time, demonstrating on the one hand the key features of Fermi’s golden rule, and on the other hand that the rule breaks down beyond the Heisenberg time , even when the first order perturbation approximation itself is still valid.

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Abstract The Golden Rule is a colloquial name for a standard formula in time-dependent quantum mechanics. It provides a way to calculate the rate of transition between quantum states. It is approximate and has limited applicability; however, it is the starting point for many treatments of rate processes. Later it will be used in the discussion of master equations. This section gives the usual derivation of the Golden Rule. The following section shows how it is used to relate optical absorption spectra to time correlation functions.

A novel and readily understandable derivation of the Golden Rule of time dependent perturbation theory is presented. The derivation is based on adiabatic turning on of the perturbation as used, for instance, in some formal developments of scattering theory. Energy conservation is expressed in terms of an intuitively and physically appealing Lorentzian line shape rather than the artificial, oscillatory sin(x)=x type line shape that appears in conventional derivations. The conditions for the derivation's validity are compactly and conveniently expressed in the frequency/energy domain rather than in the usual time domain. The derivation also highlights how, along with energy conservation, the transition rate faithfully and instantaneously follows the variations in the square of the perturbing potential as one may expect in the adiabatic limit. In the first instance, the adiabatic turning on is achieved, as usual, by a single exponential time variation. But we demonstrate that the instantaneous following of the square of the perturbing potential by the transition rate is more general and that one can derive the Golden Rule for a general slowly varying time dependent perturbation. This allows one to derive generalisations of the simple decay law, originally derived in the classic paper by Weisskopf and Wigner; a tutorial exposition of the essence of this classic work is provided. The Oppenheimer method for applying the Golden rule to problems, such as the electric field ionisation of atoms, in which the perturbing potential can also create the final states, is reviewed. No use of an energy gap condition is needed to derive our results on adiabatic behaviour in contrast to the origianl derivations of the adiabatic theorem in quantum mechanics.

A time-dependent perturbation theory which is based upon the U-matrix approach is presented using the interaction e-αt(1 + e-βt)−1V(x) with β real, α complex and β > Re α > 0. Thus the adiabatic or time-independent approximation (β → 0) and the sudden approximation (β → ∞) can be obtained using just one formalism. The usual series for the U matrix is derived and shown to converge for a semi infinite range if the interaction is of the form |t|-δ, δ > 1, for large times t. Two interesting results are (1) The derivation of the ``golden rule'' from the discrete-state time-dependent perturbation theory presented in most text books of quantum mechanics leads to erroneous physical interpretations because the energy-level shift is ignored. (2) When considering scattering between states in the continuum, it is found that a characteristic feature of time-dependent interactions is a discrete momentum spectrum of final states which, in the relativistic case, leads to a mass spectrum. These spectra cannot be obtained using the S matrix.

A simple analysis of the bound-free system shows that for a narrow continuum the Fermi golden rule can be generalised for the high-intensity regime. This generalisation has the form of a simple expression for the decay rate and helps to explain nearly perfect population trapping in the case of a narrowly peaked matrix element. Such population trapping means that the decay rate of a bound state strongly coupled to a continuum tends to zero for increasing coupling strength only if the matrix element falls off faster than the Lorentzian. The authors also find new solutions for pulsed excitation giving exact population trapping in the bound-free transition.