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American Journal of Physics

Fermi's Golden Rule—An Exercise in Quantum Field Theory

G E Stedman

Published Feb 1, 1971• 11 citations
Publisher

Abstract

The golden rule of time dependent perturbation theory is derived using the formalism of quantum field theory in statistical physics. The proof is more satisfying than that of some conventional presentations and is a simple illustration of the use of the quantum field theoretical formalism (e.g., Green's functions, linked cluster expansions, and Dyson's equation) in the context of solid state theory.

DOI
10.1119/1.1986093

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The golden rule of time dependent perturbation theory is derived using the formalism of quantum field theory in statistical physics. The proof is more satisfying than that of some conventional presentations and is a simple illustration of the use of the quantum field theoretical formalism (e.g., Green's functions, linked cluster expansions, and Dyson's equation) in the context of solid state theory.
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.
Abstract Chapter 14 covers the statistical mechanics of driven quantum systems and the non-equilibrium ensembles associated with them. The topic is presented both from the state-vector and density matrix perspectives. The chapter begins with a discussion of driving forces resulting from light-matter interactions and their treatment via time-dependent perturbation theory. The Fermi Golden Rule is derived, and it is shown how to incorporate the Golden Rule into the calculation of frequency spectra within the canonical ensemble, which then connects to canonical quantum time correlation functions. The formalism is derived again from the quantum Liouville equation. The discussion then turns to the challenge of computing quantum time correlation functions and approximate techniques based on the Feynman path integral of Chapter 12. These schemes include centroid and ring-polymer molecular dynamics as well as open-path integral methods
Diagrammatical expansion of the quantum field theory with the finite time interval is discussed by evaluating the time evolution kernel. The coherent state representation is adopted which is the most convenient formalism for this purpose. It is applied to quantum electrodynamics (QED) to derive the Feynman rule for the kernel. The rule for the matrix element of any operator is also given. Our method can also be a systematic approach to the time dependent perturbation theory. As an example, a full order formula of the transition amplitude between the number states is given in terms of the connected Green's functions.
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.
Equivalent derivations of time-dependent pertubation theory; Fermi golden rule; the Born approximation for scattering from Fermi golden rule; survival probability of a state in a time-independent perturbation; positronium in static and oscillating magnetic fields; hydrogen atom in a time-dependent electric field; a model for inelastic scattering of a projectile with a target; semiclassical treatment of the electromagnetic field; ionization of the hydrogen atom by an electromagnetic wave; cross sections for stimulated absorption and emission in hydrogen; spontaneous emission and selection rules with an application to the 2p to 1s transition in hydrogen; theory of the line width; formal scattering theory; S- and T-operators.