New Perspectives on the so-called Fermi's Golden Rule in Quantum Mechanics including Adiabatic Following

TL;DR

This paper presents a new, intuitive derivation of Fermi's Golden Rule using adiabatic switching, emphasizing energy conservation with Lorentzian line shapes and generalizing to slowly varying perturbations.

Contribution

It introduces a novel derivation of Fermi's Golden Rule based on adiabatic switching and Lorentzian line shapes, extending applicability to general slowly varying perturbations.

Findings

Derivation based on adiabatic switching with Lorentzian line shape.

Transition rate follows the square of the perturbation instantaneously.

Generalization to slowly varying time-dependent perturbations.

Abstract

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.