The Gamow and the Fermi Golden Rules

TL;DR

This paper derives a unified Gamow Golden Rule for quantum resonances, connecting decay widths, energy and angular distributions, and natural phase space factors, applicable to both short and long-lived states.

Contribution

It introduces a new formulation of the Gamow Golden Rule that unifies decay descriptions and naturally incorporates phase space factors, extending the traditional Fermi Golden Rule.

Findings

Derivation of differential and total decay widths using Gamow states

Unification of Gamow states with the Golden Rule for quantum resonances

Natural emergence of density of states and phase space factors

Abstract

By using the fact that the Gamow states in the momentum representation are square integrable, we obtain the differential and the total decay width of a two-body, non-relativistic decay. The resulting Gamow Golden Rule is well suited to describe both energy and angular decay distributions, and it becomes the Fermi Golden Rule when the resonance is long-lived and far from the energy threshold. We also show that the correct density of states and phase space factors arise naturally from the Gamow Golden Rule. The upshot is that the Gamow states and the Golden Rule can be combined into a unified description of quantum resonances.