Fermi's golden rule: its derivation and breakdown by an ideal model

TL;DR

This paper introduces an ideal model to derive Fermi's golden rule analytically, illustrating its core features and the conditions under which it breaks down, especially beyond the Heisenberg time, even within first-order perturbation theory.

Contribution

It presents an idealized model for a clear derivation of Fermi's golden rule and demonstrates its limitations beyond the Heisenberg time.

Findings

Transition probability is piecewise linear in time.

Fermi's golden rule accurately describes short-time dynamics.

The rule breaks down beyond the Heisenberg time despite valid first-order approximation.

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 $-\infty $ to $+\infty $, 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 \emph{piecewise linear} function of time, demonstrating on one hand the key features of Fermi's golden rule, and on the other hand that the rule breaks down beyond the \emph{Heisenberg time}, even when the first order perturbation approximation itself is still valid.