Multichannel decay: alternative derivation of the $i$-th channel decay probability

TL;DR

This paper presents a new detailed derivation of the decay probability functions for unstable quantum states with multiple decay channels, highlighting their non-exponential behavior and potential for new insights.

Contribution

It provides a novel joint derivation of survival and channel decay probabilities, emphasizing their non-exponential nature and ratios as tools for decay law analysis.

Findings

Decay functions are non-exponential.

Ratios of decay probabilities are not constant.

New methods to study decay laws.

Abstract

In the study of decays, it is quite common that an unstable quantum state/particle has multiple distinct decay channels. In this case, besides the survival probability $p(t)$, also the probability $w_{i}(t)$ that the decay occurs between $(0,t)$ in the $i$-th channel is a relevant object. The general form of the function $w_{i}(t)$ was recently presented in PLB \textbf{831} (2022), 137200. Here, we provide a novel and detailed `joint' derivation of both $p(t)$ and $w_{i}(t)$. As it is well known, $p(t)$ is not an exponential function; similarly, $w_{i}(t)$ is also not such. In particular, the ratio $w_{i}/w_{j}$ (for $i\neq j)$ is not a simple constant, as it would be in the exponential limit. The functions $w_{i}(t)$ and their mutual ratios may therefore represent a novel tool to study the non-exponential nature of the decay law.