Survival amplitude, instantaneous energy and decay rate of an unstable system: Analytical results

TL;DR

This paper analytically investigates the late-time behavior of unstable quantum states, revealing that their energy and decay rate tend to minimal values over time, challenging exponential decay assumptions.

Contribution

It provides explicit analytical expressions for the survival amplitude, energy, and decay rate at late times, highlighting non-exponential decay effects in unstable systems.

Findings

Survival amplitude exhibits non-exponential decay at late times.

Energy approaches the system's minimal energy as time progresses.

Decay rate tends to zero at late times, indicating slower decay than exponential models.

Abstract

We consider a model of a unstable state defined by the truncated Breit-Wigner energy density distribution function. An analytical form of the survival amplitude $a(t)$ of the state considered is found. Our attention is focused on the late time properties of $a(t)$ and on effects generated by the non--exponential behavior of this amplitude in the late time region: In 1957 Khalfin proved that this amplitude tends to zero as $t$ goes to the infinity more slowly than any exponential function of $t$. This effect can be described using a time-dependent decay rate $\gamma(t)$ and then the Khalfin result means that this $\gamma(t)$ is not a constant but at late times it tends to zero as $t$ goes to the infinity. It appears that the energy $E(t)$ of the unstable state behaves similarly: It tends to the minimal energy $E_{min}$ of the system as $t \to \infty$. Within the model considered we find two first leading time dependent elements of late time asymptotic expansions of $E(t)$ and $\gamma (t)$. We discuss also possible implications of such a late time asymptotic properties of $E(t)$ and $\gamma (t)$ and cases where these properties may manifest themselves.