Time evolution of unstable quantum states and a resolution of Zeno's paradox

Abstract

The time evolution of quantum states for unstable particles can be conveniently divided into three domains: the very short time where Zeno's paradox is relevant, the intermediate interval where the exponential decay holds more or less, and the very long time where the decay is governed by a power law. In this work, we reexamine several questions relating to the deviations from the simple exponential decay law. On the basis of general considerations, we demonstrate that deviations from exponential decay near $t=0$ are inevitable. We formulate general resonance models for the decay. From analytic solutions to specific narrow-width models, we estimate the time parameters ${T}_{1}$ and ${T}_{2}$ separating the three domains. The parameter ${T}_{1}$ is found to be much much less than the lifetime ${\ensuremath{\Gamma}}^{\ensuremath{-}1}$, while ${T}_{2}$ is much greater than the lifetime. For instance, for the charged pion decay, ${T}_{1}\ensuremath{\sim}\frac{{10}^{\ensuremath{-}14}}{\ensuremath{\Gamma}}$ and ${T}_{2}\ensuremath{\sim}\frac{190}{\ensuremath{\Gamma}}$. A resolution of Zeno's paradox provided by the present consideration and its limitaions are discussed.