Stationarity coefficients and short-time deviations from exponential decay in atomic resonance states

Abstract

By solving rigorously and accurately the time-dependent Schr\"odinger equation, we have obtained numerical results for the decay probability, $P(t),$ of real, multiparticle systems, in the time domain of $t\ensuremath{\approx}0.$ Three different types of atomic nonstationary states were examined, the ${\mathrm{He}}^{\ensuremath{-}}{1s2p}^{2}{}^{4}P,$ the Ca KLM $3d5p{}^{3}{F}^{o},$ and the ${\mathrm{He}}^{\mathrm{\ensuremath{-}}}$ $1s2s2p{}^{4}{P}_{5/2},$ the last one being metastable and decaying via spin-spin interactions. The main results are that there is a ${t}^{2}$ dependence of $P(t\ensuremath{\approx}0)$ and that a time-dependent short-time decay rate can be calculated. The computed coefficients of the ${t}^{2}$ term reflect the degree of stability of the state, (i.e., the degree of proximity to the notion of the standard stationary state of quantum mechanics), and are named the stationarity coefficients. These, together with the conventional quantity of the lifetime, corresponding to the exponential decay regime, constitute intrinsic properties of each real unstable state. For the herein studied metastable state the onset of exponential decay occurs after about $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}14}\mathrm{s},$ i.e., after a duration which is achievable experimentally with laser pulses.