Time dependence and properties of nonstationary states in the continuous spectrum of atoms

Abstract

The recently measured shape resonance, the Ca KLM 3d5p doubly excited autoionizing state and the long-lived 1s2s2p metastable level were treated as nonstationary states satisfying the time-dependent Schrödinger equation (TDSE). The lifetimes of the first two are short, of the order of s, and the solution of the TDSE well into times where nonexponential decay (NED) is established, is achievable via the state-specific expansion approach (SSEA), according to which the time-dependent solution has the form . is the square-integrable wavefunction of the localized state at t = 0 and X(t) is composed mainly of energy normalized scattering functions with time-dependent coefficients. The coefficient c(t) is related to the survival amplitude, , by , where the overlap matrix element appears when the function spaces are not completely orthonormal. For the diffuse resonance, its analysis as a decaying state has as a prerequisite the calculation of a reliable , with correlation between the two valence electrons. This has been achieved by a special procedure and a related discussion is given. The proximity of the energy E to threshold ( meV), the closeness of the ratio to unity ( is the resonance width) and the energy dependence of the bound - free matrix element, produced the result that NED should appear after only two lifetimes, when the probability of finding the system in the initial state is still non-negligible. From the exponential part of the decay curve, the width was found to be meV, in agreement with the recent width of meV derived from measured cross sections in recent collision experiments (Lee et al 1996). The shortness of the time for which exponential decay (ED) holds and the fact that the survival probability, P(t), is still significant at the beginning of the NED, does not allow the rigorous justification of the definition of the lifetime from , or the equivalence of this with the observed energy width. Thus, we propose that a mean life, , should be obtained from Calculation produces s and s. For the Ca state, whose bound - free interaction is smooth and nearly constant from zero to about 5.5 eV, NED appears after 17 lifetimes. The lifetime of Ca is deduced from the exponential decay (ED) part of P(t) to be s. From our examination of the case of the level by a number of methods based on the use of state-specific wavefunctions, we conclude that for metastable states whose lifetimes are in the range - s, the ab initio calculation of P(t) is, at present, prohibited by the huge requirements for computer time. Finally, having computed the amplitude , we obtain numerically the energy distribution function, , of the two autoionizing states. In the case of Ca it is a perfect Lorentzian.