On the violation of the exponential decay law in atomic physics: ab initio calculation of the time-dependence of the non-stationary state

Abstract

The detailed time dependence of the decay of a three-electron autoionizing state close to threshold has been obtained ab initio by solving the time-dependent Schrödinger equation (TDSE). The theory allows the definition and computation of energy-dependent matrix elements in terms of the appropriate N-electron wavefunctions, representing the localized initial state, , the stationary scattering states of the continuous spectrum, , and the localized excited states, , of the effective Hamiltonian QHQ, where . The time-dependent wavefunction is expanded over these states and the resulting coupled equations with time-dependent coefficients (in the thousands) are solved to all orders by a Taylor series expansion technique. Convergence is checked as a function of the number of the numerically obtained that span the continuous spectrum of the free electron. The robustness of the method was verified by using a model interaction in analytic form and comparing the results from two different methods for integrating the TDSE (appendix B). For the physically relevant application, the chosen state was the shape resonance, about which very accurate theoretical and experimental relevant information exists. Calculations using accurate wavefunctions and an energy grid of 20.000 points in the range 0.0 - 21.77 eV show that the effective interaction depends on energy in a state-specific manner, thereby leading to state-specific characteristics of non-exponential decay (NED). For the established energy position of 0.01 eV, the results show an exponential decay over about au of time, from which a width of meV and a lifetime of s is deduced. The experimentally obtained width is 7.16 meV (Walter, Seifert and Peterson 1994 Phys. Rev. A 50 664). After 12 lifetimes (about 1400 fs), at which time the survival probability is , NED sets in. On the other hand, due to the shape of the interaction, the NED appears at earlier times if the energy position happened to be slightly larger. For example, if E were at 0.019 eV, NED would start after nine exponential lifetimes. These facts suggest that either in this state or in other autoionizing states close to threshold, NED may have sufficient presence to make the violation of the law of exponential decay observable.