Improved boundary conditions for the decay of low lying metastable proton states in a time-dependent approach Original Research Article

Artificial boundary conditions have to be imposed on the numerical solution of the time-dependent Schrödinger equation in order to study the decay dynamics of low-lying unbound proton states. This procedure eliminates the reflections of the wave packet at the numerical boundaries. For large numerical grids this happens from the beginning. For smaller grids, these reflections are first reduced and then disappear after few oscillations. At that moment the asymptotic decay rate is reached. Due to inevitable numerical error the asymptotic decay rate depends slightly on the size of the spatial grid and attains a plateau for large grids. The convergence can be considerably improved through a modification of the transparent boundary conditions that accounts for the peculiarity of our problem: nonnegligible Coulomb and centrifugal tails beyond the numerical grid.The impact of the reflections on the time-dependent decay rate can be reduced by replacing, according to the continuity equation, the time derivative of the tunneling probability by the flux at the outer turning point. These improvements allow smaller spatial grids to be used that are also independent of the duration of time evolution. New perspectives for calculating deep tunneling of a proton through a two-dimensional time-dependent barrier are forseen.