Adaptive numerical methods for time-dependent Schrodinger equation in atomic and laser physics

Abstract

Stable adaptive methods for solving the time-dependent Schrodinger equation (TDSE)) are considered in the framework of conventional finite-element representation of smooth solutions over coordinate spaces of a projective type with long derivatives. Generalization of Cranck-Nicholson scheme of forth order in time step is implemented. Projective “hidden variable” representation of strongly oscillating solutions is realized to extract explicitly the strongly variable gauge phase factor and to evaluate only the “pilot solution” which is reduced to a smooth envelope of the solution under consideration. Such an approach corresponds to the known transformation from Euler space variables to Lagrangian ones and the inducing characteristic representation of self-similar solutions widely used in the flow propagation problems. We study both smooth and strongly oscillating solutions of TDSE describing conventional atomic models in the laser pulse field. It is shown that for short-range potentials the “pilot solution” can be naturally interpreted as the spectrum of the outgoing wave. The examples considered show the efficiency and stability of the elaborated methods.