Correlated continuum wave functions for three particles with Coulomb interactions

Abstract

We present an approximate solution of the Schr\"odinger equation for the three-body Coulomb problem. We write the Hamiltonian in parabolic curvilinear coordinates and study the possible separation of the wave equation as a system of coupled partial differential equations. When two of the particles are heavier than the others, we write an approximate wave equation that incorporates some terms of the Hamiltonian that before had been considered as a perturbation. Its solution can be expressed in terms of a confluent hypergeometric function of two variables. We show that the proposed wave function includes a correlation between the motion of the light particle relative to the heavy particles and verifies the correct asymptotic behavior when all particles are far from each other. Finally, we discuss the possible uses of this function in the calculation of transition matrices and differential cross sections in ionizing collisions.