Correlated continuum wave functions for three particles with Coulomb interactions

Abstract

We present an approximate solution of the Schrödinger 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. ͓S1050-2947͑97͒00504-0͔

Computation of a generalized Nordsieck integral

“…[3]. It has been shown in the same reference, that in some cases the coefficients b and c may, for instance, adopt the form b = 1 + 2m and c = 1 + 2n, the one observed by the Φ 2 model [6]. For those cases, the function m,n (a, b, c; x, y) can be associated with the two-body Coulomb wave function where the magnetic number is different from zero.…”

“…(6). The code was designed to integrate functions that are complicated only in a region of space by increasing the number of points taken in that interval.…”

“…The integral in Eq. (6) can be reduced to a finite sum of generalized Nordsieck integrals which were studied by Colavecchia [8].…”

Double ionization of helium by bare ions: Theoretical study of the fully differential cross sections

“…(iii) The dynamical screening (DS) models: Many studies were performed during the 1990s with the aim of improving the 3C model [31][32][33][34][35]. All these studies had the common purpose of obtaining a more accurate analytical wave function for the three-body Coulomb problem.…”

The hyperbolic–hypergeometric functions

“…[3][4][5] On the other hand, different approximated solutions for the three-body Coulomb problem are written in terms of hypergeometric functions. [6][7][8][9] The approximate solution known as C3 is given by the product of three two-body Coulomb problems and in this way its dynamical properties depend on the Kummer function properties. Recently, the authors and co-workers introduced a new approximate solution for the three-body problem that can be written in terms of the degenerate Appell functions called ⌽ 2 by Horn.…”