A 2D Formulation for the Helium Atom Using a Four-Spinor Dirac-like Equation and the Discussion of an Approximate Ground State Solution
D. L. Nascimento *
Institute of Physics and International Center of Condensed Matter, University of Brasília, 70919-970, Brasília, DF, Brazil
A. L. A. Fonseca
Institute of Physics and International Center of Condensed Matter, University of Brasília, 70919-970, Brasília, DF, Brazil
*Author to whom correspondence should be addressed.
Abstract
We present a two-dimensional analysis of the two-electron problem which comes from the classical conservation theorems and from which we obtain a version of the Dirac equation for the helium atom. Approximate solutions for this equation are discussed in two different methods, although in principle it can be solved analytically. One method is variational, of the Hylleraas type, the execution of which is left for a later communication. In contrast, the other method will have a more complete treatment, in which the set of equations will be separated into its angular and radial components. Furthermore, an exact solution for the angular component will be displayed as well as an approximate solution for the radial component, valid only for the fundamental state of the atom.
Keywords: Helium atom, Dirac equation, relativistic quantum mechanics, variational methods, Hylleraas method, semi analytic solutions