An analytical solution of the Schrodinger equation for the neutral ground state Para Helium

Abstract

This report presents the analytical solution of the Schrodinger equation and its corresponding wave function for the neutral para-helium or para-helium-like atoms in the ground state. The state functions of the two electrons for s=0 and l=0 as well as their boundary conditions are examined in detail. Furthermore, a method for describing a generic electron potential consisting of Coulomb and exchange interactions is derived, and the resulting potential function is integrated into the Schrodinger equation as a potential term. In addition, the altered electromagnetic coupling of the electrons due to vacuum polarization effects is investigated and finally the Schrodinger equation for the neutral Para-Helium is solved using Laplace transformations. The energy in the ground state is then determined , and it can be shown that this agrees with the literature values given the fact that the electron can be assumed to be a point-like particle. In the context of these investigations, an upper limit estimation for the spatial dimension of the electron can also be given as well as the existence of a minimal distance of a stable bonding state between two electrons, which can be interpreted as an entangled state; in addition, the chemical inertness of helium with regard to chemical reactions-i.e. the principle of the "closed" electron shell-can be made plausible by the quantum mechanical electron configuration and its consequences with regard to binding energy. The wave function found for the helium atom is compared with the known solutions for the hydrogen atom, and essential differences between the two are worked out.