Analytically exact solution of the Schrödinger equation for neutral helium in the ground state

This report presents the analytical solution of the Schrödinger equation and its corresponding wave function for the neutral helium, or helium-like atoms and the review of its energy levels based on literature values of the three congruent S states 1𝑆0,2𝑆0 and 3𝑆0. The entangled state function of the electrons for S,L=0 as well as their boundary conditions are examined in detail.

The basic idea is – though we treat the electron as a point like particle - not to understand its field as caused by a point charge with the rigid Coulomb field as in common approaches, but to solve the Schrödinger equation with an electric potential taking eisenberg’s uncertainty principle into account as a fundamental requirement.

Hence, a method for describing a generic electron potential is derived, and the result is integrated into the Schrödinger equation. As a direct consequence the electromagnetic coupling of the electrons was investigated by introducing an effective interaction distance 𝑑e to implement quantum electrodynamical effects.

In the next step the Schrödinger equation is solved using Laplace transformations. After determining 𝑑e iteratively from the literature ground state energy, the energies for three states 1𝑆0, 2𝑆0 and 3𝑆0 were calculated in reverse and compared again with literature values. It could be shown that with a given value 𝑑e the backward calculation gives plausible results of all three energy states within a relative error between 7.5 ∗ 10e-16 and 7.6 ∗ 10e-6, thus providing us with a convincing method to describe the helium atom analytically.

In the context of these investigations, a calculation for the spatial dimension of the interaction distance 𝑑e can be given as well as the existence of a minimal distance of a stable quasi-bonding state between two electrons in the nucleonic field. As a result, the inertness of helium regarding chemical reactions, i.e., the principle of the "closed" electron shell can be made plausible. The wave function found for the helium atom is compared with the known solutions for the hydrogen atom and Hylleraas function as well, and essential differences between those are worked out.