Analytical solution of the Schr\"odinger equation for the neutral helium atom in the ground state considering the uncertainty principle and quantum-electrodynamical effects

This report presents, presumably for the first time, the \textit{ab initio} analytical solution of the Schr\"odinger equation for neutral helium and helium-like atoms, reproducing its energy levels of the first three congruent S states. The entangled wave function of the electrons for $S = L = 0$ is examined in detail. Although the electron is treated as point-like, its field is not considered as caused by a point charge, as in common approaches, but instead respects Heisenberg's uncertainty principle, solving the Schr\"odinger equation with an adapted potential. Hence, a method for describing a generic electron potential is derived, and the result is integrated into the Schr\"odinger equation. Quantum-electrodynamical coupling effects of both electrons are identified as significantly influencing the ground energy of helium and are investigated by introducing an effective interaction length. The complete causality chain is derived via \textit{ab initio} calculations and verified with literature values. The Schr\"odinger equation is solved using Laplace transformations, and the energies for the first three S states are calculated. Accurate results for all three energy states are achieved within a relative error of $7.5 \times 10^{-16}$ to $7.6 \times 10^{-6}$ compared with literature values, thus providing a powerful method to describe helium analytically. These investigations deduce the existence of a stable quasi-bonding state between two electrons in a nucleonic field. This explains the inertness of helium in chemical reactions, i.e., the principle of the closed shell, and allows the spatial structure of helium to be calculated, yielding valid results consistent with literature values. Finally, the wave functions for helium are compared with the hydrogen solutions and Hylleraas' function as well.