Harmonic oscillator model for the helium atom

TL;DR

This paper introduces a four-dimensional harmonic oscillator model for the helium atom that estimates electron positions, angles, and energy levels using an algebraic approach without relying on trial wave functions.

Contribution

It presents a novel algebraic harmonic oscillator model with three quantum numbers to describe the two-electron helium system, deriving formulas for energy and geometric configurations.

Findings

The model estimates electron distances and angles accurately.

Energy levels converge towards ionization thresholds or extreme states.

Provides an estimate for the helium ground state energy.

Abstract

A harmonic oscillator model in four dimensions is presented for the helium atom to estimate the distance to the inner and outer electron from the nucleus, the angle between electrons and the energy levels. The method is algebraic and is not based on the choice of correct trial wave function. Three harmonic oscillators and thus three quantum numbers are sufficient to describe the two-electron system. We derive a simple formula for the energy in the general case and in the special case of the Wannier Ridge. For a set of quantum numbers the distance to the electrons and the angle between the electrons are uniquely determined as the intersection between three surfaces. We show that the excited states converge either towards ionization thresholds or towards extreme parallel or antiparallel states and provide an estimate of the ground state energy.