Elimination of angular dependency in quantum three-body problem made easy

TL;DR

This paper introduces a systematic method to eliminate angular dependency from the Schrödinger equation in three-body quantum systems, simplifying calculations of energy levels and wave functions.

Contribution

It provides a compact matrix form of the reduced Schrödinger equation and a variational approach, validated by accurate helium atom energy calculations.

Findings

Derived angular integrals using bipolar harmonics and Wigner functions.

Validated the approach with helium atom energy levels.

Provided a self-contained reference for three-particle quantum systems.

Abstract

This work presents a systematic account of elimination of angular dependency from nonrelativistic Schr\"odinger equation for a three-body quantum system with arbitrary masses, charges, angular momentum, and parity. The resulting reduced Schr\"odinger equation (RSE) for the reduced wave components, corresponding to the basis of solid bipolar harmonics, is presented in a compact matrix operator form. The variational form of RSE, providing a practical tool for calculating energy levels and wave functions, is also derived. The resulting angular integrals were derived by expanding bipolar harmonics in a basis of parity-adapted Wigner functions. The theoretical results are numerically validated by computing accurate energy levels for selected states of the helium atom in the explicitly correlated Hylleraas-type basis. The work aims to serve as a self-contained reference for the previously scattered throughout the scientific literature formulation of RSE, offering a convenient foundation for further analytical studies of three-particle quantum systems with arbitrary angular momentum and parity.