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

Abstract

We present a systematic account of the separation of the angular degrees of freedom from the nonrelativistic Schrödinger equation for a three-body quantum system with arbitrary masses, charges, total angular momentum, and parity. The resulting reduced Schrödinger equation (RSE) for the partial-wave components, expressed as functions solely of the interparticle distances, is reported in a compact matrix operator form. The remnants of the angular dependence, essential for the hermiticity of the RSE and consequently the stability of variational computations, appear in the RSE formalism as additional angular factors, derived by expanding minimal bipolar harmonics into a basis of Wigner functions \texorpdfstring{$\mathcal{D}$}{D}. We validate the final form of the RSE by computing accurate energy levels for helium states using an explicitly correlated Hylleraas-type basis. This work serves as a self-contained reference for the RSE formulation, consolidating elements previously scattered throughout the literature, thereby offering a convenient foundation for further analytical and numerical studies of general three-body quantum systems.