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

TL;DR

This work tackles the problem of removing angular dependencies in the nonrelativistic three-body Schrödinger equation for general Coulomb systems. It develops a parity-adapted MBH-based framework that yields a matrix-operator reduced Schrödinger equation (RSE) and a practical variational form, enabling efficient energy and wave-function calculations. The authors derive explicit angular integrals by expanding MBHs in parity-adapted Wigner functions, present state-specific RSEs for S^e, P^e, P^o, and D^o cases, and validate the approach by computing accurate helium energies with a Hylleraas-type basis, including finite nuclear mass effects. The results establish a self-contained reference that clarifies the RSE’s algebraic structure and lays a solid foundation for analytical and numerical studies of three-body Coulomb systems with arbitrary angular momentum and parity.

Abstract

This work presents a systematic account of elimination of angular dependency from nonrelativistic Schrödinger equation for a three-body quantum system with arbitrary masses, charges, angular momentum, and parity. The resulting reduced Schrödinger 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.

Figures (2)

• Figure 1: (a) Graphical illustration of the position $\bm{R}$ of the center of mass (CM) and the positions $\overline{\bm{r}}_1$, $\overline{\bm{r}}_2$ and $\overline{\bm{r}}_3$ of the three original particles with respect to the origin of the laboratory-fixed reference frame ($\overline{x}\,\overline{y}\,\overline{z}$). (b) Graphical illustration of the positions $\bm{r}_{1}$ and $\bm{r}_{2}$ of the two emerging quasiparticles with respect to the new laboratory-fixed reference frame $(x\,y\,z)$, obtained from ($\overline{x}\,\overline{y}\,\overline{z}$) by translating it by $\overline{\bm{r}}_3$.
• Figure 2: The orientation of the body-fixed $(X\,Y\,Z)$ reference is determined by two non-axial vectors $\bm{r}_{1}$ and $\bm{r}_{2}$ lying on the plane $XY$. The origin of the frame is located at the position of the third particle. The $X$-axis coincides with the bisector of the angle $\theta = \arccos\frac{\bm{r}_{1}\!\boldsymbol{\cdot}\bm{r}_{2}}{r_{1}r_{2}}$, while the $Z$ axis coincides with the vector $\bm{r}_{1}\times\bm{r}_{2}$.