Quantum Three-Body System: A Century-Old Problem Revisited

¹ Department of Applied Chemistry, National Yang Ming Chiao Tung University, Hsinchu City 30010, Taiwan (ROC)
² Faculty of Engineering and Technology, State Academy of Applied Sciences in Włocławek, Włocławek 87-800, Poland
³ Institute of Chemistry, University of Silesia, Szkolna 9, 40-006 Katowice, Poland

Academic Editor: Pascal Quinet

Published: 27 January 2026 by MDPI in The 1st International Online Conference on Atoms session Atomic structure and spectra: Theory and experiment

Abstract:

Introduction

The non-relativistic quantum three-body problem, despite its century-long history, continues to challenge even experienced theorists due to the intricate coupling between angular and radial degrees of freedom. A systematic and pedagogically transparent route for eliminating angular dependence from the three-body Schrodinger equation (SE) is presented, following the spirit of the exposition by Bhatia and Temkin [1,2]. This leads us to the Reduced Schrodinger Equation (RSE)—a matrix-operator formulation that fully retains the generality of arbitrary particle masses, charges, total angular momentum (L), and parity.

Methods

The derivation beginning with the elimination of center-of-mass motion, followed by an analysis of the rotational invariance of the Hamiltonian. The angular basis is constructed from solid minimal bipolar harmonics (MBHs), providing a natural way to preserve the individual partial angular momenta of the constituent particles. The resulting RSE offers immediate insight into the coupling between different partial waves. The variational counterpart is derived, enabling accurate computation of bound-state energies and wave functions. The angular integrals are evaluated through a novel technique that expresses MBHs as linear combinations of Wigner D-functions.

Results

Numerical validation is provided for the low-lying singlet and triplet states of the helium atom—both of natural parity (for L<= 7) and unnatural parity (for L<=4). Explicitly correlated multi-exponent Hylleraas-type bases are employed within the framework of the Rayleigh–Ritz variational principle.

Conclusion

This work aims to unify and clarify results that have long been scattered across the literature, offering a self-contained and conceptually cohesive framework. The formalism not only simplifies the angular reduction process for three-body systems but also provides a transparent foundation for extending such treatments to many-body quantum systems.

References

[1] A. K. Bhatia and A. Temkin. Rev. Mod. Phys., 36:1050–1064 (1964).
[2] A. K. Bhatia and A. Temkin. Phys. Rev., 137:A1335–A1343, (1965).

Keywords: Three-body problem; Bipolar harmonics; Angular momentum coupling; Variational methods.

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