The harmonic hyperspherical basis for identical particles without permutational symmetry

TL;DR

This paper introduces a hyperspherical harmonic basis approach for describing bound states in multi-particle systems without initial symmetry constraints, simplifying potential energy calculations and allowing symmetry identification after diagonalization.

Contribution

The method provides a non-symmetrized basis for multi-particle systems that simplifies potential matrix elements and enables symmetry analysis post-diagonalization, useful for systems with symmetry-breaking terms.

Findings

Successfully applied to three- and four-particle systems with short-range and Coulomb interactions.

Potential energy matrix elements are simple and compact, facilitating numerical implementation.

Eigenvectors after diagonalization reveal the physical states' symmetry properties.

Abstract

The hyperspherical harmonic basis is used to describe bound states in an $A$--body system. The approach presented here is based on the representation of the potential energy in terms of hyperspherical harmonic functions. Using this representation, the matrix elements between the basis elements are simple, and the potential energy is presented in a compact form, well suited for numerical implementation. The basis is neither symmetrized nor antisymmetrized, as required in the case of identical particles; however, after the diagonalization of the Hamiltonian matrix, the eigenvectors reflect the symmetries present in it, and the identification of the physical states is possible, as it will be shown in specific cases. We have in mind applications to atomic, molecular, and nuclear few-body systems in which symmetry breaking terms are present in the Hamiltonian; their inclusion is straightforward in the present method. As an example we solve the case of three and four particles interacting through a short-range central interaction and Coulomb potential.