On the symmetry of four particles in a one-dimensional box with harmonic interaction

TL;DR

This paper explores the symmetry properties of a four-particle system in a one-dimensional box with harmonic interactions, using group theory and variational methods to analyze energy levels across different box sizes.

Contribution

It applies group theory to analyze the symmetry of the system and develops variational approaches for different regimes, providing new insights into the energy spectrum.

Findings

Symmetry group identified as $O_h$ for the system

First-order perturbation corrections obtained for lowest states

Variational methods developed for different box sizes

Abstract

We show that a system of four particles in a one-dimensional box with a two-particle harmonic interaction can by described by means of the symmetry point group $O_h$. Group theory proves useful for the discussion of both the small-box and large-box regimes. We apply perturbation theory and obtain the corrections of first order for the lowest states. We carry out a simple Rayleigh-Ritz variational calculation with basis sets adapted to the symmetries of the system. We also obtain alternative variational results for the first three lowest energy levels that are more suitable for larger box sizes.