On the symmetry of three identical interacting particles in a one-dimensional box

TL;DR

This paper analyzes a quantum system of three identical particles in a one-dimensional box with harmonic interactions, using group theory to understand spectral properties and state behaviors across different box sizes.

Contribution

It applies group theory to simplify the analysis of a three-particle quantum system, revealing spectral features and state connections across different regimes.

Findings

Group theory classifies energy states by symmetry.

Predicts energy level crossings and avoided crossings.

Connects small and large box length regimes.

Abstract

We study a quantum-mechanical system of three particles in a one-dimensional box with two-particle harmonic interactions. The symmetry of the system is described by the point group $D_{3d}$. Group theory greatly facilitates the application of perturbation theory and the Rayleigh-Ritz variational method. A great advantage is that every irreducible representation can be treated separately. Group theory enables us to predict the connection between the states for the small box length and large box length regimes of the system. We discuss the crossings and avoided crossings of the energy levels as well as other interesting features of the spectrum of the system.