Symmetries of Three Harmonically-Trapped Particles in One Dimension

TL;DR

This paper introduces a symmetry-based method for solving three-particle quantum systems in a one-dimensional harmonic trap, revealing a ${ m C}_{6v}$ symmetry that simplifies energy calculations and offers insights into few-body physics.

Contribution

The paper uncovers a ${ m C}_{6v}$ symmetry in a 1D three-particle system and demonstrates its utility in simplifying the calculation of energy eigenstates.

Findings

Discovery of ${ m C}_{6v}$ symmetry in the system

Simplification of energy eigenstate calculations

Revelation of the system's rich structure similar to 3D systems

Abstract

We present a method for solving trapped few-body problems and apply it to three equal-mass particles in a one-dimensional harmonic trap, interacting via a contact potential. By expressing the relative Hamiltonian in Jacobi cylindrical coordinates, i.e. the two-dimensional version of three-body hyperspherical coordinates, we discover an underlying ${\rm C}_{6v}$ symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of ${\rm C}_{6v}$. We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.