Two and three particles interacting in a one-dimensional trap

TL;DR

This paper presents a matrix mechanics approach to compute energy levels and states for two and three particles in a one-dimensional trap, linking undergraduate quantum mechanics to cold-atom research and illustrating fermionization phenomena.

Contribution

It introduces a practical method for few-particle quantum systems in one dimension, enhancing educational tools and connecting to current cold-atom research.

Findings

Demonstrates fermionization in excited states under strong interactions

Provides a computational framework bridging undergraduate and research-level quantum physics

Shows how strong interactions affect energy spectra in few-particle systems

Abstract

We outline a procedure for using matrix mechanics to compute energy eigenvalues and eigenstates for two and three interacting particles in a confining trap, in one dimension. Such calculations can bridge a gap in the undergraduate physics curriculum between single-particle and many-particle quantum systems, and can also provide a pathway from standard quantum mechanics course material to understanding current research on cold-atom systems. In particular we illustrate the notion of "fermionization" and how it occurs not only for the ground state in the presence of strong repulsive interactions, but also for excited states, in both the strongly attractive and strongly repulsive regimes.