Two interacting fermions in a 1D harmonic trap: matching the Bethe ansatz and variational approaches

TL;DR

This paper combines the Bethe ansatz and variational methods to accurately compute the ground state energy of two interacting fermions in a 1D harmonic trap, bridging analytical and numerical approaches.

Contribution

It introduces a novel combined approach of Bethe ansatz and variational principles for solving few-body quantum systems in traps.

Findings

Good agreement with analytical solutions

Provides a foundation for studying more complex systems

Demonstrates effectiveness of combined methods

Abstract

In this work, combining the Bethe ansatz approach with the variational principle, we calculate the ground state energy of the relative motion of a system of two fermions with spin up and down interacting via a delta-function potential in a 1D harmonic trap. Our results show good agreement with the analytical solution of the problem, and provide a starting point for the investigation of more complex few-body systems where no exact theoretical solution is available.