# Four fermions in a one-dimensional harmonic trap: Accuracy of a variational-ansatz approach

## TL;DR

This paper evaluates a simple variational approach to model four interacting ultra-cold fermions in a one-dimensional harmonic trap, showing it provides surprisingly accurate predictions validated against exact numerical methods.

## Contribution

The study demonstrates that a basic variational ansatz can reliably predict properties of four-fermion systems, offering a simple yet effective tool for experimental analysis.

## Key findings

- Variational ansatz yields accurate ground state predictions.
- Method is effective for both equal and different mass mixtures.
- Validated against exact diagonalization results.

## Abstract

Detailed analysis of the system of four interacting ultra-cold fermions confined in a one-dimensional harmonic trap is performed. The analysis is done in the framework of a simple variational ansatz for the many-body ground state and its predictions are confronted with the results of numerically exact diagonalization of the many-body Hamiltonian. Short discussion on the role of the quantum statistics, i.e. Bose-Bose and Bose-Fermi mixtures is also presented. It is concluded that the variational ansatz, although seemed to be oversimplified, gives surprisingly good predictions of many different quantities for mixtures of equal as well as different mass systems. The result may have some experimental importance since it gives quite simple and validated method for describing experimental outputs.