A multi particle toy system with analytic solutions to investigate composite bosons in a harmonic potential

TL;DR

This paper presents an analytically solvable three-dimensional toy model of two fermion types forming composite bosons in a harmonic potential, enabling detailed study of their transition from unbound to bound states.

Contribution

It introduces a novel toy system with analytical solutions that models composite bosons, allowing exploration of their properties and behaviors in a harmonic trap.

Findings

Analytical solutions for all integrals in the model.

High symmetry simplifies calculations.

The model can tune composite bosons from unbound to bound states.

Abstract

We construct a three dimensional toy systems with two types of fermions forming a composite boson. They are hold in a harmonic potential. The basis functions are constructed from an internal and an external Gauss function. All integrals have analytical solutions. The high symmetry reduces the number of integrals to be calculated for the symmetrized wave functions. With the internal Gauss function the composite bosons can be tuned from fermionic unbound behavior to bosonic bound behavior.