Two trapped particles interacting by a finite-ranged two-body potential in two spatial dimensions

TL;DR

This paper investigates the energy spectrum of two particles in a 2D harmonic trap interacting via a finite-range Gaussian potential, analyzing the effects of interaction strength and potential range, and connecting finite-range and zero-range models.

Contribution

It derives an approximate transcendental equation for the energy spectrum and explores the impact of potential range, providing a link between finite-range and zero-range interaction models in two dimensions.

Findings

Finite-range potential significantly affects ground-state energy.

Zero-range limit reproduces non-interacting results without Hilbert space truncation.

Established connection between finite-range and regularized zero-range models.

Abstract

We examine the problem of two particles confined in an isotropic harmonic trap, which interact via a finite-ranged Gaussian-shaped potential in two spatial dimensions. We derive an approximative transcendental equation for the energy and study the resulting spectrum as a function of the interparticle interaction strength. Both the attractive and repulsive systems are analyzed. We study the impact of the potential's range on the ground-state energy. Complementary, we also explicitly verify by a variational treatment that in the zero-range limit the positive delta potential in two dimensions only reproduces the non-interacting results, if the Hilbert space in not truncated. Finally, we establish and discuss the connection between our finite-range treatment and regularized zero-range results from the literature.