Two Particles in a Trap

TL;DR

This paper derives an approximate expression for the energy spectrum shift of two particles in a 3D harmonic trap based on the Busch-formula, comparing it to other trap geometries and discussing its implications for in-medium interactions.

Contribution

It provides a new approximation for the energy shift in a harmonic trap using the Busch-formula, and analyzes its accuracy relative to other trap geometries and previous theoretical work.

Findings

The energy shift is approximately proportional to the scattering phase shift.

The approximation is highly accurate (~1%) for most states, except the lowest energy state.

The validity of phase shift approximation depends on the trap geometry, differing between harmonic and square-well traps.

Abstract

The Busch-formula relates the energy-spectrum of two point-like particles interacting in a 3-D isotropic Harmonic Oscillator trap to the free scattering phase-shifts of the particles. This formula is used to find an expression for the \it shift \rm in the spectrum from the unperturbed (non-interacting) spectrum rather than the spectrum itself. This shift is shown to be approximately $\Delta=-\delta(k)/\pi\times dE$, where $dE$ is the spacing between unperturbed energy levels. The resulting difference from the Busch-formula is typically 1/2% except for the lowest energy-state and small scattering length when it is 3%. It goes to zero when the scattering length $\rightarrow \pm \infty$. The energy shift $\Delta$ is familiar from a relatedproblem, that of two particles in a spherical infinite square-well trap of radius $R$ in the limit $R\rightarrow \infty$. The approximation ishowever as large as 30% for finite values of $R$, a situation quite different from the Harmonic Oscillator case. The square-well results for $R\rightarrow \infty$ led to the use ofin-medium (effective) interactions in nuclear matter calculations that were $\propto \Delta$ and known as the \it phase shift approximation \rm.Our results indicate that the validity of this approximation depends on the trapitself, a problem already discussed by DeWitt more than 50 years ago for acubical vs spherical trap.