Analytical results for the Morse potential in s-wave ultracold scattering: three dimensional vs. one dimensional problem

TL;DR

This paper derives explicit formulas for scattering length and effective range in ultracold atom s-wave scattering using the Morse potential, comparing one-dimensional and three-dimensional models.

Contribution

It provides analytical expressions for scattering parameters based on the Morse potential, highlighting differences between 1D and 3D treatments in ultracold scattering.

Findings

Explicit formulas for $a$ and $r_e$ derived from Morse eigenfunctions.

Significant effects observed when considering radial coordinate intervals.

Comparison between 1D and 3D models elucidates dimensional influences.

Abstract

Taking advantage of the known analytic expression of the eigenfunctions and eigenenergies of the Morse Hamiltonian, explicit expressions are found for the scattering length $a$ and the effective range $r_e$ which determine the s-wave scattering of ultracold atoms. The effects on $a$ and $r_e$ of considering the radial coordinate in the interval $[0,\infty)$ or in the extended region $(-\infty, \infty)$ are studied in detail.