Example Exact Solutions of the Time-independent Gross-Pitaevskii and Schr\"odinger Equations

TL;DR

This paper presents a method to find external potentials in the Gross-Pitaevskii and Schrödinger equations that produce specific probability density functions, aiding experimental programming of potentials in Bose-Einstein condensates.

Contribution

It introduces a novel approach to determine potentials corresponding to desired wavefunction distributions, rather than solving for wavefunctions given potentials.

Findings

Exact potentials for specific probability densities are derived.

Examples include well-known PDFs and the hydrogen atom in momentum space.

The method facilitates experimental design of potentials in Bose-Einstein condensates.

Abstract

A prescription is given to obtain some exact results for certain external potentials $V\left({\vec r}\right)$ of the time-independent Gross-Pitaevskii and Schr\"odinger equations. The study motivation is the ability to program $V\left({\vec r}\right)$ experimentally in Bose-Einstein condensates. Rather than derive wavefunctions that are solutions for a given $V\left({\vec r}\right)$, we ask which $V\left({\vec r}\right)$ will have a given pdf (probability density function) $P\left({\vec r}\right)$. Several examples in 1D and 2D are presented for well-known pdfs and for the hydrogen atom in momentum space.