Solution of the Gross-Pitaevskii equation in terms of the associated non-linear Hartree potential

TL;DR

This paper presents an iterative numerical method to solve the Gross-Pitaevskii equation by computing the associated non-linear Hartree potential, achieving high precision in energy eigenvalues.

Contribution

The authors introduce a physically intuitive iterative approach starting from the Thomas-Fermi approximation to solve the GP equation with high accuracy.

Findings

Converges to stable energy eigenvalues after ~35 iterations

Achieves eight significant figures in eigenvalue stability

Method can be extended to nuclear shell-model potentials

Abstract

The Gross-Pitaevskii equation (GP), that describes the wave function of a number of coherent Bose particles contained in a trap, contains the cube of the normalized wave function, times a factor proportional to the number of coherent atoms. The square of the wave function, times the above mentioned factor, is defined as the Hartree potential. A method implemented here for the numerical solution of the GP equation consists in obtaining the Hartree potential iteratively, starting with the Thomas Fermi approximation to this potential. The energy eigenvalues and the corresponding wave functions for each successive potential are obtained by a method described previously. After approximately 35 iterations a stability of eight significant figures for the energy eigenvalues is obtained.This method has the advantage of being physically intuitive, and could be extended to the calculation of a shell-model potential in nuclear physics, once the Pauli exclusion principle is allowed for.