Analytical Solution for the Gross-Pitaevskii Equation in Phase Space and Wigner Function

A X Martins,T M R Filho,S C Ulhoa,R A S Paiva,R G G Amorim,G Petronilo,R R Luz

doi:10.1155/2020/7010957

Abstract

In this work, we study symplectic unitary representations for the Galilei group. As a consequence a nonlinear Schrödinger equation is derived in phase space. The formalism is based on the noncommutative structure of the star product, and using the group theory approach as a guide a physically consistent theory is constructed in phase space. The state is described by a quasi-probability amplitude that is in association with the Wigner function. With these results, we solve the Gross-Pitaevskii equation in phase space and obtained the Wigner function for the system considered.

Highlights

A relevant equation that describes a variety physical phenomena, as a Bose-Einstein condensed, is the Gross-Pitaevskii equation [1]
Wigner introduced his formalism by using a kind of Fourier transform of the density matrix, ρðq, q′Þ, giving rise to what is nowadays called the Wigner function, f Wðq, pÞ, where ðq, pÞ are the coordinates of a phase space manifold (Γ) [4,5,6,7]
We find an analytical solution for the wave function but the Wigner function is calculated up to a given order of approximation of the star product

Summary

Introduction

A relevant equation that describes a variety physical phenomena, as a Bose-Einstein condensed, is the Gross-Pitaevskii equation [1]. An important case for the Gross-Pitaevskii system is their approach in quantum phase space, the calculation of Wigner function for this system, in which it is not known in the literature In this context, the first formalism to quantum mechanics is phase space which was introduced by Wigner notion of phase space in 1932 [4]. In terms of nonrelativistic quantum mechanics, the proposed formalism has been used to treat a nonlinear oscillator perturbatively, to study the notion of coherent states and to introduce a nonlinear Schrödinger equation from the point of view of phase space In this context, there are a few examples of analytical solutions such as the harmonic oscillator [33], the Hydrogen atom [34], and some spin systems [35,36,37].

Nonlinear Schrödinger Equation in Phase Space

Solution of Gross-Pitaevskii Equation and Wigner Function

Analysis of Solution

Concluding Remarks