GrossPitaevskiiGleichung

The Gross-Pitaevskii equation (GPE) is a nonlinear Schrödinger equation that describes the mean-field dynamics of a Bose-Einstein condensate at low temperature. It represents the condensate by a macroscopic wavefunction ψ(r,t) whose squared amplitude corresponds to the local particle density, and whose norm is related to the total number of particles.

The equation was developed independently by Eugene Gross and Lev Pitaevskii in the early 1960s and has since become the standard mean-field description of weakly interacting bosons in a trap. It captures the collective behavior of the condensate under external confinement and interactions.

The time-dependent Gross-Pitaevskii equation is iħ ∂ψ/∂t = [ -ħ^2/(2m) ∇^2 + V_ext(r) + g|ψ|^2 ] ψ, where m is the particle mass and g = 4π ħ^2 a / m with a the s-wave scattering length. The wavefunction is normalized such that ∫ |ψ|^2 d^3r = N, the total number of particles.

A stationary form yields μ ψ = [ -ħ^2/(2m) ∇^2 + V_ext(r) + g|ψ|^2 ] ψ, with μ the chemical potential. This form is used to compute ground-state and excited-state properties of trapped condensates.

Applications include modeling trapped atomic BECs, vortex configurations, solitons, and collective modes. The model is valid for dilute, weakly interacting systems at temperatures well below the critical temperature; it neglects thermal and quantum fluctuations and may require extensions at finite temperature or beyond mean field.

Extensions and variants include multi-component (spinor) condensates, dipolar interactions, reduced dimensionality leading to effective 1D/2D equations, and dissipative or stochastic forms to account for finite-temperature effects.