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GrossPitaevskii equationThe GrossPitaevskii equation is a nonlinear model equation for the order parameter or wavefunction of a BoseEinstein condensate. It is similar in form to the GinzburgLandau equation and is sometimes referred to as a nonlinear Schrödinger equation. A BoseEinstein condensate (BEC) is a gas of bosons that are in the same quantum state, and thus can be described by the same wavefunction. A free quantum particle is described by a singleparticle Schrödinger equation. Interaction between particles in a real gas is taken into account by a pertinent manybody Schrödinger equation. If the average spacing between the particles in a gas is greater than the scattering length (that is, in the socalled dilute limit), then one can approximate the true interaction potential that features in this equation by a pseudopotential. The nonlinearity of the GrossPitaevskii equation has its origin in the interaction between the particles. This becomes evident by equating the coupling constant of interaction, , in the GrossPitaevskii equation with zero (see the following section), on which the singleparticle Schrödinger equation describing a particle inside a trapping potential is recovered. Additional recommended knowledge
Form of EquationThe equation has the form of the Schrödinger equation with the addition of an interaction term. The coupling constant, g, is proportional to the scattering length of two interacting bosons:
where is Planck's constant and m is the mass of the boson. The energy density is where Ψ is the wavefunction, or order parameter, and V is an external potential. The timeindependent GrossPitaevskii equation, for a conserved number of particles, is where μ is the chemical potential. The chemical potential is found from the condition that the number of particles is related to the wavefunction by
From the timeindependent GrossPitaevskii equation, we can find the structure of a BoseEinstein condensate in various external potentials (e.g. a harmonic trap). The timedependent GrossPitaevskii equation is
From the timedependent GrossPitaevskii equation we can look at the dynamics of the BoseEinstein condensate. It is used to find the collective modes of a trapped gas. SolutionsSince the GrossPitaevskii equation is a nonlinear, partial differential equation, exact solutions are hard to come by. As a result, solutions have to be approximated via a myriad of techniques. Exact SolutionsFree ParticleThe simplest exact solution is the free particle solution, with , . This solution is often called the Hartree solution. Although it does satisfy the GrossPitaevskii equation, it leaves a gap in the energy spectrum due to the interaction: . According to the HugenholtzPines theorem,^{[1]} an interacting bose gas does not exhibit an energy gap (in the case of repulsive interactions). SolitonA onedimensional soliton can form in a BoseEinstein condensate, and depending upon whether the interaction is attractive or repulsive, there is either a light or dark soliton. Both solitons are local disturbances in a condensate with a uniform background density If the BEC is repulsive, so that g > 0, then a possible solition of the GrossPitaevskii equation is,
where ψ_{0} is the value of the condensate wavefuntion at , and , is the coherence length. This solution represents the dark soliton, since there is a deficit of condensate in a space of nonzero density. The dark soliton is also a type of topological defect, since ψ flips between positive and negative values across the origin, corresponding to a π phase shift. For g < 0 where the chemical potential is . This solution represents the bright soliton, since there is a concentration of condensate in a space of zero density. 1D Square Well PotentialVariational SolutionsIn systems where an exact analytical solution may not be feasible, one can make a variational approximation. The basic idea is to make a variational ansatz for the wavefunction with free parameters, plug it into the free energy, and minimize the energy with respect to the free parameters. ThomasFermi ApproximationBogoliubov ApproximationNotes
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "GrossPitaevskii_equation". A list of authors is available in Wikipedia. 