The Gross–Pitaevskii equation (GPE, named after Eugene P. Gross and Lev Petrovich Pitaevskii) describes the ground state of a quantum system of identical bosons using the Hartree–Fock approximation and the pseudopotential interaction model. A Bose–Einstein 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 single-particle Schrödinger equation. Interaction between particles in a real gas is taken into account by a pertinent many-body Schrödinger equation. In the Hartree–Fock approximation, the total wave-function Ψ {\displaystyle \Psi } of the system of N {\displaystyle N} bosons is taken as a product of single-particle functions ψ {\displaystyle \psi } :
Ψ ( r 1 , r 2 , … , r N ) = ψ ( r 1 ) ψ ( r 2 ) … ψ ( r N ) , {\displaystyle \Psi (\mathbf {r} _{1},\mathbf {r} _{2},\dots ,\mathbf {r} _{N})=\psi (\mathbf {r} _{1})\psi (\mathbf {r} _{2})\dots \psi (\mathbf {r} _{N}),}
where r i {\displaystyle \mathbf {r} _{i}} is the coordinate of the i {\displaystyle i} -th boson. If the average spacing between the particles in a gas is greater than the scattering length (that is, in the so-called dilute limit), then one can approximate the true interaction potential that features in this equation by a pseudopotential. At sufficiently low temperature, where the de Broglie wavelength is much longer than the range of boson–boson interaction, the scattering process can be well approximated by the s-wave scattering (i.e. ℓ = 0 {\displaystyle \ell =0} in the partial-wave analysis, a.k.a. the hard-sphere potential) term alone. In that case, the pseudopotential model Hamiltonian of the system can be written as
H = ∑ i = 1 N ( − ℏ 2 2 m ∂ 2 ∂ r i 2 + V ( r i ) ) + ∑ i < j 4 π ℏ 2 a s m δ ( r i − r j ) , {\displaystyle H=\sum _{i=1}^{N}\left(-{\frac {\hbar ^{2}}{2m}}{\frac {\partial ^{2}}{\partial \mathbf {r} _{i}^{2}}}+V(\mathbf {r} _{i})\right)+\sum _{i<j}{\frac {4\pi \hbar ^{2}a_{s}}{m}}\delta (\mathbf {r} _{i}-\mathbf {r} _{j}),}
where m {\displaystyle m} is the mass of the boson, V {\displaystyle V} is the external potential, a s {\displaystyle a_{s}} is the boson–boson s-wave scattering length, and δ ( r ) {\displaystyle \delta (\mathbf {r} )} is the Dirac delta-function. The variational method shows that if the single-particle wavefunction satisfies the following Gross–Pitaevskii equation
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