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 Ψ of the system of N bosons is taken as a product of single-particle functions ψ ψ : Ψ Ψ ( r 1 , r 2 , … … , r N ) = ψ ψ ( r 1 ) ψ ψ ( r 2 ) … … ψ ψ ( r N ) , where r i is the coordinate of the 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 ℓ = 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 ℏ 2 2 m ∂ ∂ 2 ∂ ∂ r i 2 + V ( r i ) ) + ∑ ∑ i < j 4 π π ℏ ℏ 2 a s m δ δ ( r i − − r j ) , where m is the mass of the boson, V is the external potential, a s is the boson–boson s -wave scattering length, and δ δ ( r ) is the Dirac delta-function.