Within the formalism of the mean-field theory it is easy to obtain
the ground state energy from the stationary solution of the
Gross-Pitaevskii equation (1.11). To this purpose
the condensate wave function should be written as
, where is the chemical
potential and the function is real and normalized
to the total number of particles
. Then the Gross-Pitaevskii equation becomes

(1.12)

It has the form of a nonlinear Schrödinger equation. In the
absence of interactions () it reduces to the usual
single-particle Schrödinger equation with the Hamiltonian
.
In the uniform case, , is a constant
and the kinetic term in () disappears. The chemical potential is given by

(1.13)

At the chemical potential is the derivative of the energy
with respect to the number of particles
. Substitution of (1.2) into (1.13) and simple
integration gives the ground state energy per particle