 Solution

At fixed particle number $N_p=\int |\psi {|}^{2}d\mathbf{x}$, an equilibrium wavefunction is a constrained critical point of the energy functional. Introduce the chemical potential $\mu$ as the Lagrange multiplier for $N_p$, and impose $\delta (E-\mu N_p)=0$. The resulting stationary cubic-quintic Gross–Pitaevskii equation isThe time-dependent equilibrium has $\psi (\mathbf{x},t)=e^{-i\mu t/\hslash }{\psi }_0(\mathbf{x})$, so its number density is stationary even though its quantum phase changes uniformly. The specified particle number fixes $\mu$ together with the boundary conditions. In an infinite uniform bulk, particle number is understood through a finite volume or a prescribed bulk density.