 Solution

The Hamiltonian functional isTreating the field and its complex conjugate as independent variables, and integrating the gradient term by parts with periodic, decaying or fixed-field boundary conditions, givesThus $i\hslash {\psi }_t=\delta H/\delta {\psi }^{\ast }$ is the Gross–Pitaevskii equation. The factor $1/2$ in the interaction energy avoids counting each pair twice.

At prescribed particle number, minimize $H$ subject to $N=\int |\psi {|}^2{d}^{3}x$. Introducing the chemical potential as a Lagrange multiplier, the variation of $H-\mu N$ vanishes at a ground-state profile ${\psi }_0$, provingThe corresponding time-dependent field is $\psi (\mathbf{x},t)=e^{-i\mu t/\hslash }{\psi }_0(\mathbf{x})$. A solution of this stationary equation is only a candidate ground state; the ground state is the minimizer among all fields satisfying the normalization. For the uniform repulsive gas in a periodic volume, the gradient energy is nonnegative and $\int n^2{d}^{3}x\ge N^2/\mathcal{V}$, so the constant-density, constant-phase state is indeed the minimum. The uniform-background solutions below assume $U>0$ and a positive background density.