= Grand-canonical Hamiltonian of a Bose field
{title2=$K=H-\mu N$}
For a repulsive contact-interacting <Bose-Einstein condensate>, the classical <Hamiltonian density> generating evolution in a frame rotating at the <chemical potential> is $\mathcal K=\hbar^2|\nabla\psi|^2/(2M)-\mu|\psi|^2+g|\psi|^4/2$. Its integral is the <grand canonical ensemble> generator $H-\mu N$, not the physical energy $H$ alone. For $g,\mu>0$ the uniform <mean-field theory> minima satisfy $|\psi_0|^2=\mu/g$, with a freely chosen constant phase. In the physical frame, the corresponding field has time factor $e^{-i\mu t/\hbar}$.
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