= One-body coherence of a Hartree condensate
{title2=$G^{(1)}(x,y)=N\phi^*(x)\phi(y)$}
For the normalized <Hartree product state> $|N:\phi\rangle$, applying the <bosonic annihilation operator> gives $\psi(y)|N:\phi\rangle=\sqrt N\phi(y)|N-1:\phi\rangle$. Taking the inner product proves the displayed factorization of the <one-body density matrix>. Its diagonal is the total <number density> in this approximation. For a real positive <orbital>, $G^{(1)}(0,r)=\sqrt{n(0)n(r)}$. Restoring the <kinetic energy> in a <Gross–Pitaevskii equation> can smooth a <Thomas–Fermi approximation for a condensate> edge while retaining this factorization. Describing <quantum depletion of a Bose condensate> additionally requires noncondensate mode occupations and their contribution to the <one-body density matrix>.
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