= Coherent-field condensate variational state
{title2=$|\psi\rangle=e^{-\bar N/2}\exp(\int\psi\widehat\Psi^\dagger)|0\rangle$}
A normalized bosonic field coherent state with $\bar N=\int|\psi|^2$ has $\langle\widehat\Psi(\mathbf r)\rangle=\psi(\mathbf r)$. For a normal-ordered contact-interacting Hamiltonian, its <energy> expectation is $\int[\hbar^2|\nabla\psi|^2/(2m)+V|\psi|^2+g|\psi|^4/2]d^3r$. Minimizing at fixed mean particle number gives the stationary <Gross–Pitaevskii equation>. This mean-field state has number fluctuations and does not include the pair correlations responsible for quantum depletion.
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