Coherent-field condensate variational state ($|\psi ⟩=e^{-\bar{N}/2}\exp (\int \psi {\hat{\Psi }}^{†})|0⟩$)

A normalized bosonic field coherent state with $\bar{N}=\int |\psi {|}^2$ has $⟨\hat{\Psi }(\mathbf{r})⟩=\psi (\mathbf{r})$. For a normal-ordered contact-interacting Hamiltonian, its energy expectation is $\int [{\hslash }^2|\nabla \psi {|}^2/(2m)+V|\psi {|}^2+g|\psi {|}^4/2]d^{3}r$. 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.