= Solution
A bosonic <coherent state> is an annihilation-operator eigenstate,
$$
|\alpha\rangle=e^{-|\alpha|^2/2}\sum_{N=0}^{\infty}\frac{\alpha^N}{\sqrt{N!}}|N\rangle,
\qquad a|\alpha\rangle=\alpha|\alpha\rangle.
$$
It has $\langle N\rangle=\operatorname{Var}N=|\alpha|^2$, a selected complex amplitude, and Poisson-distributed particle number. A <coherent-field condensate variational state> extends this construction to a spatial field with $\langle\widehat\psi(\mathbf r)\rangle=\Psi(\mathbf r)$. Evaluating a normal-ordered interacting Bose Hamiltonian in that state gives a classical-field <energy> functional; its variational equation is the <Gross–Pitaevskii equation>. This connects a macroscopic phase to <interference>, <quantized circulation> and neutral <Josephson effects>. A fixed-number condensate instead has vanishing one-point field expectation; <off-diagonal long-range order> identifies the condensate without requiring exact number fluctuations.
The <BCS ground state> has the pair-coherent form
$$
|\mathrm{BCS}\rangle=\prod_{\mathbf k}
\left(u_{\mathbf k}+v_{\mathbf k}c_{\mathbf k\uparrow}^\dagger c_{-\mathbf k\downarrow}^\dagger\right)|0\rangle.
$$
It is a product of <Cooper-pair pseudospin coherent states>: each paired <momentum> sector mixes its empty and doubly occupied configurations. The anomalous pair expectation is $\langle c_{-\mathbf k\downarrow}c_{\mathbf k\uparrow}\rangle=u_{\mathbf k}^*v_{\mathbf k}$, while an ordinary single-electron expectation vanishes by <fermion> parity. Pair operators are not unrestricted canonical <boson> operators, so this state must not be identified with a naive oscillator coherent state of individual <electrons>. Number projection gives a fixed-even-number description with pair correlations. Phase differences between condensates enter the <Josephson effect>; in a charged condensate that phase difference must be gauge invariant.
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