  We improve the Bardeen-Cooper-Schrieffer wave function with a fixed particle
number so as to incorporate many-body correlations beyond the mean-field
treatment. It is shown that the correlations lower the ground-state energy far
more than Cooper-pair condensation in the weak-coupling region. Moreover, they
naturally bring a superposition over the number of condensed particles. Thus,
Cooper-pair condensation is special among the various bound-state formations of
quantum mechanics in that number fluctuations are necessarily present in the
condensate through the dynamical exchange of particles with the non-condensate
reservoir. On the basis of this result, we propose $\varDelta N_{\rm con}\cdot
\varDelta \phi\gtrsim 1$ as the uncertainty relation relevant to the
number-phase fluctuations in superconductors and superfluids, where the number
of condensed particles $N_{\rm con}$ is used instead of the total particle
number $N$. The formula implies that a macroscopic phase $\phi$ can be
established even in number-fixed superconductors and superfluids since
$\varDelta N_{\rm con}\gg 1$.
