  The variance and uncertainty product of the position and momentum
many-particle operators of structureless bosons interacting by a long-range
inter-particle interaction and trapped in a single-well potential are
investigated. In the first example, of an out-of-equilibrium interaction-quench
scenario, it is found that, despite the system being fully condensed, already
when a fraction of a particle is depleted differences with respect to the
mean-field quantities emerge. In the second example, of the pathway from
condensation to fragmentation of the ground state, we find out that, although
the cloud's density broadens while the system's fragments, the position
variance actually decreases, the momentum variance increases, and the
uncertainty product is not a monotonous function but has a maximum. Implication
are briefly discussed.
