  We develop the Fock-space many-body quantum approach for large number of
bosons, when the bosons occupy significantly only few modes. The approach is
based on an analogy with the dynamics of a single particle of either positive
or negative mass in a quantum potential, where the inverse number of bosons
plays the role of Planck constant. As application of the method we consider the
Bose-Einstein condensate in a double-well trap. The ground state of positive
mass particle corresponds to the mean-field fixed point of lower energy, while
that of negative mass to the excited fixed point. In the case of attractive BEC
above the threshold for symmetry breaking, the ground state is a cat state and
we relate this fact to the double-well shape of the quantum potential for the
positive mass quantum particle. The tunneling energy splitting between the
local Fock states of the cat state is shown to be extremely small for the
nonlinearity parameter just above the critical value and exponentially
decreasing with the number of atoms. In the repulsive case, the phase locked
($\pi$-phase) macroscopic quantum self-trapping of BEC is related to the
double-well structure of the potential for the negative mass quantum particle.
We also analyze the running phase macroscopic quantum self-trapping state and
show that it is subject to quantum collapses and revivals. Moreover, the
quantum dynamics just before the first collapse of the running phase shows
exponential growth of dispersion of the quantum phase distribution which may
explain the growth of the phase fluctuations seen in the experiment on the
macroscopic quantum self-trapping.
