  The flat, homogeneous, and isotropic universe with a massless scalar field is
a paradigmatic model in Loop Quantum Cosmology. In spite of the prominent role
that the model has played in the development of this branch of physics, there
still remain some aspects of its quantization which deserve a more detailed
discussion. These aspects include the kinematical resolution of the
cosmological singularity, the precise relation between the solutions of the
densitized and non-densitized versions of the quantum Hamiltonian constraint,
the possibility of identifying superselection sectors which are as simple as
possible, and a clear comprehension of the Wheeler-DeWitt (WDW) limit
associated with the theory in those sectors. We propose an alternative operator
to represent the Hamiltonian constraint which is specially suitable to deal
with these issues in a satisfactory way. In particular, with our constraint
operator, the singularity decouples in the kinematical Hilbert space and can be
removed already at this level. Thanks to this fact, we can densitize the
quantum Hamiltonian constraint in a rigorous manner. Besides, together with the
physical observables, this constraint superselects simple sectors for the
universe volume, with a support contained in a single semiaxis of the real line
and for which the basic functions that encode the information about the
geometry possess optimal physical properties. Namely, they provide a
no-boundary description around the cosmological singularity and admit a
well-defined WDW limit in terms of standing waves. Both properties explain the
presence of a generic quantum bounce replacing the singularity at a fundamental
level, in contrast with previous studies where the bounce was proved in
concrete regimes and focusing on states with a marked semiclassical behavior.
