  The foundations of Quantum Mechanics (QM) and its interpretation have been
debated since the beginning of its formulation, but still many items have not
yet reached a common consensus. In particular it is well known that Niels Bohr
and Albert Einstein were of quite different opinions on the possible
interpretation of QM. The focus of their debate was on the possibility that QM
is incomplete because of its feature of probabilistic theory. Bohr was denying
that, while Einstein tried to prove its incompleteness. The underlying question
of this debate is if one can have a 'realistic' physical picture of QM, or, on
the contrary, we have to remain at the level of macroscopic description of
phenomena and experiments. In this paper we take the line of thought that QM is
incomplete, but that the wave function is still the proper representation of
physical systems. The starting point is the introduction of an hyperfinite
lattice in the nonstandard real axis \Rns\ for the space coordinates. This
extension of the standard real axis \Rs\ introduces additional nonstandard
degrees of freedom. With additional assumptions the model is able to formulate
a physical description of several so called QM 'paradoxes', as well as to
handle the measurement problem. The latter is described by a stochastic process
that arises from the model itself, and it is in line with the general scheme by
Pearle. Furthermore the entanglement between distant systems gets a natural
representation. In this sense the measurement process and the entanglement turn
out to have a common physical origin.
