  By probability theory the probability space to underlie the set of
statistical data described by the squared modulus of a coherent superposition
of microscopically distinct (sub)states (CSMDS) is non-Kolmogorovian and, thus,
such data are mutually incompatible. For us this fact means that the squared
modulus of a CSMDS cannot be unambiguously interpreted as the probability
density and quantum mechanics itself, with its current approach to CSMDSs, does
not allow a correct statistical interpretation. By the example of a 1D
completed scattering and double slit diffraction we develop a new
quantum-mechanical approach to CSMDSs, which requires the decomposition of the
non-Kolmogorovian probability space associated with the squared modulus of a
CSMDS into the sum of Kolmogorovian ones. We adapt to CSMDSs the presented by
Khrennikov ({\it Found. of Phys., 35, No. 10, p.1655 (2005)}) concept of real
contexts (complexes of physical conditions) to determine uniquely the
properties of quantum ensembles. Namely we treat the context to create a
time-dependent CSMDS as a complex one consisting of elementary (sub)contexts to
create alternative subprocesses. For example, in the two-slit experiment each
slit generates its own elementary context and corresponding subprocess. We show
that quantum mechanics, with a new approach to CSMDSs, allows a correct
statistical interpretation and becomes compatible with classical physics.
