  Quantum nanosystems such as graphene nanoribbons or superconducting
nanoparticles are studied via a multiscale approach. Long space-time dynamics
is derived using a perturbation expansion in the ratio of the nearest-neighbor
distance to a nanometer-scale characteristic length, and a theorem on the
equivalence of long-time averages and expectation values. This dynamics is
shown to satisfy a coarse-grained wave equation (CGWE) which takes a
Schr\"odinger-like form with modified masses and interactions. The scaling of
space and time is determined by the orders of magnitude of various
contributions to the N-body potential. If the spatial scale of the
coarse-graining is too large, the CGWE would imply an unbounded growth of
gradients; if it is too short, the system's size would display uncontrolled
growth inappropriate for the bound states of interest, i.e., collective motion
or migration within a stable nano-assembly. The balance of these two extremes
removes arbitrariness in the choice of the scaling of space-time. Since the
long-scale dynamics of each fermion involves its interaction with many others,
we hypothesize that the solutions of the CGWE have mean-field character to good
approximation, i.e., can be factorized into single-particle functions. This
leads to a Coarse-grained Mean-field (CGMF) approximation that is distinct in
character from traditional Hartree-Fock theory. A variational principle is used
to derive equations for the single-particle functions. This theme is developed
and used to derive an equation for low-lying disturbances from the ground state
corresponding to long wavelength density disturbances or long-scale migration.
An algorithm for the efficient simulation of quantum nanosystems is suggested.
