  We derive the main physical galaxy properties: mass, halo radius, phase space
density and velocity dispersion from a semiclassical gravitational approach in
which fermionic WDM is treated quantum mechanically. They turn out to be
compatible with observations. Pauli Principle implies for the fermionic DM
phase-space density Q(r) = rho(r)/sigma^3(r) the quantum bound Q(r) < K
m^4/hbar^3, where m is the DM particle mass, sigma(r) is the DM velocity
dispersion and K is a pure number of order one which we estimate. N-body galaxy
simulations produce a divergent Q(r) at r = 0 violating this quantum bound.
Combining this bound with the behaviour of Q(r) from simulations, the virial
and galaxy data on Q implies lower bounds on the halo radius and a minimal
distance r_{min} at which classical dynamics for DM fermions breaks down. This
quantum bound rules out the presence of galaxy cusps for fermionic WDM, in
agreement with astronomical observations, which show that the DM halos are
cored. We show that compact dwarf galaxies are quantum objects supported
against gravity by the fermionic WDM quantum pressure. Quantum mechanical
calculations become necessary to compute galaxy structures at kpc scales and
below. Classical N-body simulations are not valid at scales below r_{min}. We
apply the Thomas-Fermi semiclassical approach to fermionic WDM galaxies and
find the physical magnitudes: mass, halo radius, phase-space density, velocity
dispersion, consistent with observations especially for compact dwarf galaxies.
Namely, fermionic WDM treated quantum mechanically, as it must be, reproduces
the sizes of the observed cores. The lightest known galaxy Willman I implies a
lower bound for the WDM particle mass m > 0.96 keV. These results and the
observed galaxies with halo radius > 30 pc and halo mass > 4 10^5 M_sun provide
further indication that the WDM particle mass m is in the range 1-2 keV.
