  Quantum mechanics is necessary to compute galaxy structures at kpc scales and
below. This is so because near the galaxy center, at scales below 10 - 100 pc,
warm dark matter (WDM) quantum effects are important: observations show that
the interparticle distance is of the order of, or smaller than the de Broglie
wavelength for WDM. This explains why all classical (non-quantum) WDM N-body
simulations fail to explain galactic cores and their sizes. We describe
fermionic WDM galaxies in an analytic semiclassical framework based on the
Thomas-Fermi approach, we resolve it numerically and find the main physical
galaxy magnitudes: mass, halo radius, phase-space density, velocity dispersion,
fully consistent with observations, including compact dwarf galaxies. Namely,
fermionic WDM treated quantum mechanically, as it must be, reproduces the
observed galaxy DM cores and their sizes. [In addition, as is known, WDM
simulations produce the right DM structures in agreement with observations for
scales > kpc]. We show that compact dwarf galaxies are natural quantum
macroscopic objects supported against gravity by the fermionic WDM quantum
pressure (quantum degenerate fermions) with a minimal galaxy mass and minimal
velocity dispersion. Interestingly enough, the minimal galaxy mass implies a
minimal mass m_{min} for the WDM particle. The lightest known dwarf galaxy
(Willman I) implies m > m_{min} = 1.91 keV. These results and the observed halo
radius and mass of the compact galaxies provide further indication that the WDM
particle mass m is approximately around 2 keV.
