The Heisenberg dynamics of the energy, momentum, and particle densities
for fermions with short-range pair interactions is shown to converge to
the compressible Euler equations in the hydrodynamic limit. The pressure
function is given by the standard formula from quantum statistical
mechanics with the two-body potential under consideration. Our derivation
is based on a quantum version of the entropy method and a suitable quantum
virial theorem. No intermediate description, such as a Boltzmann equation
or semi-classical approximation, is used in our proof.