  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. We require some technical conditions on
the dynamics, which can be considered as interesting open problems in their own
right.
