  The theory of electron holes is extended into the quantum regime. The
Wigner--Poisson system is solved perturbatively based in lowest order on a
weak, standing electron hole. Quantum corrections are shown to lower the
potential amplitude and to increase the number of deeply trapped electrons.
They, hence, tend to bring this extreme non--equilibrium state closer to
thermodynamic equilibrium, an effect which can be attributed to the tunneling
of particles in this mixed state system.
