  Traditional plasma physics has mainly focused on regimes characterized by
high temperatures and low densities, for which quantum-mechanical effects have
virtually no impact. However, recent technological advances (particularly on
miniaturized semiconductor devices and nanoscale objects) have made it possible
to envisage practical applications of plasma physics where the quantum nature
of the particles plays a crucial role. Here, I shall review different
approaches to the modeling of quantum effects in electrostatic collisionless
plasmas. The full kinetic model is provided by the Wigner equation, which is
the quantum analog of the Vlasov equation. The Wigner formalism is particularly
attractive, as it recasts quantum mechanics in the familiar classical phase
space, although this comes at the cost of dealing with negative distribution
functions. Equivalently, the Wigner model can be expressed in terms of $N$
one-particle Schr{\"o}dinger equations, coupled by Poisson's equation: this is
the Hartree formalism, which is related to the `multi-stream' approach of
classical plasma physics. In order to reduce the complexity of the above
approaches, it is possible to develop a quantum fluid model by taking
velocity-space moments of the Wigner equation. Finally, certain regimes at
large excitation energies can be described by semiclassical kinetic models
(Vlasov-Poisson), provided that the initial ground-state equilibrium is treated
quantum-mechanically. The above models are validated and compared both in the
linear and nonlinear regimes.
