The dynamics of a quantum plasma can be described self-consistently by the
nonlinear Schrodinger-Poisson system. We consider a multistream model repre
senting a statistical mixture of N pure states, each described by a wave fu
nction. The one-stream and two-stream cases are investigated. We derive the
dispersion relation for the two-stream instability and show that a new, pu
rely quantum, branch appears. Numerical simulations of the complete Schrodi
nger-Poisson system confirm the linear analysis, and provide further result
s in the strongly nonlinear regime. The stationary states of the Schrodinge
r-Poisson system are also investigated.-These can be viewed as the quantum
mechanical counterpart of the classical Bernstein-Greene-Kruskal modes, and
are described by a set of coupled nonlinear differential equations for the
electrostatic potential and the stream amplitudes.