The quasilinear theory of the Wigner-Poisson system in one spatial dimension
is examined. Conservation laws and properties of the stationary solutions are
determined. Quantum effects are shown to manifest themselves in transient
periodic oscillations of the averaged Wigner function in velocity space. The
quantum quasilinear theory is checked against numerical simulations of the
bump-on-tail and the two-stream instabilities. The predicted wavelength of the
oscillations in velocity space agrees well with the numerical results