Self-consistent fluid model for a quantum electron gas

It is shown that, for a large class of statistical mixtures, the Wigner-Poisson (or Hartree) system can be reduced to an effective Schr\"odinger-Poisson system, in which the Schr\"odinger equation contains a new nonlinearity. For the case of a zero-temperature one-dimensional electron gas, this additional nonlinearity is of the form |Ψ|⁴. In the long-wavelength limit, the results obtained from the effective Schr\"odinger-Poisson system are in agreement with those of the Wigner-Poisson system. The reduced model is further used to describe the stationary states of a quantum electron gas and the two-stream instability.