Negative energy modes and the quantum two-stream instability

The quantum two-stream instability is a benchmark displaying many of the particularities of quantum plasmas, including a new unstable branch of the dispersion relation for large wave- numbers and almost stationary, quasineutral, nonlinear oscillations without analog in classical plasmas. The unexpected features of the two-stream instability in electrostatic quantum plasmas are interpreted in terms of the coupling of approximate fast and slow waves. This is accomplished thanks to the factorization of the dispersion relation into different sectors carrying positive or negative energy. The concept of negative and positive energy waves, therefore, is shown to be useful not only for classical, but for quantum plasmas as well. The analysis can in principle be carried on in similar problems, like for the quantum beam- plasma instability or the case of parallel propagating quantum beams.