Quantum hydrodynamic models for the two-band Kane system

We consider Kane's quantum model for two-band charged particles, composed of two Schr\"odinger-like equations coupled by a $\kp$ term which describes interband tunneling. Using the WKB method we derive the hydrodynamic equations both for a zero-temperature and for a nonzero-temperature two-band quantum fluid. In the latter case, we introduce a family of closure relations and write in full details the simplest two-band, isentropic, fluiddynamical model. Finally, introducing appropriate relaxation terms and performing drift-diffusive limits, we obtain the corresponding quantum drift-diffusion models for zero and nonzero-temperature quantum fluids.