Long wavelength oscillations of a quantum plasma in a uniform magnetic field. II

Abstract The self-sustained oscillations of a quantum mechanical electron gas in a uniform magnetic field are discussed in the time-dependent Hartree approximation, or random phase approximation. The full electromagnetic interaction between electrons is taken into account, and spin effects are included. The resulting dispersion relation reduces to the Appleton-Hartree equation of classical magnetohydrodynamics in the limit of long wavelengths. It is found that this classical equation holds even for a highly degenerate electron gas or for magnetic fields so strong that only a few Landau oscillator levels are appreciably occupied. The validity of long wavelength expansions is discussed for the various normal modes, particular attention being given to the low frequency transverse mode, a particular case of which is the helicon or whistler. It is also shown that the normal modes with appreciable longitudinal currents are correctly described over a wide range of wavelengths by the electrostatic approximation, in which only Coulomb forces between electrons are retained.