Quantum Kinetic Equations for Plasmas and Radiation; Part II. Cyclotron Instabilities in a Bounded Plasma

The procedure for obtaining "kinetic" equations for internal distribution functions of a system was first developed by Bogoliubov* (1) in his study of.the properties of un-ionized gases. Bogoliubov also indicated some of the problems which would be encountered in a similar development for systems interacting through long-range Coulomb forces . Born and Green (3) , Kirkwood and collaborators (13, 25) and Yvon (37 ) also studied classical and quantum systems, using techniques similar to those developed by Bogolilibov. However, they too were primarily interested in un-ionized gases and liquids. Recently, Rosenbluth and Rostoker (26) derived kinetic equations for a classical plasma, assuming only Coulomb interactions . Simon and Harris (30) extended the theory to include transverse electromagnetic interactions .

Most of the investigations of quantum plasmas have employed techniques differing somewhat from those used here . Several texts have been devoted to the methods appropriate for various many-body problems, but some of them most often employed in plasma studies will be indicated here. Perhaps the best known treatment is due to Bohm and Pines (2). Here, "collective " variables replace the usual coordinates of the system, facilitating the solution of problems in which the individual particle nature is not as important as the gross features of the system. In particular, Bohm and Pines obtained a dispersion relation for the frequencies of collective oscillations of a quantum plasma. This same relation has been obtained by several other authors (8, 15, 29, 39) in different ways and will also be derived in this investigation. Of especial interest is the work of Klimontovich and Selin (7), in which kinetic equations for the quantum plasma were obtained and applied to several problems, including the small-amplitude Coulomb disturbances. Ehrenreich and Cohen (8) have also studied this problem, obtaining the quantum dispersion relation by means of the one-particle Liouville equation and the self-consistent field approximation for the Coulomb potential. Finally, von Roos (36), formulating the problem in terms of a quantum mechanical distribution function similar to that used first by Wigner (37), obtained the dispersion relation mentioned above and showed how exchange affects the relation. None of the above treatments have included a development of kinetic equations for particles and the electromagnetic field, although Osborn and Klevans (24) initiated an investigation of this problem at about the same time that the present study was begun. However, the direction of these authors' work seems to be somewhat different from this dissertation