Quantum-Mechanical Theory of Nonlinear Plasma Phenomena in a Strong Magnetic Field

A quantum-mechanical description of a cold plasma in a strong magnetic field is developed from the hydrodynamic equations. The plasmon-particle and plasmon-plasmon interaction Hamiltonians are derived and expressed in terms of second quantized operators for plasmons and particles. The quantum-mechanical plasmon growth rate due to stimulated emission is obtained. The classical limit is taken and compared with the growth rate predicated by the linearized Vlasov equations. The possibility of a spacial gradient instability, that of Mikhailovskii and Timofeev, arises and is interpreted in terms of emission and absorption of plasmons by the energetic particles. The plasmon-particle interaction Hamiltonian is used to determine the time development of the particle distribution function. Finally, the plasmon-plasmon coupling is included in the time development of the plasmon distribution.