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.