Motion of an electron in the field of a standing electromagnetic wave

The classical equations of motion are analyzed for a relativistic electron in the superposition of a standing wave and a steady magnetic field, which gives an integral of motion relating the energy to the radius of the cyclotron orbit. The cases of finite and infinite motion are examined. The quantum approach is based on solving the Klein-Gordon equation and is supplemented by analysis of the energy spectrum as affected by the type of boundary conditions.