Electron in the field of a quantized electromagnetic wave and in a homogeneous magnetic field

The exact solution is found to the eigenfunetion and eigenvatue problem for the Hamiltonian of a Dirae electron that interacts with the quantized field of a monochromatic electromagnetic wave and an external homogeneous magnetic field, the direction of propagation of the wave coinciding with the direction of the homogeneous magnetic field. It is shown that the energy spectrum of the system contains a forbidden region, which disappears when the electron-photon interaction is switched off; the boundaries of this region correspond to the phenomenon of cyclotron resonance, at which the electron and photon oscillators have the same frequencies.