Electron in the field of a plane quantized electromagnetic wave

Wave functions are found for a system consisting of a charge plus the second-quantized electromagnetic field of a plane wave. The plane wave is represented by an expansion with respect to photons with definite momentum and polarization. The interaction between the electron and all the photons of the wave is allowed for exactly. The problem is solved by the method of canonical transformations of creation and annihilation operators without explicit specification of the form of these operators. The properties of the solutions are studied and their relation to the Volkov solutions is found.