Electron in the field of two monochromatic electromagnetic waves

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Abstract
The motion of the electron in the classical as well as in the quantized field of two circularly polarized waves which move in opposite directions is investigated. The spin characteristics of the electron are neglected and it is assumed that the electron and one of the waves move in the same direction. In the classical case the problem is reduced to the solution of the general Mathieu equation, but in the quantum case to the solution of an ordinary second‐order differential equation with two irregular points. For particular values of the parameters the analytic expressions of the solution are found. The character of the solution essentially depends on the mutual polarization of waves. The conserved quantity for opposite polarization of waves is the energy of the electron, but for the same polarization it is the momentum of the electron. In the latter case the effective potential depends on the time, and the electron‐positron pair creation is allowed.

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