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Non-Volkov solutions for a charge in a plane wave

We focus our attention, once again, on the Klein--Gordon and Dirac equations with a plane-wave field. We recall that for the first time a set of solutions of these equations was found by Volkov. The Volkov solutions are widely used in calculations of quantum effects with electrons and other elementary particles in laser beams. We demonstrate that one can construct sets of solutions which differ from the Volkov solutions and which may be useful in physical applications. For this purpose, we show that the transversal charge motion in a plane wave can be mapped by a special transformation to transversal free particle motion. This allows us to find new sets of solutions where the transversal motion is characterized by quantum numbers different from Volkov's (in the Volkov solutions this motion is characterized by the transversal momentum). In particular, we construct solutions with semiclassical transversal charge motion (transversal squeezed coherent states). In addition, we demonstrate how the plane-wave field can be eliminated from the transversal charge motion in a more complicated case of the so-called combined electromagnetic field (a combination of a plane-wave field and constant colinear electric and magnetic fields). Thus, we find new sets of solutions of the Klein--Gordon and Dirac equations with the combined electromagnetic field.