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We study two mathematical descriptions of a charged particle interacting withit's self-generated electromagnetic field. The first model is the one-bodyMaxwell-Schr\"odinger system where the interaction of the spin with themagnetic field is neglected and the second model is the related one-bodyMaxwell-Pauli system where the spin-field interaction is included. We provethat there exist travelling wave solutions to both of these systems providedthat the speed $|v|$ of the wave is not too large. Moreover, we observe thatthe energies of these solutions behave like $\frac{mv^2}{2}$ for smallvelocities of the particle, which may be interpreted as saying that theeffective mass of the particle is the same as it's bare mass.

Existence of Travelling Wave Solutions to the Maxwell-Pauli and Maxwell-Schr\"odinger Systems