Charged particles in crossed and longitudinal electromagnetic fields and beam guides

We consider a class of electromagnetic fields that contains crossed fields combined with longitudinal electric and magnetic fields. We study the motion of a classical particle (solutions of the Lorentz equations) in such fields. Then, we present an analysis that allows one to decide which fields from the class act as a beam guide for charged particles, and we find some time-independent and time-dependent configurations with beam guiding properties. We demonstrate that the Klein-Gordon and Dirac equations with all the fields from the class can be solved exactly. We study these solutions, which were not known before, and prove that they form complete and orthogonal sets of functions.