Propagation of vortex electron wave functions in a magnetic field

The physics of coherent beams of photons carrying axial orbital angular momentum (OAM) is well understood, and such beams, sometimes known as vortex beams, have found applications in optics and microscopy. Recently electron beams carrying very large values of axial OAM have been generated. In the absence of coupling to an external electromagnetic field, the propagation of such vortex electron beams is virtually identical mathematically to that of vortex photon beams propagating in a medium with a homogeneous index of refraction. But when coupled to an external electromagnetic field, the propagation of vortex electron beams is distinctly different from photons. Here we use the exact path integral solution to Schrodinger's equation to examine the time evolution of an electron wave function carrying axial OAM. Interestingly we find that the nonzero OAM wave function can be obtained from the zero OAM wave function, in the case considered here, simply by multipling it by an appropriate time and position dependent prefactor. Hence adding OAM and propagating it can in this case be replaced by first propagating then adding OAM. Also, the results shown provide an explicit illustration of the fact that the gyromagnetic ratio for OAM is unity. We also propose a novel version of the Bohm-Aharonov effect using vortex electron beams.