  We consider the quantum theory of paraxial non-relativistic electron beams in
non-uniform magnetic fields, such as the Glaser field. We find the wave
function of an electron from such a beam and show that it is a joint eigenstate
of two ($z$-dependent) commuting gauge-independent operators. This generalized
Laguerre-Gaussian vortex beam has a phase that is shown to consist of two
parts, each being proportional to the eigenvalue of one of the two conserved
operators and each having different symmetries. We also describe the dynamics
of the angular momentum and cross-sectional area of any mode and how a varying
magnetic field can split a mode into a superposition of modes. By a suitable
change in frame of reference all of our analysis also applies to an electron in
a quantum Hall system with a time-dependent magnetic field.
