  Scalar theory of quantum electron beam optics, at the single-particle level,
derived from the Dirac equation using a Foldy-Wouthuysen-like transformation
technique is considered. Round magnetic electron lenses with Glaser and power
law models for the axial magnetic field $B(z)$ are studied. Paraxial quantum
propagator for the Glaser model lens is obtained in terms of the well known
fundamental solutions of its paraxial equation of motion. In the case of lenses
with the power law model for $B(z)$ the well known fundamental solutions of the
paraxial equations, obtained by solving the differential equation, are
constructed using the Peano-Baker series also. Quantum mechanics of aberrations
is discussed briefly. Role of quantum uncertainties in aberrations, and in the
nonlinear part of the equations of motion for a nonparaxial beam, is pointed
out. The main purpose of this article is to understand the quantum mechanics of
electron beam optics though the influence of quantum effects on the optics of
present-day electron beam devices might be negligible.
