Quantum mechanics of round magnetic electron lenses with Glaser and power law models of $B(z)$

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.