Axial and Landau gauge for a continuum electron in a homogeneous magnetic field

In an attempt to obtain a suitable access to the quantum description of a continuum electron in a combined Coulomb and homogeneous magnetic field, we first illustrate the complexity of the problem by selected classical trajectories. Subsequently, we examine the properties and consequences of two particular gauges for representing a homogeneous magnetic field, in particular, their usefulness in describing Coulombic bound and continuum states. We show that in the absence of a Coulomb field, the Landau gauge but not the axial gauge allows for a simple construction of asymptotic states that satisfy the requirement of tending to plane waves in the limit of vanishing field strength.