  The rotation of electron vortex beams (EVBs) presents a complex interplay of
the Gouy phase characterizing free-space behavior and Landau states or Larmor
rotation observed in magnetic fields. Despite being studied separately, these
phenomena manifest within a single beam during its propagation in magnetic
fields, lacking a comprehensive description. We address this by utilizing exact
solutions of the relativistic paraxial equation in magnetic fields, termed
"paraxial Landau modes". The paraxial Landau modes describe the quantum states
of EVBs in magnetic fields. Our study of rotation angles demonstrates
consistency with experimental data, supporting the practical presence of these
modes. We provide a unified description of different regimes under generalized
Gouy rotation, linking the Gouy phase to EVB rotation angles. This connection
enhances our understanding of the Gouy phase and can be extended to nonuniform
magnetic fields. Our theoretical analysis is validated through numerical
simulations using the Chebyshev method. This work offers new insights into the
dynamics of EVBs in magnetic fields and suggests practical applications in beam
manipulation and beam optics of vortex particles.
