  We consider an electron constrained to move on a surface with revolution
symmetry in the presence of a constant magnetic field $B$ parallel to the
surface axis. Depending on $B$ and the surface geometry the transverse part of
the spectrum typically exhibits many crossings which change to avoided
crossings if a weak symmetry breaking interaction is introduced. We study the
effect of such perturbations on the quantum propagation. This problem admits a
natural reformulation to which tools from molecular dynamics can be applied. In
turn, this leads to the study of a perturbation theory for the time dependent
Born-Oppenheimer approximation.
