  A well known example in quantum electrodynamics (QED) shows that Coulomb
scattering of unpolarized electrons, calculated to lowest order in perturbation
theory, yields a results that exactly coincides (in the non-relativistic limit)
with the Rutherford formula. We examine an analogous example, the classical and
perturbative quantum scattering of an electron by a magnetic field confined in
an infinite solenoid of finite radius. The results obtained for the classical
and the quantum differential cross sections display marked differences. While
this may not be a complete surprise, one should expect to recover the classical
expression by applying the classical limit to the quantum result. This turn not
to be the case. Surprisingly enough, it is shown that the classical result can
not be recuperated even if higher order corrections are included. To recover
the classic correspondence of the quantum scattering problem a suitable
non-perturbative methodology should be applied.
