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

Perturbative quantum analysis and classical limit of the electron scattering by a solenoidal magnetic field