We derive a four-component spinor wave function for an electron in a helical undulator which, in the relativistic limit, successfully reproduces all of the results of the classical calculation for the radiation angular distribution, polarization, and energy spectrum. This wave function also allows the nonclassical calculation of the spin flip. For electron energies below the several hundred GeV range, the spin-flip probability is negligible, but for higher energies and high undulator strengths it cannot be neglected if beam polarization is to be preserved, even though nonflip radiation still greatly dominates the radiation intensity. The anomalous magnetic moment ${a}_{e}$ is seen to play a dominant role in the helical undulator spin-flip process. The probability of spin flip is shown to have a ${\ensuremath{\gamma}}^{5}$ dependence on electron energy. For high energy electrons, the direction of spin flip is independent of the handedness of the undulator. As a result, at sufficiently high energy, a polarized electron or positron beam rapidly depolarizes by spontaneous radiation in the undulator. Because of the high correlation between the direction of spin flip and the handedness of the spin-flip radiation, we conjecture that it may be possible to polarize electrons by using the intense circularly polarized photons in the helical undulator to stimulate spin flip.