  We investigate the radiation emitted by an ultrarelativistic electron
traveling in a 1-dimensional parabolic potential. Having in mind a simplified
model for beamstrahlung, we consider the realistic case of the electron motion
being highly directional, with the transverse momentum being much smaller than
the longitudinal one. In this case we can find solutions of the Dirac equation
and we calculate exactly the radiation emission using first-order perturbation
theory. We compare the results obtained to that obtained via the semi-classical
method of Baier and Katkov which includes quantum effects due to photon recoil
in the radiation emission but ignores the quantum nature of the electron
motion. On the one hand, we confirm a prediction of the semi-classical method
that the emission spectrum should coincide with that in the case of a linearly
polarized monochromatic wave. On the other hand, however, we find that the
semi-classical method does not yield the exact result when the quantum number
describing the transverse motion becomes small. In this way, we address
quantitatively the problem of the limits of validity of the semi-classical
method, which is known, generally speaking, to be applicable for large quantum
numbers. Finally, we also discuss which beam conditions would be necessary to
enter the studied regime where also the motion of the particles must be
considered quantum mechanically to yield the correct spectrum.
