The impact of the quantized transverse motion on radiation emission in a Dirac harmonic oscillator

Abstract

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.