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Robb and Bonifacio (2011) claimed that a previously neglected quantum effectresults in noticeable changes in the evolution of the energy distributionassociated with spontaneous emission in long undulators. They revisitedtheoretical models used to describe the emission of radiation by relativisticelectrons as a continuous diffusive process, and claimed that in the asymptoticlimit for a large number of undulator periods the evolution of the electronenergy distribution occurs as discrete energy groups according to Poissondistribution. We show that these novel results have no physical sense, becausethey are based on a one-dimensional model of spontaneous emission and assumethat electrons are sheets of charge. However, electrons are point-likeparticles and, as is well-known, the bandwidth of the angular-integratedspectrum of undulator radiation is independent of the number of undulatorperiods. If we determine the evolution of the energy distribution using athree-dimensional theory we find the well-known results consistent with acontinuous diffusive process. The additional pedagogical purpose of this paperis to review how quantum diffusion of electron energy in an undulator withsmall undulator parameter can be simply analyzed using the Thomsoncross-section expression, unlike the conventional treatment based on theexpression for the Lienard-Wiechert fields.

On quantum effects in spontaneous emission by a relativistic electron beam in an undulator