Compton Scattering of an Intense Photon Beam

The fully quantum-mechanical description of the Compton scattering of a photon beam is shown to be equivalent under certain conditions to a semiclassical description, thereby confirming the prediction by Brown and Kibble and by Goldman of an intensity-dependent increment to the frequency shift. If the incident beam is in a coherent state, the equivalence is exact under all conditions. If it contains a definite number of photons, the equivalence is approximate, and requires many photons and convergence of the expansion in powers of the photon density. The demonstration is based upon the explicit use of wave packets to introduce the boundary conditions, and the equivalence is shown to hold in the sense that the transition probabilities, but not the transition amplitudes, are the same in the quantum and semiclassical treatments. The apparent failure of energy-momentum conservation implied by the incremental shift is seen to be accounted for by the energy-momentum uncertainty of the ensemble of localized wave packets even in the monochromatic limit. It is proved in the Appendix that the coherent states are the only kind for which the equivalence is exact.