Classical and Quantum Description of the Kapitza-Dirac Effect

In this work we explore the classical and quantum descriptions of the Kapitza-Dirac effect, the scattering of charged particles by a standing wave electromagnetic field. In the classical case, initially we integrate numerically the covariant Lorentz force equation, for a statistical ensemble of electrons interacting with a superposition of two linearly polarized plane waves. We then employ the relativistic ponderomotive force approximation to the same initial conditions, and compare the two approaches across different field intensities. We present a graphical representation of the final momentum distribution as a function of the particles' initial positions for different field intensities, and we study the dependence of the magnitude of the momentum transfer on the field intensity. In the quantum case we consider the Klein-Gordon equation for a charged particle interacting with the same electromagnetic field. We look for a solution of a particular form, inspired by the well known Klein-Gordon Volkov states, and we show that in this case the Klein-Gordon equation reduces to a linear Goursat equation, which we solve numerically. The exact quantum results are compared both with the classical predictions and with an analytical approximation. We find that, in the intensity domain considered, the quantum and classical approximations are in agreement.