  Many calculations in strong field quantum field theory are carried out by
using a simple field geometry, often neglecting the spacial field envelope. In
this article, we simulate the electron diffraction quantum dynamics of the
Kapitza-Dirac effect in a Gaussian beam standing light wave. The
two-dimensional simulation is computed in a relativistic framework, by solving
the Dirac equation with the fast Fourier transform split operator method.
Except the numerical propagation method, our results are obtained without
applying approximations and demonstrate that a spin-flip in the Kapitza-Dirac
effect is possible. We further discuss properties, such as the validity of a
plane wave approach for the theoretical description, the influence of the
longitudinal polarization component due to laser beam focusing and higher order
diffraction peaks in Kapitza-Dirac scattering.
