Abstract
We propose a method to study the interaction of a single-mode laser beam (as an external field) containing a finite number of photons with a free electron. In particular, we give some numerical results of how the photon number ${l}_{k,\ensuremath{\varepsilon}}$ affects the differential scattering cross section when the initial state of an external field is a Fock state $|{l}_{k,\ensuremath{\varepsilon}}\ensuremath{\rangle}$. Besides, for arbitrarily polarized light, we prove that our model will return to the semiclassical theory as ${l}_{k,\ensuremath{\varepsilon}}\ensuremath{\rightarrow}\ensuremath{\infty}$.
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