The Thomson scattering describes photon scattering by free electrons in the classical electricity and magnetism. This is an approximation in the low energy limit where the quantum effect is no longer significant.
Let me first remind you of some fundamental equations of the classical electricity and magnetism involving the photon radiation. In the system with a charge density and its current (see the FIG. 1), the Maxwell equations give the electric scalar potential and the vector potential as
The solutions of the equations above describe the famous
When , then
, and the radiation
fields are considered to be plane waves. Then only the vector potential
describes everything because
Now let us consider the system that consist of an electron in the external field and . In the quantum picture, these fields correspond to a photon, which implies the interactions between a photon and an electron. When non-relativistic case, , the electrical force is by far larger than the Lorentz force. Then the equation of motion of an electron gives