Mutual Refraction of Electrons and Photons

Abstract

The equation of motion governing the time-averaged motion of a classical electron in an oscillating electromagnetic field is derived under very general conditions. It is shown that the motion is that of a relativistic particle of variable rest mass $m{(1+{\mu }^2)}^{\frac{1}{2}}$, where ${\mu }^2$ is the parameter proportional to the field intensity, introduced earlier. Nonrelativistically, it is that of a particle with the effective potential-energy function $\frac{1}{2}m{c}^2{\mu }^2$. The complete analogy between the processes of refraction of light by electrons and of electrons by light is emphasized. It is shown that in the case of focused laser beams, effects substantially larger than those originally predicted are to be expected. The interaction of electrons with standing waves is discussed with particular reference to the Kapitza-Dirac effect, and it is shown that a modified effect may perhaps be expected at high intensities and low electron velocities. The use of classical electrodynamics is justified by showing, with the help of the WKB approximation, that specifically quantum effects should normally be negligible. A model which helps to explain the complementarity between the two refraction effects is presented in which the electrons and photons are treated as classical relativistic fluids. The relationship of this model to quantum electrodynamics is pointed out.