Trajectory and Mass Shift of a Classical Electron in a Radiation Pulse

Abstract

We solve the Hamilton-Jacobi equations of motion of a classical charge in the presence of a traveling pulse of electromagnetic radiation. The orbit solutions for arbitrary radiation pulse shape are given in terms of one-dimensional integrals, and for a particular choice of singly peaked pulse shape are given in detail as a function of the particle's proper time. The question of the particle mass is then investigated, and an expression for the classical interacting mass is given as an explicit function of the time width of the radiation pulse. The average shift in the square of the mass, $\ensuremath{\Delta}{m}^{2}$, varies smoothly from zero to a maximum and then to zero again as the pulse overtakes and passes the charged particle. The maximum value is $\ensuremath{\Delta}{m}^{2}=\frac{1}{2}{e}^{2}{a}^{2}$ (where $e$ and $a$ are the particle charge and the field amplitude). We conclude that for the free-electron-photon scattering experiments currently being contemplated at optical frequencies the maximum figure is likely to be the relevant one.