Note on the Radiation Field of the Electron

Previous methods of treating radiative corrections in non-stationary processes such as the scattering of an electron in an atomic field or the emission of a $\ensuremath{\beta}$-ray, by an expansion in powers of $\frac{{e}^{2}}{\ensuremath{\hbar}c}$, are defective in that they predict infinite low frequency corrections to the transition probabilities. This difficulty can be avoided by a method developed here which is based on the alternative assumption that $\frac{{e}^{2}\ensuremath{\omega}}{m{c}^{3}}$, $\frac{\ensuremath{\hbar}\ensuremath{\omega}}{m{c}^{2}}$ and $\frac{\ensuremath{\hbar}\ensuremath{\omega}}{c\ensuremath{\Delta}p}$ ($\ensuremath{\omega}$ = angular frequency of radiation, $\ensuremath{\Delta}p$ = change in momentum of electron) are small compared to unity. In contrast to the expansion in powers of $\frac{{e}^{2}}{\ensuremath{\hbar}c}$, this permits the transition to the classical limit $\ensuremath{\hbar}=0$. External perturbations on the electron are treated in the Born approximation. It is shown that for frequencies such that the above three parameters are negligible the quantum mechanical calculation yields just the directly reinterpreted results of the classical formulae, namely that the total probability of a given change in the motion of the electron is unaffected by the interaction with radiation, and that the mean number of emitted quanta is infinite in such a way that the mean radiated energy is equal to the energy radiated classically in the corresponding trajectory.