Theory of Bremsstrahlung and Pair Production. I. Differential Cross Section

The differential cross sections for bremsstrahlung and pair production are calculated without the use of the Born approximation, assuming the energy of the electron to be large compared to $m{c}^2$ both in initial and final state. The wave functions in initial and final state are essentially those previously proposed by Furry (Sec. II). It is proved in Sec. III that our wave functions agree with the exact ones of Darwin except for terms of relative order $\frac{a^2}{l^2}$, where $a=\frac{Z{e}^2}{\hslash c}$ and $l$ the angular momentum, and that this agreement holds for any energy of the electron. An independent proof is given in Sec. IX, showing that the Furry wave functions give the matrix element correctly except for terms of relative order $\frac{1}{\epsilon }$.