Soft-Photon Theorem for Bremsstrahlung in a Potential Model

The bremsstrahlung matrix element calculated from the Schrödinger equation for two particles interacting with each other via a local potential and with the electromagnetic field in the standard gauge-invariant manner is shown to satisfy the soft-photon theorem. That is, the first two terms of the expansion of the radiative matrix element in powers of the photon's energy are calculable from a knowledge of the nonradiative matrix element. The proof is given first without, then with, spin for arbitrary order of perturbation theory in the strong interaction. While the derivation is very different from the one which Low used for a relativistic theory, the final formulas are similar in appearance and agree in the nonrelativistic limit. For low particle momenta, it is found that there is a relation between the on- and off-energy-shell derivatives of the nonradiative $T$ matrix to the leading order in the momentum. Furthermore, the $p$-wave contribution to the internal-emission matrix element is of the same order in the momentum as the $s$-wave part. For particle momenta much less than the reciprocal of the $s$-wave scattering length, the second term of the expansion of the bremsstrahlung matrix element in powers of the photon's energy is negligible compared with the first term.