Sum rule and classical limit for scattering in a low-frequency laser field

A previously derived low-frequency approximation for scattering in a laser field represents the amplitude for scattering with the absorption or emission of a specified number of photons in terms of the physical field-free scattering amplitude, a result which holds even with the inclusion of a correction term of first order in the frequency of the field. In the general case, where the polarization of the field is arbitrary and the dipole approximation is not assumed, the cross section depends on the phase as well as the magnitude of the field-free scattering amplitude. It is shown here that this phase dependence disappears when the cross section is summed over all possible final states of the field. The result of this summation is identical (within the limits set by the domain of validity of the low-frequency approximation itself) to that which would be predicted from a classical description of the motion of the electron in the field, the collision taking place instantaneously and without influence from the field. This sum rule and its classical interpretation, obtained here in a nonrelativistic potential scattering model, is of the form derived some time ago by Brown and Goble on the basis of general field-theoretic considerations.