While the exact amplitude for scattering in the presence of an external electromagnetic field is invariant under a change of gauge, the invariance property will not necessarily be preserved in approximations, and this may lead to inaccuracies and ambiguities in the results of numerical calculations. A method for removing this undesirable feature through a reformulation of the scattering problem is proposed here. The equations which determine the time evolution of the system are rewritten in a form which involves not the (gauge-dependent) vector and scalar potentials associated with the external field but rather effective potentials which are gauge independent; this allows for the introduction of gauge-invariant approximations in a systematic way. To focus on the essential features the method is described in the context of the relatively simple problem of nonrelativistic potential scattering but greater generality is possible. As an illustration of the method the problem of scattering in a slowly varying external field is considered and an approximation of the ‘‘low-frequency’’ type is obtained. The result, whose gauge invariance is verified explicitly, represents an improvement over earlier versions, having been obtained through a more accurate treatment of the interaction of the projectile with the external field in intermediate states of the scattering process.

• Received 24 February 1986

DOI:https://doi.org/10.1103/PhysRevA.34.978