SMatrix and Causality Condition. II. Nonrelativistic Particles

The application of the "causality condition" to the $S$ matrix for nonrelativistic particles encounters several difficulties: (a) there is no maximum velocity; (b) the interference of ingoing and outgoing waves has to be taken into account; (c) wave packets with a sharp front do not exist. The condition is therefore reformulated as follows: At any time the total probability of finding the particle outside the scattering center shall not be greater than 1, for every form of the incident wave packet. From this follows for spherical waves that $S$, as a function of the momentum $p$, is analytic and holomorphic in the first quadrant and that $e^{2iap}S\left(p\right)\mathrm{}$ (where $a$ is the radius of the scattering center) has an imaginary part ⩽ 1. That suffices to give an explicit integral representation and a product expansion for $S$, but these permit a more general form for $S$ than is usually envisaged. If, however, the usual symmetry relation $S(-p)=S{\mathrm{}\left(p\right)\mathrm{}}^{*}$ is assumed in addition to the causality condition, more specific equations can be derived, which are direct generalizations of those in Part I. In particular, integral relations between the real and imaginary parts of $S$, and the properties that Wigner found for the $R$ matrix can be deduced.