A causality condition for Schrödinger scattering

Dimitri R. Yafaev

A. Matzkin, S. V. Mousavi, Mordecai Waegell

David Park

The Schrödinger equation for the scattered particle is converted into an integral equation, in which the kernel is obtained by the use of the condition that the events described by the solutions of the equation must be causally related, i.e., that causes must always precede the effects which they produce. The integral equation is iterated once, and it is found that its solution corresponds to the usual description of a scattering process in terms of an undeflected wave traversing the system plus a radially scattered wave moving outwards. In the second part, it is shown that a simple relation exists between solutions of Schrödinger's equation of the sort just described and the solution of the initial-value problem associated with the equation of motion for a free particle. By this means a solution of the scattering problem is set up which describes how the wave function develops as the wave which it describes propagates through the system. The series which results was first given by Feynman. As an example, the first Born approximation for the scattering problem is developed by each approach.

V.Ya. Fainberg

H. M. Nussenzveig

A new formulation of the causality condition, in terms of the total wave function at a finite distance from the scatterer, is given. This condition is employed to derive all known analytic properties of the scattering amplitude in both variables (energy and scattering angle or momentum transfer) for a scatterer of finite radius. It is shown that all the assumptions are valid in the particular case of scattering by a local potential. The relations between causality, Wigner's inequality, and completeness are discussed.

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