It is shown that if the wave function of a nonrelativistic particle satisfies the Schr\"odinger equation with a velocity-independent potential, then its scattering amplitudes (and the
S matrix in general) satisfy the same dispersion formulas as those derived for the scattering of light. In the present derivation, the validity of a perturbation expansion and certain integrability of the potential are assumed, and the requirement of the outgoing wave Green's function replaces the condition of strict causality for the scattering of light. The scattering amplitude for fixed momentum transfer is also shown to satisfy the dispersion relations. Together with the unitarity of the
S matrix, the complete
S matrix is determined by the Fourier transform of the potential through an iterative procedure using the dispersion relations. If the potential possesses bound states, then all the dispersion formulas are modified to include residue terms corresponding to singularities on the positive imaginary axis of the momentum plane. The necessity of these modifications is related to the divergence of the perturbation series for small wave numbers. Essential singularities of the
S matrix due to exponentially damped potentials give no additional contribution to the dispersion formulas.
With the aid of a Coulomb ``free propagator'' Uc(±)(t) constructed from the asymptotic form of the conventional time-independent solution to the scattering problem in a pure Coulomb field, Mo/ller wave operators are shown to exist for general Coulomb-like (Coulomb + short-range) potentials. It is emphasized that when scattering occurs in such long-range fields, one should distinguish three classes of states: the prepared and detected states, the scattering states, and certain asymptotic states which are peculiar to long-range potentials. The existence of the third class as a class distinct from the first suggests a connection between Uc(±)(t) and the quantum theory of measurements.
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
e2iapS(p) (where
a is the radius of the scattering center) has an imaginary part \ensuremath{\leqslant} 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 $S(\ensuremath{-}p)=S{(p)}^{*}$ 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.
Scattering theory provides a convenient framework for the solution of a variety of problems. In this thesis we focus on the combination of boundary conditions and scattering potentials and the combination of non-overlapping scattering potentials within the context of scattering theory. Using a scattering t-matrix approach, we derive a useful relationship between the scattering t-matrix of the scattering potential and the Green function of the boundary, and the t-matrix of the combined system, effectively renormalizing the scattering t-matrix to account for the boundaries. In the case of the combination of scattering potentials, the combination of t-matrix operators is achieved via multiple scattering theory. We also derive methods, primarily for numerical use, for finding the Green function of arbitrarily shaped boundaries of various sorts. These methods can be applied to both open and closed systems. In this thesis, we consider single and multiple scatterers in two dimensional strips ...
In this paper we extend our previous analysis of the scattering of wave packets in one dimension to the case of the square-well potential. The analytic properties of the general scattering solution are emphasized thereby making the analysis useful as introductory material for a more sophisticated S-matrix treatment. The square-well model is particularly interesting because of its application to the deuteron problem. Resonance scattering, barrier penetration, time delay, and line shape are discussed at the level of the first-year graduate student.