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Journal of Mathematical Physics

Schrödinger Scattering Amplitude. I

Alex Grossmann, Tai Tsun Wu

Published Sep 1, 1961• 54 citations
Publisher

Abstract

The Schrödinger scattering amplitude for a fixed potential is studied as a function of the three components of the initial momentum, the three components of the final momentum, and the square root of the energy.
DOI
10.1063/1.1703760

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Schrödinger Scattering Amplitude. II

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The results of the preceding paper are used to indicate procedures of calculation of the scattering amplitude, obtain several expansions, find bounds on the variation of the amplitude under a change in the potential, and study multiple scattering.

Journal of Mathematical Physics•Sep 1, 1961•15 citations
The concept of a generalized scattering amplitude is introduced and applied to quantum potential scatterings. After an outgoing spherical wave is factored out from a wave function, the remaining wave envelope is defined as the generalized scattering amplitude. The transformed Schr\"odinger equation, in terms of the generalized scattering amplitude, can be solved numerically by using a finite-difference method over the entire scattering domain. Example problems of scatterings by a spherical potential well and the (${\mathit{e}}^{\mathrm{\ensuremath{-}}}$,H) static field interaction potential are solved to validate the theoretical formulation and numerical method. The far-field solutions for the ordinary scattering amplitude and differential cross section agree very well with those obtained from the partial-wave analysis. The radial profiles of the generalized scattering amplitude and particle density function over the entire scattering region are also presented, and their properties are discussed. These results demonstrate that the method can be used to yield a complete and accurate solution to scattering problems involving arbitrary interaction potentials.
One-dimensional quantum scattering from a local potential barrier is considered. Analytical properties of the scattering amplitudes have been investigated by means of the integral equations equivalent to the Schrodinger equations. The transition and reflection amplitudes are expressed in terms of two complex functions of the incident energy, which are similar to the Jost function in the partial-wave scattering. These functions are entire for finite-range potentials and meromorphic for exponentially decreasing potentials. The analytical properties result from locality of the potential in the wave equation and represent the effect of causality in time dependence of the scattering process.
One-dimensional quantum scattering from a local potential barrier is considered. Analytical properties of the scattering amplitudes have been investigated by means of the integral equations equivalent to the Schrödinger equations. The transition and reflection amplitudes are expressed in terms of two complex functions of the incident energy, which are similar to the Jost function in partial-wave scattering. These functions are entire for finite-range potentials and meromorphic for exponentially decreasing potentials. The analytical properties result from the locality of the potential in the wave equation and represent the effect of causality in the time dependence of the scattering process.