Journal of Mathematical Physics

On stochastic scattering

Piero Bassanini

Published Mar 1, 1973• 0 citations

Abstract

The Schrödinger equation for two-particle scattering with random potential is considered, including both the cases of a local and a nonlocal potential with convolution kernel. It is shown that the statistical fluctuations of the potential give rise to additional terms which increase the scattering cross section, and an evaluation of the additional terms in the Born approximation is given. A brief discussion about analyticity properties of the scattering amplitude under suitable assumptions is included.

DOI: 10.1063/1.1666329

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Journal of Mathematical Physics

On stochastic scattering

Piero Bassanini

Published Mar 1, 1973• 0 citations

Publisher

Abstract

DOI: 10.1063/1.1666329

Related papers

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E/\hbar{\lesssim}\gamma{\lesssim}\hbar/(2ma^{2})

because of the large inelastic cross section, where E is the energy of the incident particle with a mass m and a is the range of the potential.

v

with an associated energy-dependent nonlocal potential

{\mathcal{V}}_{k}

that does not couple to the evanescent waves. The scattering solutions $\ensuremath{\psi}(\mathbf{r})$ of the Schr\"odinger equation, $(\ensuremath{-}{\ensuremath{\nabla}}^{2}+{\mathcal{V}}_{k})\ensuremath{\psi}(\mathbf{r})={k}^{2}\ensuremath{\psi}(\mathbf{r})$, have the remarkable property that their Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\psi}}(\mathbf{p})$ vanishes unless

\mathbf{p}

corresponds to the momentum of a classical particle whose magnitude equals

k

. We construct a transfer matrix for this class of nonlocal potentials and explore its representation in terms of the evolution operator for an effective nonunitary quantum system. We show that the above approximation reduces to the first Born approximation for weak potentials and, similar to the semiclassical approximation, becomes valid at high energies. Furthermore, we identify an infinite class of complex potentials for which this approximation scheme is exact. We also discuss the appealing practical and mathematical aspects of this scheme.