Quantum-mechanical scattering in one dimension

Summary

The purpose of this mainly pedagogical review is to fill a lacuna in the usual treatment of scattering in quantum mechanics, by showing the essential of it in the simplest, one-dimensional setting. We define in this situation amplitudes and scattering coefficients and deal with optical and Levinson’ theorems as consequences of unitarity in coordinate or momentum space. Parity waves en lieu of partial waves, integral equations and Born series, etc., are defined naturally in this frame. Several solvable examples are shown. Two topics best studied in 1d are transparent potentials and supersymmetric quantum mechanics. Elementary analytical properties and general behaviour of amplitudes give rise to study inverse problems, that is, recovering the potential from scattering data. Isospectral deformations of the wave equation give relations with some nonlinear evolution equations (Lax), solvable by the inverse scattering method (Kruskal), and we consider the KdV equation as an example. We also refer briefly to some singular potentials, where, e.g., the essence of renormalization can be read off again in the simplest setting. The whole paper emphasizes the tutorial and introductory aspects.

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Acknowledgments

This work was supported by MEC grant FPA 2006-02315 & D.G.A. grant 2006/E24/2. I thank also J. de Lucas for technical help.

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Correspondence to Luis J. Boya.

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Boya, L.J. Quantum-mechanical scattering in one dimension. Riv. Nuovo Cim. 31, 75–139 (2008). https://doi.org/10.1393/ncr/i2008-10030-4

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Key words

  • PACS 03.65.Nk–Scattering theory