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A new finite difference scheme adapted to the one-dimensional Schrödinger equation

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Abstract

We present a new discretisation scheme for the Schrödinger equation based on analytic solutions to local linearisations. The scheme generates the normalised eigenfunctions and eigenvalues simultaneously and is exact for piecewise constant potentials and effective masses. Highly accurate results can be obtained with a small number of mesh points and a robust and flexible algorithm using continuation techniques is derived. An application to the Hartree approximation for SiGe heterojunctions is discussed in which we solve the coupled Schrödinger-Poisson model problem selfconsistently.

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Author notes

  1. Bernard J. Geurts

    Present address: Department of Applied Mathematics, Twente University, P.O. Box 217, 7500, AE Enschede, The Netherlands

Authors and Affiliations

  1. Philips Research Laboratories, P.O. Box 80 000, 5600, JA Eindhoven, The Netherlands

    Bernard J. Geurts

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  1. Bernard J. Geurts
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To the memory of my good friend Erik van Loon, taken from us so soon

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Geurts, B.J. A new finite difference scheme adapted to the one-dimensional Schrödinger equation. Z. angew. Math. Phys. 44, 654–672 (1993). https://doi.org/10.1007/BF00948481

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  • DOI: https://doi.org/10.1007/BF00948481

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