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The One-Dimensional Stationary Schrödinger Equation

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Basic Concepts in Computational Physics

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Abstract

After giving a short survey of basic quantum mechanics the eigenvalue problem of the stationary one-dimensional Schrödinger equation is solved analytically for the quantum mechanical problem of a particle in a box. This eigenvalue problem is then solved numerically using Numerov’s shooting method. Analytical and numerical results are compared. The problem is then augmented by the introduction of three different potentials. The numerical solution of the new eigenvalue problems allows to investigate how the eigenvalues and eigenfunctions are influenced by these potentials.

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Notes

  1. 1.

    It depends on the problem on hand whether or not the index n will be continuous or discrete. For simplicity, we assume here n to be discrete.

  2. 2.

    This is only possible because the Schrödinger equation is linear.

  3. 3.

    Here we make use of

    $$\displaystyle{ \int \mathrm{d}u\sin ^{2}(u) = \frac{1} {2}\left [u -\cos (u)\sin (u)\right ]\;. }$$
    (10.42)

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Authors and Affiliations

  1. Faculty of Physics, University of Duisburg-Essen, Duisburg, Germany

    Benjamin A. Stickler

  2. Institute of Theoretical and Computational Physics, Graz University of Technology, Graz, Austria

    Ewald Schachinger

Authors
  1. Benjamin A. Stickler
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  2. Ewald Schachinger
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© 2016 Springer International Publishing Switzerland

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Stickler, B.A., Schachinger, E. (2016). The One-Dimensional Stationary Schrödinger Equation. In: Basic Concepts in Computational Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-27265-8_10

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