Riccati differential equation for quantum mechanical bound states: Comparison of numerical integrators

Authors

  • Chia-Chun Chou,

    1. Institute for Theoretical Chemistry and Department of Chemistry and Biochemistry, The University of Texas at Austin, Austin, TX 78712
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  • Robert E. Wyatt

    Corresponding author
    1. Institute for Theoretical Chemistry and Department of Chemistry and Biochemistry, The University of Texas at Austin, Austin, TX 78712
    • Institute for Theoretical Chemistry and Department of Chemistry and Biochemistry, The University of Texas at Austin, Austin, TX 78712
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Abstract

Computational comparison is presented of several integrators for the Riccati differential equation for the one-dimensional quantum mechanical bound state problem. The computational method for the quantum Hamilton-Jacobi equation for stationary states is briefly reviewed. An interpolation formula for the quantum momentum function is proposed for the evaluation of the phase integral. The Möbius integrator, the R-matrix propagation method, the log derivative method, and the symplectic integrator can accurately pass through singularities in the solution. These four integrators are actually Möbius transformations with different coefficients. Analytical analysis and numerical results demonstrate that the classical momentum is a good choice for the initial condition to obtain the bound state solution of the Riccati differential equation as long as the starting point of the propagation is far enough from the origin. The computational comparison is demonstrated for the harmonic oscillator. The numerical results indicate that the symplectic integrators achieve highest accuracy, and the simplicity of these algorithms shows an advantage over the other methods. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2008

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