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Minimum-error method for scattering problems in quantum mechanics: Two stable and efficient implementations.

Burcin Temel1, Greg Mills, Horia Metiu

  • 1Department of Chemistry and Biochemistry, University of California at Santa Barbara, Santa Barbara, California 93106-9510, USA.

The Journal of Physical Chemistry. A
|September 1, 2006
PubMed
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The minimum error method (MEM) offers a numerically stable and efficient alternative for quantum mechanics scattering problems. This method accurately calculates wave functions, outperforming the Kohn variational principle (KVP).

Area of Science:

  • Quantum mechanics
  • Computational physics
  • Scattering theory

Background:

  • Kohn variational principle (KVP) is a common method for scattering problems.
  • KVP can suffer from numerical instability.
  • An efficient and stable alternative is needed for quantum mechanical scattering calculations.

Purpose of the Study:

  • To examine two implementations of the minimum error method (MEM).
  • To demonstrate MEM as a numerically stable and efficient alternative to KVP.
  • To assess MEM's accuracy in calculating wave functions for scattering problems.

Main Methods:

  • Least-squares minimization of an error-functional (HPsi - EPsi)^2.
  • Representing the wave function using Chebyshev polynomials.

Related Experiment Videos

  • Minimizing error by varying expansion coefficients and R-matrix.
  • Utilizing Fast Chebyshev Transforms for derivative calculations.
  • Employing a conjugate-gradient procedure for error minimization.

    • Main Results:

    • Both MEM implementations provide efficient and numerically stable solutions.
    • MEM demonstrates superior stability compared to KVP, especially with R-matrix boundary conditions.
    • Accurate wave functions are obtained in the interaction region using MEM.
    • Chebyshev polynomials enable efficient derivative calculations.

    Conclusions:

    • The minimum error method (MEM) is a viable and robust alternative to KVP for quantum scattering.
    • MEM's numerical stability and accuracy make it suitable for complex scattering problems.
    • The Chebyshev polynomial implementation of MEM is particularly effective.