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Solving the bound-state Schrodinger equation by reproducing kernel interpolation

Hu1, Ho, Rabitz

  • 1Department of Chemistry, Princeton University, Princeton, New Jersey 08544-1009, USA.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|October 25, 2000
PubMed
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A novel grid method accurately solves the N-dimensional bound-state Schrodinger equation using reproducing kernel theory. This approach efficiently determines energy levels for quantum systems, outperforming existing methods.

Area of Science:

  • Quantum Mechanics
  • Computational Physics
  • Numerical Analysis

Background:

  • The Schrodinger equation is fundamental to quantum mechanics.
  • Solving the N-dimensional bound-state Schrodinger equation numerically presents significant challenges.
  • Existing methods like distributed Gaussian basis and distributed approximating functional methods have limitations.

Purpose of the Study:

  • To develop a novel grid method for solving the N-dimensional bound-state Schrodinger equation.
  • To utilize reproducing kernel Hilbert space and radial basis approximation theories.
  • To accurately and efficiently determine all energy levels up to the dissociation limit.

Main Methods:

  • Construction of a bounded reproducing kernel (RK) from the linear operator -nabla(2)(r)+lambda(2).

Related Experiment Videos

  • The RK, Lambda(alpha)(r,r(')), is a positive-definite radial basis function.
  • Application of the method to 1D Morse and 2D Henon-Heiles potentials.

    • Main Results:

    • The developed grid method accurately solves the N-dimensional bound-state Schrodinger equation.
    • The method successfully yields all energy levels up to the dissociation limit.
    • Numerical tests demonstrate high accuracy and efficiency.

    Conclusions:

    • The proposed reproducing kernel-based grid method offers an accurate and efficient solution for the N-dimensional bound-state Schrodinger equation.
    • This method shows promise for various quantum mechanical systems.
    • It provides a competitive alternative to existing numerical techniques.