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Fourier grid hamiltonian method and lagrange-mesh calculations

Semay1

  • 1Universite de Mons-Hainaut, Place du Parc, 20, B-7000 Mons, Belgium.

Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
|January 4, 2001
PubMed
Summary

The Fourier grid Hamiltonian (FGH) method accurately computes quantum mechanical solutions. This study reveals FGH is a specific case of the Lagrange-mesh (LM) method, enhancing its applicability.

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Area of Science:

  • Quantum Mechanics
  • Computational Physics
  • Theoretical Chemistry

Background:

  • Solving the Schrödinger equation and spinless Salpeter equation is crucial for understanding bound states.
  • Existing methods like the Lagrange-mesh (LM) method offer simplified approaches to these quantum mechanical problems.
  • The Fourier grid Hamiltonian (FGH) method has emerged as an effective technique for eigenvalue and eigenfunction computation.

Purpose of the Study:

  • To demonstrate that the Fourier grid Hamiltonian (FGH) method is a specialized instance of the Lagrange-mesh (LM) method.
  • To establish a theoretical foundation for the FGH method by linking it to the established LM framework.
  • To extend the capabilities of the FGH method, enabling the evaluation of eigenfunctions at arbitrary points.

Main Methods:

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  • The study analyzes the mathematical underpinnings of both the FGH and LM methods.
  • It identifies the treatment of the kinetic energy operator in FGH as a discrete Fourier transform within the LM context.
  • This involves comparing the computational steps and theoretical basis of both approaches.

Main Results:

  • The FGH method is mathematically shown to be a special case of the LM method.
  • This connection is established through the specific treatment of the kinetic energy operator using discrete Fourier transforms in FGH.
  • The eigenfunctions computed via FGH can now be accurately evaluated at any desired point, not just grid points.

Conclusions:

  • The Fourier grid Hamiltonian (FGH) method is validated as a specific, effective implementation within the broader Lagrange-mesh (LM) framework.
  • This provides a robust theoretical basis for the FGH method.
  • The findings significantly enhance the flexibility and utility of the FGH method for solving quantum mechanical equations.