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Discrete variable representation for singular Hamiltonians.

Barry I Schneider1, Nicolai Nygaard

  • 1Physics Division, National Science Foundation, Arlington, Virginia 22230, USA. bschneid@nsf.gov

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
PubMed
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This study presents a simplified discrete variable representation (DVR) method for solving quantum mechanics problems with singular Hamiltonians. The approach effectively handles singularities using orthogonal polynomials and specific quadrature rules, ensuring accurate bound and continuum state calculations.

Area of Science:

  • Quantum Mechanics
  • Computational Chemistry
  • Theoretical Physics

Background:

  • Solving the Schrödinger equation with singular Hamiltonians is computationally challenging.
  • Existing methods often involve complex transformations to manage singularities.
  • Accurate treatment of both bound and continuum states is crucial in quantum problems.

Purpose of the Study:

  • To develop a more efficient and less complex method for applying the discrete variable representation (DVR) to Schrödinger problems with singular Hamiltonians.
  • To demonstrate the adequacy of an orthogonal polynomial basis with specific quadrature rules for handling singularities.
  • To validate the method's accuracy and applicability to diverse quantum systems.

Main Methods:

  • Application of the discrete variable representation (DVR) using an orthogonal polynomial basis.

Related Experiment Videos

  • Utilization of Gauss-Lobatto or Gauss-Radau quadrature rules to include singular points in the mesh.
  • Exclusion of DVR functions corresponding to singular points to ensure well-behaved matrix elements.
  • Direct application to the hydrogen atom as a test case.

    • Main Results:

    • The proposed DVR method effectively handles singular Hamiltonians without complex transformations.
    • Matrix elements remain well-behaved, boundary conditions are met, and calculations exhibit rapid convergence.
    • The method accurately describes both bound states and continuum solutions.
    • The hydrogen atom calculation confirms the method's accuracy.

    Conclusions:

    • The orthogonal polynomial basis with Gauss-Lobatto/Radau quadrature offers a robust and efficient approach to singular quantum problems.
    • This simplified DVR method avoids the complexity of alternative singularity treatments.
    • The technique is versatile, applicable to both bound and continuum states in various quantum systems.