Fourier grid hamiltonian method and lagrange-mesh calculations

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:

• 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.