Solution of the singular Dirac equation using mapped trigonometric functions.

Area of Science:

• Quantum Mechanics
• Computational Physics
• Theoretical Chemistry

Background:

• Solving the Dirac equation is crucial for understanding relativistic quantum systems.
• The singular Coulomb potential presents significant challenges for numerical methods.
• Existing spectral methods for the Dirac equation have limitations in efficiency and precision.

Purpose of the Study:

• To develop a novel, highly effective spectral method for the Dirac equation with a singular Coulomb potential.
• To leverage the fast Fourier transform for direct application in the numerical solution.
• To achieve high-precision results comparable to state-of-the-art methods.

Main Methods:

• Utilized trigonometric sine functions as the basis set.
• Introduced a specialized scaling function to handle the Coulomb potential's singularity.
• Employed Chebyshev-Gauss quadrature within a discrete variable representation framework.
• Applied fast Fourier transformation for computational efficiency.

Main Results:

• The proposed spectral method demonstrates excellent convergence properties.
• Achieved remarkable precision of approximately 10^{-32} using quadruple precision arithmetic.
• The method's numerical convergence is on par with the established Lagrange mesh method for Dirac equation solutions.

Conclusions:

• The developed spectral method offers an effective and efficient approach for solving the Dirac equation with singular potentials.
• This technique provides a viable alternative to existing methods, particularly for problems requiring high accuracy.
• The method's success highlights the potential of spectral techniques combined with appropriate basis functions and quadrature rules for relativistic quantum problems.