P-stable eighth algebraic order methods for the numerical solution of the Schrödinger equation

Area of Science:

• Numerical analysis
• Quantum mechanics
• Computational physics

Background:

• The Schrödinger equation is fundamental in quantum mechanics, but its numerical integration can be computationally intensive.
• Existing methods often require small step sizes, limiting efficiency for large-scale simulations.
• P-stable methods offer advantages in maintaining accuracy with larger integration steps.

Purpose of the Study:

• To develop a new P-stable numerical integration method for the Schrödinger equation.
• To achieve a higher algebraic order of accuracy (order eight).
• To improve the efficiency of numerical solutions for quantum mechanical problems.

Main Methods:

• Development of a novel P-stable method with algebraic order eight.
• Integration of the new method with a previously developed sixth-order P-stable method (Simos, 1997).
• Creation of a new variable step size integration scheme.

Main Results:

• The developed eighth-order P-stable method enables the use of larger step sizes.
• Numerical tests on the radial Schrödinger equation phase-shift problem demonstrated efficiency.
• Application to coupled differential equations from the Schrödinger equation confirmed the method's effectiveness.

Conclusions:

• The new eighth-order P-stable method is efficient for solving the Schrödinger equation.
• The variable step method derived from it further enhances computational performance.
• This advancement offers a more efficient approach to numerical quantum mechanics.