Hydrogen Atom Structures: An Exploratory Study using Numerov-Cooley’s Solution of Schrödinger’s Equation
In this paper, we present a computationally efficient method to solve the Schrödinger equation for hydrogen atoms using atomic units. The method incorporates the Numerov-Cooley Shooting Method, a well-known fourth-order method, to attain high precision solutions. Coupled with the Cooley's Energy correction formula, the method achieves rapid convergence to accurate energy eigenvalues. With defined boundary conditions, this method enables efficient computation of the radial probability density for atoms. The method is implemented using FORTRAN 90/95, a language favored for its computational efficiency. The results are represented through various hydrogen states, providing a detailed understanding of their energy associations. Additionally, data visualization tools like matplotlib are employed for effective representation of these outputs. The method presented introduces a proficient approach for solving complex equations in quantum mechanics and can potentially be extended to other atomic systems.
