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The Schrödinger equation in a 2D cylindrical coordinate system is numerically solved for the ground state and a few excited states of the hydrogen atom in arbitrary magnetic fields. The second order discretization of the PDEs on finite volumes results in a set of algebraic equations that are solved simultaneously using Gauss-Seidel Algebraic Multi-Grid (AMG) solver. The modified Stodola-Vianello method is implemented using Gram-Schmidt orthogonalization process to extract the first few energy states and their wave functions concurrently. A detailed mesh convergence study suggests that both energies and wave functions correctly approach toward the unknown exact solutions.
}, issn = {2079-7346}, doi = {https://doi.org/10.4208/jams.110813.021414a}, url = {http://global-sci.org/intro/article_detail/jams/8306.html} }The Schrödinger equation in a 2D cylindrical coordinate system is numerically solved for the ground state and a few excited states of the hydrogen atom in arbitrary magnetic fields. The second order discretization of the PDEs on finite volumes results in a set of algebraic equations that are solved simultaneously using Gauss-Seidel Algebraic Multi-Grid (AMG) solver. The modified Stodola-Vianello method is implemented using Gram-Schmidt orthogonalization process to extract the first few energy states and their wave functions concurrently. A detailed mesh convergence study suggests that both energies and wave functions correctly approach toward the unknown exact solutions.