On Computation for a Hydrogen Atom in Arbitrary Magnetic Fields Using Finite Volume Method (Journal of Atomic and Molecular Sciences, 2014)
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.
Keywords: Schrödinger equation; hydrogen atom; magnetic field; finite volume method; eigenvalues; eigenvectors.