FEM solver for confined quantum system: hydrogen atom or ion in uniform magnetic field
Oleg O. Kovalev1,2, Vladimir L. Derbov3, Alexander A. Gusev2,3,4,
Luong Le Hai 5, Sergue I. Vinitsky1,2,6; 1Joint Institute for Nuclear Research, Dubna, Russian Federation; 2Dubna State University, Dubna, Russian Federation; 3N.G. Chernyshevsky Saratov State University, Saratov, Russian
Federation; 4School of Applied Sciences, Mongolian University of Science and
Technology, Ulaanbaatar, Mongolia; 5Ho Chi Minh city University of Education, Ho Chi Minh City, Viet Nam; 6RUDN University, Moscow, Russian Federation
Abstract
A magnetically compressed hydrogen atom or ion have axial symmetry,
meaning that a 3-dimensional boundary value problem (BVP) for a fixed
magnetic quantum number is reduced to a 2-dimensional BVP for the
partial differential equations with non-separable variables in the spherical,
cylindrical, parabolic or spheroidal coordinate systems, respectively. For
small values of the radial or quasiradial variable, the Coulomb interaction
predominates, while for large values, the potential energy of interaction
with the magnetic field predominates. Accordingly, the energy eigenvalues
are characterized by spherical quantum numbers at small magnetic fields
and cylindrical or parabolic quantum numbers at large magnetic fields.
Therefore, solving the problem using the Galerkin method requires
introducing a composite basis taking into account these properties, that
are revealed in the corresponding coordinate systems, respectively, as well
as stitching them together, which is cumbersome procedure. Thus, the
solving of the 2D BVP by finite element method is more relevant. Here it’s
demonstrated by calculating the eigenvalues and eigenfunctions of the
lower part of the energy spectrum of Hydrogen atom and ion in uniform
magnetic field by means of author’s program GCMFEM 8 [O.O. Kovalev et
al, Lecture Notes in Computer Sci. 16235, 210 (2026).] for solving up to six
dimensional BVPs that is also applicable to a wide class of the quantum
confined systems.
Speaker
Vladimir L. Derbov
Saratov State University
Russia
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