Programme

O.39 -- Oral, VCTP-51

Date: Thursday, 6 August 2026

Time: 15:00 - 15:20

Magnetically Compressed Hydrogen Atom in Spherical Coordinates

Luong Le Hai (1), Oleg O. Kovalev (2,3), Alexander A. Gusev (2,3,4), Sergue I. Vinitsky (2,3,5)

(1) Ho Chi Minh city University of Education, Ho Chi Minh City, Viet Nam (2) Joint Institute for Nuclear Research, Dubna, Russian Federation (3) Dubna State University, Dubna, Russian Federation (4) School of Applied Sciences, Mongolian University of Science and Technology, Ulaanbaatar, Mongolia (5) RUDN University, Moscow, Russian Federation

A magnetically compressed hydrogen atom has axial symmetry, meaning that a 3-dimensional boundary value problem for a fixed magnetic quantum number is reduced to a 2-dimensional boundary value problem for partial differential equations with non-separable variables in a spherical or cylindrical coordinate system. For small values of the radial 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 quantum numbers at large magnetic fields. Therefore, solving the problem using the Galerkin method requires introducing a composite basis in both the spherical and cylindrical coordinate systems, as well as stitching them together, which is not a satisfactory procedure. A more general Kantorovich method for reducing a two-dimensional boundary-value problem to a second-order ODE system uses an orthogonal basis of oblate spheroidal functions as an expansion of the desired solution. These functions depend on the radial variable as a parameter, ensuring the correct asymptotic behavior of the solution to the original boundary-value problem. The boundary-value problem for the ODE system is solved using the finite element method. The performance of the computational scheme, implemented in Maple, which provides a user-friendly interface for developing algorithms for solving problems with non-separable variables, such as the Stark effect for a magnetically compressed Hydrogen atom in spherical coordinates with a complex-valued spectrum, is demonstrated by calculating the energy eigenvalues and eigenfunctions of the lower part of the spectrum.

Presenter: Luong Le Hai