A symbolic-numerical algorithm for solving the eigenvalue problem for a hydrogen atom in the magnetic field: cylindrical coordinates

Authors:
Ochbadrakh Chuluunbaatar;Alexander Gusev;Vladimir Gerdt;Michail Kaschiev;Vitaly Rostovtsev;Valentin Samoylov;Tatyana Tupikova;Sergue Vinitsky
Affiliations:
Joint Institute for Nuclear Research, Moscow, Russia;Joint Institute for Nuclear Research, Moscow, Russia;Joint Institute for Nuclear Research, Moscow, Russia;Institute of Mathematics and Informatics, BAS, Sofia, Bulgaria;Joint Institute for Nuclear Research, Moscow, Russia;Joint Institute for Nuclear Research, Moscow, Russia;Joint Institute for Nuclear Research, Moscow, Russia;Joint Institute for Nuclear Research, Moscow, Russia
Venue:
CASC'07 Proceedings of the 10th international conference on Computer Algebra in Scientific Computing
Year:
2007

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Abstract

The boundary problem in cylindrical coordinates for the Schrödinger equation describing a hydrogen-like atom in a strong homogeneous magnetic field is reduced to the problem for a set of the longitudinal equations in the framework of the Kantorovich method. The effective potentials of these equations are given by integrals over transversal variable of a product of transverse basis functions depending on the longitudinal variable as a parameter and their first derivatives with respect to the parameter. A symbolic-numerical algorithm for evaluating the transverse basis functions and corresponding eigenvalues which depend on the parameter, their derivatives with respect to the parameter and corresponded effective potentials is presented. The efficiency and accuracy of the algorithm and of the numerical scheme derived are confirmed by computations of eigenenergies and eigenfunctions for the low-excited states of a hydrogen atom in the strong homogeneous magnetic field.