Two-dimensional hydrogen-like atom in a weak magnetic field

Radosław Szmytkowski

doi:10.1140/epjp/i2018-12126-7

Radosław Szmytkowski

Open Access

https://doi.org/10.1140/epjp/i2018-12126-7

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Abstract

.We consider a non-relativistic two-dimensional (2D) hydrogen-like atom in a weak, static, uniform magnetic field perpendicular to the atomic plane. Within the framework of the Rayleigh-Schrödinger perturbation theory, using the Sturmian expansion of the generalized radial Coulomb Green function, we derive explicit analytical expressions for corrections to an arbitrary planar hydrogenic bound-state energy level, up to the fourth order in the strength of the perturbing magnetic field. In the case of the ground state, we correct an expression for the fourth-order correction to energy available in the literature.

Highlights

We have come across the need to know exact analytical representations for low-order perturbation theory corrections to an arbitrary energy level of a two-dimensional analogue of a hydrogen-like atom placed in a weak and uniform magnetic field perpendicular to the atomic plane
Exact values of the second-order corrections for states with the principal quantum numbers 1 n 4 may be derived from a table provided in ref
Neither of the publications invoked above, nor any other related one we have had in hands in the course of browsing the literature, contains the general formulas we have been seeking for

Summary

Introduction

Theoretical studies on elementary two-dimensional quantum structures in magnetic fields have been carried out for several decades [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]. Neither of the publications invoked above, nor any other related one we have had in hands in the course of browsing the literature, contains the general formulas we have been seeking for This is a bit astonishing in view of the fact that for a similar problem of the planar one-electron atom placed in a weak, uniform, in-plane electric field, closed-form analytical expressions for Stark-Lo Surdo corrections to energies of discrete parabolic eigenstates are known up to the sixth order in the perturbing field [37,38]. We believe they may be of some interest, in particular because the result for the fourth-order correction to the ground state given in ref. [25], and repeated in ref. [31], has been found to be incorrect

Preliminaries

Basics and the zeroth-order problem

The second-order corrections to Coulomb energies

The fourth-order corrections to Coulomb energies

Summary and concluding remarks