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Acta Physica Polonica A

Relativistic Paschen-Back Effect for the Two-Dimensional H-Like Atoms

A. Poszwa, Andrzej Rutkowski

Published Mar 1, 2010• 7 citations
Publisher

Abstract

The classification of states based on good quantum numbers for the two-dimensional Coulomb problem is proposed.The first order magnetic energy corrections are calculated using exact field-free analytic solutions of the Dirac equation as a zero-order approximation.
DOI
10.12693/aphyspola.117.439

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Abstract The (2+1)-dimensional Dirac equation for an electron in a constant magnetic field is considered. Exact solutions of this equation along with the corresponding energy spectrum are derived in a concise form. Among these solutions, we noticed that the lowest Landau level wavefunction corresponds to an anti-electron (an electron with a negative energy). The latter appears because special relativity allows solutions with negative energies. This means that a relativistic electron (matter) in a uniform magnetic field, at the lowest Landau level, is mutated to an anti-electron (anti-matter). This is a good finding for those that are interested in anti-matter applications.
In 1928, the brilliance of Paul Dirac granted him the insight to develop a particle equation that was fully consistent with both the principles of quantum mechanics and the theory of special relativity. In this bachelor thesis we have studied the fundamentals of relativistic quantum mechanics including the derivation of the Dirac equation in a similar fashion to the way originally done by Paul Dirac. Additionally we have studied the analytical solution for hydrogen-like atoms. Furthermore we have numerically analysed the energy spectrum, radial characteristics and correctional terms using programs written in Python and MATLAB. Finally we have investigated the implications and applications ofthe Dirac Hamiltonian and its use to effectively describe the attributes of gold atoms.We find that relativistic contribution scales approximately by Z2 for large Z. In addition, we conclude that the Dirac Hamiltonian is of vital importance for an accurate description of chemical and physical properties of large atoms. Furthermore, we have completed calculations of the radial solution for the Coulomb potential and also for a free particle.
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