The electron is known to possess an anomalous magnetic moment, which interacts
with the gradient of the electric field. This makes it necessary to compute its
effects on the energy spectrum. Even though the Coulomb Dirac equation can be
solved in closed form, this is no longer possible when the anomalous magnetic moment is included. In fact the interaction due to this term is so strong that it changes the domain of the Hamiltonian. From a differential equation point of view, the anomalous magnetic moment term is strongly singular near the origin. As usual, one has to resort to perturbation theory. This, however, only makes sense if the eigenvalues are stable. To prove stability is therefore a challenge one has to face before actually computing the energy shifts. The first stability results in this line were shown by Behncke for angular momenta κ ≥ 3, because the eigenfunctions of the unperturbed Hamiltonian decay fast enough near the origin. He achieved this by decoupling the system and then using the techniques available for second order differential equations. Later, Kalf and Schmidt extended
Behncke’s results basing their analysis on the Prüfer angle technique and a comparison result for first order differential equations. The Prüfer angle method is particularly useful because it shows a better stability and because it obeys a first
order differential equation. Nonetheless, Kalf and Schmidt had to exclude some
coupling constants for κ > 0. This I believe is an artefact of their method. In this
study, I make increasing use of asymptotic integration, a method which is rather
well adapted to perturbation theory and is known to give stability results to any
level of accuracy. Together with the Prüfer angle technique, this lead to a more general stability result and even allows for an energy shifts estimate. Hamiltonians traditionally treated in physics to describe the spin-orbit effect are not self adjoint i.e. they are not proper observables in quantum mechanics. Nonetheless, naive perturbation theory gives correct results regarding the spectrum. To solve this mystery, one has to study the nonrelativistic limit of the Dirac operator. In the second part of this study, I have not only given the higher order correction to the Dirac operator but also shown the effects of the spin-orbit term.
| Identifer | oai:union.ndltd.org:uni-osnabrueck.de/oai:repositorium.ub.uni-osnabrueck.de:urn:nbn:de:gbv:700-2015071713311 |
| Date | 17 July 2015 |
| Creators | Ambogo, David Otieno |
| Contributors | Prof. em. Horst Behncke Ph.D., Prof. Don Hinton, Prof. Dr. Karl Michael Schmidt |
| Source Sets | Universität Osnabrück |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf, application/zip |
| Rights | Namensnennung-NichtKommerziell-KeineBearbeitung 3.0 Unported, http://creativecommons.org/licenses/by-nc-nd/3.0/ |
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