Electronic Theses and Dissertations

3-10-2019

Master Thesis

M.Sc.

Physics

Gordon Drake

Keywords

Foldy-Wouthuysen, Relativistic, Sturmian, Transition, Wave Function

Rights

info:eu-repo/semantics/openAccess

The Dirac equation provides a fully relativistic covariant equation which can be used to calculate relativistic transition rates but only for one-electron systems. For the two-electron case, one can either use approximate relativistic wave functions or obtain equivalent nonrelativistic operators that can be used with Schr{\"o}dinger wave functions; an approach that is preferred for low atomic number (Z) atoms. By using equivalent nonrelativistic operators obtained from the Foldy-Wouthuysen transformation and relativistically corrected Schr{\"o}dinger wave functions, we show that we obtain the same transition amplitude as in Dirac Theory up to order $\alpha^2$, where $\alpha$ is the fine structure constant. We show this for the one-electron case and provide a theoretical framework for the two-electron case. For the one-electron case we obtain analytic first order corrected wave functions for the $2p$ states which have not been published before. For the two-electron case we obtain first order corrected wave functions using a variational method and compare two different Sturmian basis sets, which we label triangular and linear basis sets. We show that the triangular basis set provides a significant advantage over the linear basis set, increasing the precision by two orders of magnitude. We also compare the wave functions obtained using pseudostates with those obtained analytically and give some suggestions to improve the agreement near zero.