Numerical investigations of the Dirac equation and bound state quantum electrodynamics in atoms : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Physics at Massey University, Albany, New Zealand

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Massey University
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This thesis addresses, from a computational perspective, several open questions in relativistic atomic structure theory, which is the theoretical description of atoms based on the Dirac equation and quantum electrodynamics (QED). The first part of this thesis investigates several fundamental problems of the Dirac equation with the help of a novel numerical solver based on the one-dimensional finite element (FEM) basis set. Significant effort is made to validate and benchmark the solver, which is reliably able to converge to accurate results at numerical floating-point precision, including when nuclear potentials derived from nuclear models with finite spacial extent (as opposed to a point nucleus) are used. The solver is then applied to the Dirac equation in the challenging high nuclear charge regime where the Dirac equation exhibits several mathematical difficulties. In particular, the problem of the 1s bound state diving into the sea of negative energy continuum states is studied and the diving resonance state is numerically traced and analysed. As a type of workaround, a modified version of the Dirac equation where the negative energy plane-wave states are projected out of the Hilbert space is also solved and studied in the high nuclear charge regime. The second part of the thesis involves expanding the QED self-energy treatment in the atomic structure software GRASP. The configuration interaction (CI) portion of the code is significantly refactored to allow for the implementation of new additional effective operators that provide a more modern multi-electron treatment of QED self-energy effects. The implementation is tested by evaluating the QED and other post-Dirac-Coulomb corrections for the ground states of the beryllium-like isoelectronic sequence, which was also discovered to exhibit an interesting ground state configuration transition at high nuclear charge.
Atomic structure, Mathematical models, Dirac equation, Quantum electrodynamics, Bound states (Quantum mechanics)