Computeralgebraische Herleitung und Auswertung atomarer Störungsreihen und deren Anwendung auf geschlossenschalige und einfache offenschalige Systeme

In der vorliegenden Arbeit wurde gezeigt, wie mit Hilfe der atomaren Vielteilchenstörungstheorie totale Energien und auch Anregungsenergien von Atomen und Ionen berechnet werden können. Dabei war es zunächst erforderlich, die Störungsreihen mit Hilfe computeralgebraischer Methoden herzuleiten. Mit Hilfe des hierbei entwickelten Maple-Programmpaketes APEX wurde dies für geschlossenschalige Systeme und Systeme mit einem aktiven Elektron bzw. Loch bis zur vierten Ordnung durchgeführt, wobei die entsprechenden Terme aufgrund ihrer großen Anzahl hier nicht wiedergegeben werden konnten. Als nächster Schritt erfolgte die analytische Winkelreduktion unter Anwendung des Maple-Programmpaketes RACAH, was zu diesem Zwecke entsprechend angepasst und weiterentwickelt wurde. Erst hier wurde von der Kugelsymmetrie des atomaren Referenzzustandes Gebrauch gemacht. Eine erhebliche Vereinfachung der Störungsterme war die Folge. Der zweite Teil dieser Arbeit befasst sich mit der numerischen Auswertung der bisher rein analytisch behandelten Störungsreihen. Dazu wurde, aufbauend auf dem Fortran-Programmpaket Ratip, ein Dirac-Fock-Programm für geschlossenschalige Systeme entwickelt, welches auf der in Kapitel 3 dargestellen Matrix-Dirac-Fock-Methode beruht. Innerhalb dieser Umgebung war es nun möglich, die Störungsterme numerisch auszuwerten. Dabei zeigte sich schnell, dass dies nur dann in einem angemessenen Zeitrahmen stattfinden kann, wenn die entsprechenden Radialintegrale im Hauptspeicher des Computers gehalten werden. Wegen der sehr hohen Anzahl dieser Integrale stellte dies auch hohe Ansprüche an die verwendete Hardware. Das war auch insbesondere der Grund dafür, dass die Korrekturen dritter Ordnung nur teilweise und die vierter Ordnung gar nicht berechnet werden konnten. Schließlich wurden die Korrelationsenergien He-artiger Systeme sowie von Neon, Argon und Quecksilber berechnet und mit Literaturwerten verglichen. Außerdem wurden noch Li-artige Systeme, Natrium, Kalium und Thallium untersucht, wobei hier die niedrigsten Zustände des Valenzelektrons betrachtet wurden. Die Ionisierungsenergien der superschweren Elemente 113 und 119 bilden den Abschluss dieser Arbeit.

On the Computation of Fourier Coefficients

In this paper we derive an identity for the Fourier coefficients of a differentiable function f(t) in terms of the Fourier coefficients of its derivative f'(t). This yields an algorithm to compute the Fourier coefficients of f(t) whenever the Fourier coefficients of f'(t) are known, and vice versa. Furthermore this generates an iterative scheme for N times differentiable functions complementing the direct computation of Fourier coefficients via the defining integrals which can be also treated automatically in certain cases.

Entanglement analysis of atomic processes and quantum registers

During recent years, quantum information processing and the study of N−qubit quantum systems have attracted a lot of interest, both in theory and experiment. Apart from the promise of performing efficient quantum information protocols, such as quantum key distribution, teleportation or quantum computation, however, these investigations also revealed a great deal of difficulties which still need to be resolved in practise. Quantum information protocols rely on the application of unitary and non–unitary quantum operations that act on a given set of quantum mechanical two-state systems (qubits) to form (entangled) states, in which the information is encoded. The overall system of qubits is often referred to as a quantum register. Today the entanglement in a quantum register is known as the key resource for many protocols of quantum computation and quantum information theory. However, despite the successful demonstration of several protocols, such as teleportation or quantum key distribution, there are still many open questions of how entanglement affects the efficiency of quantum algorithms or how it can be protected against noisy environments. To facilitate the simulation of such N−qubit quantum systems and the analysis of their entanglement properties, we have developed the Feynman program. The program package provides all necessary tools in order to define and to deal with quantum registers, quantum gates and quantum operations. Using an interactive and easily extendible design within the framework of the computer algebra system Maple, the Feynman program is a powerful toolbox not only for teaching the basic and more advanced concepts of quantum information but also for studying their physical realization in the future. To this end, the Feynman program implements a selection of algebraic separability criteria for bipartite and multipartite mixed states as well as the most frequently used entanglement measures from the literature. Additionally, the program supports the work with quantum operations and their associated (Jamiolkowski) dual states. Based on the implementation of several popular decoherence models, we provide tools especially for the quantitative analysis of quantum operations. As an application of the developed tools we further present two case studies in which the entanglement of two atomic processes is investigated. In particular, we have studied the change of the electron-ion spin entanglement in atomic photoionization and the photon-photon polarization entanglement in the two-photon decay of hydrogen. The results show that both processes are, in principle, suitable for the creation and control of entanglement. Apart from process-specific parameters like initial atom polarization, it is mainly the process geometry which offers a simple and effective instrument to adjust the final state entanglement. Finally, for the case of the two-photon decay of hydrogenlike systems, we study the difference between nonlocal quantum correlations, as given by the violation of the Bell inequality and the concurrence as a true entanglement measure.

Elimination in Operator Algebras

A large class of special functions are solutions of systems of linear difference and differential equations with polynomial coefficients. For a given function, these equations considered as operator polynomials generate a left ideal in a noncommutative algebra called Ore algebra. This ideal with finitely many conditions characterizes the function uniquely so that Gröbner basis techniques can be applied. Many problems related to special functions which can be described by such ideals can be solved by performing elimination of appropriate noncommutative variables in these ideals. In this work, we mainly achieve the following: 1. We give an overview of the theoretical algebraic background as well as the algorithmic aspects of different methods using noncommutative Gröbner elimination techniques in Ore algebras in order to solve problems related to special functions. 2. We describe in detail algorithms which are based on Gröbner elimination techniques and perform the creative telescoping method for sums and integrals of special functions. 3. We investigate and compare these algorithms by illustrative examples which are performed by the computer algebra system Maple. This investigation has the objective to test how far noncommutative Gröbner elimination techniques may be efficiently applied to perform creative telescoping.