Thesis (MSc (Physics))--University of Stellenbosch, 2005. / It was Wigner that in the 1950’s first introduced the idea of modelling physical reality with an ensemble of random matrices while studying the energy levels of heavy atomic nuclei. Since then, the field of Random Matrix Theory has grown tremendously, with applications ranging from fluctuations on the economic markets to M-theory. It is the purpose of this thesis to discuss the basic concepts of Random Matrix Theory, using the ensembles of random matrices originally introduced by Wigner, the Gaussian ensembles, as a starting point. As Random Matrix Theory is classically concerned with the statistical properties of levels sequences, we start with a brief introduction to the statistical analysis of a level sequence before getting to the introduction of the Gaussian ensembles. With the ensembles defined, we move on to the statistical properties that they predict. In the light of these predictions, a few of the classical applications of Random Matrix Theory are discussed, and as an example of some of the important concepts, the Anderson model of localization is investigated in some detail.

Thesis (MSc)--University of Stellenbosch, 2003. / ENGLISH ABSTRACT: The efficient manipulation of various nonlinear optical processes frequently requires the shaping of the laser beams used for these processes. Three beam shaping techniques were investigated in this thesis. The focussing of Gaussian laser beams was investigated analytically, in order to efficiently manipulate the focussed beam characteristics. The beam-shaping characteristics of a diffractive optical element (DOE) was investigated numerically, which illustrates the beamshaping capability of the DOE, and identifies the critical parameters in experimental situations. The use of a waveguide as beam shaping device was investigated analytically and experimentally, and characterized for use with the available tunable laser sources. A Raman laser, or Raman shifter, employs stimulated vibrational Raman scattering to generate laser radiation at shifted frequencies. The waveguide was successfully applied as a beam shaping device in the Raman laser system, for optimisation of the process. The Raman laser system was investigated experimentally and characterized for use with the available tunable laser sources. The successful generation of laser radiation at shifted frequencies illustrates the usefulness of the system for generating tunable red-shifted frequencies. The results of this work allow the simple and efficient application of the Raman laser to generate laser radiation at shifted frequencies, in particular tunable infrared laser radiation which is desirable for molecular spectroscopy. / AFRIKAANSE OPSOMMING: Nie-liniêre optiese prosesse kan meer effektief benut word deur die vervorming van die laserbundels wat gebruik word in die prosesse. In hierdie tesis word drie laserbundel-vervormings tegnieke ondersoek. Die fokussering van Gaussiese laserbundels word analities ondersoek, om die gefokusseerde bundel se eienskappe effektief te manipuleer. Die bundel-vervormings eienskappe van ’n diffraktiewe optiese element word numeries ondersoek, wat die effektiwiteit van die bundelvervorming en die sensitiewe parameters in die sisteem uitwys. Die gebruik van ’n golfgeleier as ’n bundel-vervormings tegniek word ook analities en eksperimenteel ondersoek, en gekarakteriseer vir gebruik met die gegewe golflengte-verstelbare laser sisteme. ’n Raman laser, wat gestimuleerde vibrasionele Raman verstrooiing gebruik om laser lig te genereer by Stokes-verskuifde frekwensies, word ondersoek. Die golfgeleier word effektief gebruik as ’n bundel-vervormings tegniek in die Raman laser, om die bogenoemde nie-liniêre proses te optimeer. Die Raman laser was eksperimenteel ondersoek en gekarakteriseer vir gebruik met die gegewe golflengte-verstelbare lasers. Laser lig by verskuifde golflengtes is suksesvol gegenereer, wat die bruikbaarheid van die sisteem illustreer. Van belang is spesifiek verstelbare infrarooi laser lig, wat gebruik kan word in die laser-spektroskopie van molekules. Die resultate van hierdie werk lei tot die eenvoudige en effektiewe gebruik van die Raman laser, om langer golflengtes in die infrarooi gebied te genereer met ’n gegewe laser in die sigbare gebied.

Thesis (MSc)--University of Stellenbosch, 2006. / ENGLISH ABSTRACT: static density and magnetic structure functions of a melt of flexible polymers whose monomers possess bond-directed dipoles which interact with each other. In order to observe the effect of screening of the dipolar interaction on the structure functions we obtain results for cases with and without steric interactions and also for cases with and without Debye-H¨uckel screening of the dipole moments. / AFRIKAANSE OPSOMMING: Approximation” RPA) om die statiese digtheids- en magnetiese struktuurfunksies te bereken vir ’n smelt van hoogsbuigsame polimere, waarvan die monomere dipole langs die verbindings besit wat met mekaar in wisselwerking tree. Om die effek van afskerming op die dipolare wisselwerking en die struktuurfunksies te kan waarneem, bepaal ons resultate vir die gevalle met en sonder steriese wisselwerkings en ook vir gevalle met en sonder die Debye-H¨uckel afskerming van die dipoolmomente.

Thesis (MSc)--University of Stellenbosch, 2006. / ENGLISH ABSTRACT: Entropy bounds arise from Black hole thermodynamics and are a significant departure from the conventional understanding of the information in a given region. This shift in paradigm is a consequence of the the fact that there is an unexpected relationship between the area and the entropy of a given region of spacetime. Entropy bounds are simplified formulations which are ultimately attempting to be developed into the complete and broad conjecture of the Holographic Principle. This hasn’t been achieved successfully as yet. In this thesis the aim is to introduce how the notion of an entropy bound was first suggested and it’s subsequent development into more robust formulations. The shortcomings of these conjectures are highlighted along with their strengths. A foundational introduction of the mathematical requirements for General Relativity is addressed along with an overview of Einstein’s theory of gravity. This is illustrated by showing the curvature of relative geodesics as being a consequence of gravity. This is contrasted with Newtonian theory where gravity is also shown to manifests as the curvature of relative geodesics. The working background is concluded with a discussion of Einstein’s field equations along with simple and common solutions often used and required. / AFRIKAANSE OPSOMMING: Swartgat Termodinamika impliseer grense op die entropie, en dus inligting, in ’n gegewe ruimtetyd volume, wat ’n drastiese afwyking van die tradisionele denkwyse oor inligting impliseer. Hierdie paradigma skuif het sy oorsprong in ’n onverwagte verband tussen die oppervlakte van, en entropie bevat, in ’n gegewe ruimte tyd volume. Entropie grense is eenvoudige formulerings van hierdie verwantskap wat uiteindelik beslag moet kry in die vollediger en wyer holografiese beginsel. Hierdie doelwit is nog nie bereik nie. Die doel van hierdie tesis is om die oorsprong en verdere formalisering van entropie grense te verduidelik. Beide die sterk en swak punte van die formulerings word bespreek. Algemene relatiwiteits teorie as ’n teorie van gravitasie, sowel as die wiskundige onderbou daarvan, word oorsigtelik bespreek. Die geometries onderbou van gravitasie word geillustreer aan die hand van die buiging van relatiewe geodete. Dit word met Newton se gravitasie teorie vergelyk wat ook in die buiging van relatiewe geodete gemanifesteer word. Hierdie oorsigtelike agtergrond word afgesluit met ’n oorsig van Einstein se vergelykings, asook eenvoudige en algemene oplossings wat dikwels nodig is en gebruik word.

Thesis (MSc)--Stellenbosch University, 2011. / ENGLISH ABSTRACT: The goal of this thesis is to investigate the effects of distorting a graphene lattice and connect- ing this distorted graphene sheet to a superconductor. At low energies the possible excitation states in graphene are restricted to two distinct regions in momentum space called valleys. Many electronic applications are possible if one can design a graphene system where excitations can be forced to occupy a single valley in a controllable way. Investigating the spectrum of the distorted graphene sheet reveals that, if the chemical potential is chosen to coincide with a bulk Landau level, the normal-superconductor interface always supports propagating modes in both directions. Excitations from opposite valleys travel in opposite directions along the interface. The spectrum of a distorted graphene sheet terminated by an armchair edge, in contrast, is dis- persionless. We verify this insulating nature of the armchair edge for finite samples by numerical means. Furthermore, we verify previous analytical results pertaining to a graphene sheet with NS interface and an applied perpendicular real magnetic field numerically. In the process, it is shown that considering graphene sheets of perfect width is not necessary, as long as the width a few magnetic lengths away from the interface is well-defined. By then considering a finite graphene sheet, terminated by armchair edges, that is distorted and connected to a superconductor, we find bound states near the NS interface that can be changed by distorting the graphene lattice further. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is om die uitwerking van die vervorming van ’n grafeenrooster te ondersoek wanneer die met ’n supergeleier verbind word. By lae energieë word die moontlike opwekkings in grafeen beperk tot twee aparte gebiede van momentumruimte — die sogenaamde valleie. Verskeie elektroniese toepassings is moontlik indien ’n grafeenstelsel ontwerp kan word waar opwekkings slegs ’n enkele vallei beset en die besetting beheer kan word. Deur die spektrum van die vervormde grafeenrooster te ondersoek word daar gevind dat, indien die chemiese potensiaal gekies word om saam te val met ’n Landauvlak, die NS-tussenvlak geleiding in beide rigtings ondersteun. Opwekkings van verskillende valleie beweeg in teenoorgestelde rigtings langs die tussenvlak. Daarteenoor is die spektrum van ’n vervormde grafeenroster met ’n leunstoelrand dispersieloos. Ons bevestig hierdie insulerende gedrag van ’n leunstoelrand vir eindige grafeen- roosters deur middel van ’n numeriese berekening. Verder word vorige analitiese resultate wat verband hou met ’n grafeenrooster met normaal-supergeleiertussenvlakstelsel en loodregte mag- neetveld op die vlak bevestig deur middel van numeriese berekeninge. In die proses word dit ook aangedui dat die grafeenrooster nie ’n perfekte wydte hoef te hˆe nie, solank die wydte goed gedefinieer is vir ’n paar magnetiese lengtes in die omgewing van die tussenvlak. Deur dan die eindige grafeenrooster met leunstoelrande te koppel aan ’n supergeleier word gebonde toe- stande naby aan die NS tussenvalk gevind. Hierdie toestande kan gemanipuleer word deur die grafeenrooster verder te vervorm.

Thesis (MSc)--University of Stellenbosch, 2007. / ENGLISH ABSTRACT: The dielectric-barrier discharge (DBD) is a source of non-equilibrium plasma that has seen widespread industrial application in recent years. A high-voltage arbitrary-waveform generator has been designed, built and characterised for the purpose of investigating the influence that the applied voltage waveform has on the operation of a DBD. The developed arbitrary-waveform generator is based on the principle of Fourier synthesis. Up to twenty Fourier components are generated by means of a digital circuit board, and then separately amplified by Class-AB amplifiers. Twenty step-up transformers are subsequently used to transform the Fourier components to higher voltages; the summation of the Fourier components are realised by the series connection of the transformer secondary sides. It was found that the digital generation of the Fourier components is very accurate and provides for the easy configuration of arbitrary waveforms. Furthermore, the amplification of the Fourier components by the Class-AB amplifiers introduces very little distortion. The principle of adding the Fourier components via the step-up transformers has been demonstrated; however, the large distributed capacitances of the transformers adversely affect the operation of the Class-AB amplifiers, leading to the introduction of distortion into the generated waveform. Furthermore, it was found that care had to be taken to limit the introduction of EMI through the system’s large ground plane. / AFRIKAANSE OPSOMMING: Die di¨elektriese versperringsontlading (DVO) is ’n bron van nie-ekwilibrium plasma wat in die afgelope jare wye toepassing in die nywerheid gevind het. ’n Arbitrˆere-golfvorm hoogspanningskragbron is ontwerp, gebou en gekarakteriseer, met die doel om die invloed wat die aangewende spanningsgolfvorm het op die werking van die DVO, te ondersoek. Die ontwikkelde arbitrˆere golfvormgenerator is gebaseer op die beginsels van Fourier samestelling. Tot twintig Fourier komponente word digitaal gegenereer, en dan afsonderlik versterk deur Klas-AB versterkers. Twintig transformators word dan gebruik om die Fourier komponente na ho¨er spannings te transformeer. Die sommasie van die Fourier komponente geskied deur die serieskakeling van die transformators se sekondˆere windings. Daar is bevind dat die digitale generasie van die Fourier komponente baie akkuraat is, en dat die arbitrˆere golfvorms maklik verstel kan word. Verder versterk die Klas-AB versterkers die Fourier komponente sonder enige noemenswaardige vervorming. Die gebruik van die transformators om die Fourier komponente saam te voeg, is gedemonstreer. Die groot verspreide kapasitansies van die transformators be¨ınvloed egter die funksioneering van die Klas-AB versterkers, wat lei tot ’n vervorming van die uittree. Daar is ook bevind dat die toetrede van EM versteurings deur die grondvlak van die sisteem problematies kan wees.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT:In this thesis we discuss some of the anomalies present in systems with long-range interactions, for instance negative speci c heat and negative magnetic susceptibility, and show how they can be related to the convexity properties of the thermodynamic potentials and nonequivalence of ensembles. We also discuss the possibility of engineering long-range quantum spin systems with cold atoms in optical lattices to experimentally verify the existence of nonequivalence of ensembles. We then formulate an expression for the density of states when the energy and magnetisation correspond to a pair of non-commuting operators. Finally we analytically compute the entropy s( ;m) as a function of energy, , and magnetisation, m, for the anisotropic Heisenberg model with Curie-Weiss type interactions. The results show that the entropy is non-concave in terms of magnetisation under certain circumstances which in turn indicates that the microcanonical and canonical ensembles are not equivalent and that the magnetic susceptibility is negative. After making an appropriate change of variables we show that a second-order phase transition can be present at negative temperatures in the microcanonical ensemble which cannot be represented in the canonical ensemble. / AFRIKAANSE OPSOMMING: In hierdie tesis bespreek ons van die onverwagte eienskappe wat sisteme met lang afstand wisselwerkings kan openbaar, byvoorbeeld negatiewe spesi eke warmte en negatiewe magnetiese suseptibiliteit. Ons dui ook die ooreenkoms tussen hierdie gedrag en die konveksiteit van die termodinamiese potensiale en nie-ekwivalente ensembles aan. Hierna bespreek ons die moontlikheid om lang afstand kwantum spin sisteme te realiseer met koue atome in 'n optiese rooster. Daarna wys ons hoe dit moontlik is om 'n uitdrukking vir die digtheid van toestande te formuleer vir sisteme waar die energie en magnetisasie ooreenstem met operatore wat nie met mekaar kommuteer nie. Uiteindelik bepaal ons die entropie, s( ;m), in terme van die energie, , en magnetisasie, m, vir die anisotropiese Heisenberg model met Curie-Weiss tipe interaksies. Die resultate wys dat die entropie onder sekere omstandighede nie konkaaf in terme van magnetisasie is nie. Dit, op sy beurt, dui aan dat die mikrokanoniese en kanoniese ensembles nie ekwivalent is nie en dat die magnetiese suseptibiliteit negatief kan wees. Nadat ons 'n toepaslike transformasie van veranderlikes maak, wys ons dat 'n tweede orde fase-oorgang by negatiewe temperature kan plaasvind in die mikrokanoniese ensemble wat nie verteenwoordig kan word in die kanoniese ensemble nie.

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The goal of this thesis is to present evolutionary algorithms, and demonstrate their applicability in quantum computing. As an introduction to evolutionary algorithms, it is applied to the simple but still challenging (from a computational viewpoint) Travelling Salesman Problem (TSP). This example is used to illustrate the e ect of various parameters like selection method, and maximum population size on the accuracy and e ciency of the evolutionary algorithms. For the sample problem, the 48 continental state capitals of the USA, solutions are evolved and compared to the known optimal solution. From this investigation tournament selection was shown to be the most e ective selection method, and that a population of 200 individuals per generation gave the most e ective convergence rates. In the next part of the thesis, evolutionary algorithms are applied to the generation of optimal quantum circuits for the following cases: The identity transformation : Picked for its simplicity as a test of the correct implementation of the evolutionary algorithm. The results of this investigation showed that the solver program functions correctly and that evolutionary algorithms can indeed nd valid solutions for this kind of problem. The work by Ding et al. [16] on optimal circuits for the two-qubit entanglement gate, controlled-S gate as well as the three qubit entanglement gate are solved by means of EA and the results compared. In all cases similar circuits are produced in fewer generations than the application of Ding et al. [16]. The three qubit quantum Fourier transform gate was also attempted, but no convergence was attained. The quantum teleportation algorithm is also investigated. Firstly the nature of the transformation that leads to quantum teleportation is considered. Next an e ective circuit is sought using evolutionary algorithms. The best result is one gate longer than Brassard [11], and seven gates longer than Yabuki [61]. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is om evolusionêre algoritmes te ondersoek en hulle toepaslikheid op kwantumkomputasie te demonstreer. As 'n inleiding tot evolusionêre algoritmes is die eenvoudige, maar steeds komputasioneel uitdagende handelsreisigerprobleem ondersoek. Die invloed van die keuse van 'n seleksie metode, sowel as die invloed van die maksimum aantal individue in 'n generasie op die akkuraatheid en e ektiwiteit van die algoritmes is ondersoek. As voorbeeld is die 48 kontinentale hoofstede van die state van die VSA gekies. Die oplossings wat met evolusionêre algoritmes verkry is, is met die bekende beste oplossings vergelyk. Die resultate van hierdie ondersoek was dat toernooi seleksie die mees e ektiewe seleksie metode is, en dat 200 individue per generasie die mees e ektiewe konvergensie tempo lewer. Evolusionêre algoritmes word vervolgens toegepas om optimale oplossings vir die volgende kwantumalgoritmes te genereer: Die identiteitstransformasie: Hierdie geval is gekies as 'n eenvoudige toepassing met 'n bekende oplossing. Die resultaat van hierdie toepassing van die program was dat dit korrek funksioneer, en vinnig by die korrekte oplossings uitkom. Vervolgens is daar ondersoek ingestel na vier van die gevalle wat in Ding et al. [16] bespreek word. Die spesi eke transformasies waarna gekyk is, is 'n optimale stroombaan vir twee kwabis verstrengeling, 'n beheerde-S hek, 'n drie kwabis verstrengelings hek, en 'n drie kwabis kwantum Fourier transform hek. In die eerste drie gevalle stem die oplossings ooreen met die van Ding et al. [16], en is die konvergensie tempo vinniger. Daar is geen oplossing vir die kwantum Fourier transform verkry nie. Laastens is daar na die kwantumteleportasiealgoritme gekyk. Die eerste stap was om te kyk na die transformasie wat in hierdie geval benodig word, en daarna is gepoog om 'n e ektiewe stroombaan te evolueer. Die beste resultaat was een hek langer as Brassard [11], en sewe hekke langer as Yabuki [61].

Thesis (MSc)--Stellenbosch University, 2012. / ENGLISH ABSTRACT: The nite size of nucleons should play an important role in the description of high density nuclear matter as found in astro-physical objects. Yet we see that the Walecka model, which is generally used to describe these systems, treats the nucleons as point particles. Here we argue that a non-commutative version of the Walecka model may be a consistent and appropriate framework to describe nite nucleon size e ects. In this framework the length scale introduced through the non-commutative parameter plays the role of the nite nucleon size. To investigate the consequences of this description, the equations of motion and energy-momentum tensor for the non-commutative Walecka model are derived. We also derived an expression for the total energy of the system, as a function of the non-commutative parameter, in a spatially non-uniform matter approximation. The non-commutative parameter, as a variable dependent on the dynamics of the system, remains to be solved self-consistently. / AFRIKAANSE OPSOMMING: Die eindige grootte van nukleone moet 'n belangrike rol speel in die beskrywing van ho e-digtheid kern materie soos gevind in astro- siese voorwerpe. Tog sien ons dat die Walecka model, wat in die algemeen gebruik word om hierdie stelsels te beskryf, die nukleone as punt deeltjies hanteer. Ons redeneer dus dat 'n nie-kommutatiewe weergawe van die Walecka model 'n konsistente en gepaste raamwerk is om die e ekte van eindige nukleon grootte te beskryf. In hierdie raamwerk speel die lengte-skaal wat ingevoer word deur die nie-kommutatiewe parameter die rol van eindige grootte vir nukleone. Om die gevolge van hierdie beskrywing te ondersoek, word die vergelykings van beweging en die energie-momentum tensor afgelei vir die nie-kommutatiewe Walecka model. Ons het ook 'n uitdrukking vir die totale energie van die stelsel, as 'n funksie van die niekommutatiewe parameter, afgelei in 'n ruimtelik nie-uniforme materie benadering. Die niekommutatiewe parameter, as 'n veranderlike afhanklik van die dinamika van die stelsel, bly steeds om self-konsistent opgelos te word.

Thesis (MSc)--Stellenbosch University, 2012. / AFRIKAANSE OPSOMMING: Berekeninge met Density Functional Theory (DFT) is ’n nuttige tegniek om die dinamika van molekules op potensiële energievlakke te verstaan. Beginnende met ’n prototipe molekuul formaldimien, wat die kern vorm van die groter fotochromiese molekuul dithizonatophenyl kwik (DPM), word die modellering van die molekuul meer ingewikkeld tot laasgenoemde bestudeer kan word asook sy fotochromiese afgeleides wat vervanging van elektronryk en elektronarm radikale by orto, meta en para posisies van die phenyl ringe insluit. DFT berekeninge word met spektra van Absorpsiespektroskopie met UV en sigbare lig asook tyd opgeloste spektra, verkry dmv femtosekondespektroskopie, vergelyk. In pol^ere aprotiese, pol^ere protiese en nie-pol^ere oplosmiddels, isomeriseer die molekuul om die C=N dubbelbinding. Daar kan tussen die twee isomere onderskei word deur dat die een in oplossing in sy grondtoestand blou en die ander een oranje voorkom. Die isomerisering is’n fotogeinduseerde proses. Die optimering van die molekul^ere struktuur, absorpsiespektra, oplosmiddel-afhanklikheid, en potensiële energievlak metings van die molekuul word bestudeer. Die sterk/swak wisselwerking wat in pol^ere protiese/aprotiese oplosmiddels verskyn word geopenbaar deur die hoe/lae absorpsie van die sekond^ere bande van die molekules. Daar is gevind dat die absorpsiespektra van DPM bathochromies in oplosmiddels met hoë diëlektriese konstantes is. Vir die potensiële energievlak berekeninge van die grondtoestand word rigiede en ontspanne metodes gebruik waar laasgenoemde met gebroke simmetrie berekeninge verkry word. Van alle metodes wat vir berekeninge gebruik was, gee die B3LYP/CEP-31G metode die beste benadering aan eksperimentele data. Alle berekeninge word gedoen met twee bekende sagteware pakkette; Amsterdam Density Functional (ADF) en Gaussian, wat op twee verskillende DFT metodes gebaseer is. / ENGLISH ABSTRACT: Density functional theory is a useful computational tool in the understanding of molecular dynamics on potential energy surfaces. Starting with a prototype molecule formaldimine, the photochromic molecule dithizonatophenylmercury II (DPM) and a set of its photochromic derivatives, (involving substitutions of electron donating and electron withdrawing substituents at ortho, meta and para positions of the dithizonato phenyl rings), are studied through density functional calculation in comparison with steady state absorption spectra obtained from UV-Visible and femto second spectroscopy experiments. In polar aprotic, polar protic and non-polar solvents these molecules isomerise around C=N double bond chromophore, from orange electronic ground states to blue electronic ground states upon photo-excitation. We investigate the structural optimisations, the absorption spectra, the solvent dependence and the potential energy surface (PES) of these molecules. The strong (weak) interactions exhibited by the polar protic (aprotic) solvents used are revealed through high (low) absorbance in the secondary bands of these molecules. The absorption spectra of DPM are found to be bathochromic in solvents with high dielectric constants. For the ground state PES calculation we make use of rigid and relaxed methods, and the latter is obtained through broken symmetry calculation. Of all the methods used in calculation, B3LYP/CEP-31G method gives the best approximation to the experimental data. All calculations are done using the two renown software, Amsterdam Density Functional (ADF) and Gaussian, availing their different density functional methods.