Thesis (MSc (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The overall goal of this project is to study neutron star properties and locate the transition density from the core to the crust using fifteen parameter sets of the effective Skyrme nucleon-nucleon interaction within a method called the dynamical method. Although another approach used to describe nucleon-nucleon interactions called the modified Gogny interaction is briefly discussed in this work, along with a second method for locating the transition density called the thermodynamical method, results using this interaction and method were not generated, but lays some foundation for a PhD project to be undertaken and potentially showing the relation between the interactions and results. The importance of results depends on how well other theoretical approaches to the problem can reproduce those results and to what accuracy. For models to be valid there also has to be good agreement between the theoretical results and known observables. In this project some properties of neutron stars, such as the equation of state, saturation density, binding energy, symmetry energy, slope and incompressibility parameters of symmetry energy are studied. The transition density is located using the dynamical method. Results of the fifteen Skyrme parameter sets show excellent agreement with the published values of the properties of neutron stars and are consistent with their empirical values inferred from nuclear laboratory data, thus validating the use of the Skyrme interactions for describing nuclear matter. / AFRIKAANSE OPSOMMING: Die hoofdoel van hierdie projek is om neutron ster eienskappe te bestudeer en die oorgangsdigtheid vanaf die kors na die kern te vind deur gebruik te maak van vyftien parameter stelle van die effektiewe Skyrme nukleon-nukleon interaksie binne ‘n metode genaamd die dinamiese metode. Alhoewel ‘n ander benadering vir die beskrywing van nukleon-nukleon interaksies, genaamd die gewysigde Gogny interaksie kortliks in hierdie werk beskryf word, asook ‘n tweede metode, genaamd die termodinamiese metode om die oorgangsdigtheid te bepaal, was resultate vir hierdie interaksie en metode nie gegenereer nie, maar l die fondasie vir verdere werk aan ‘n PhD projek wat die verband tussen die twee interaksies en resultate kan wys. Die belangrikheid van resultate hang af van hoe goed ander teoretiese benaderinge tot die problem daardie resultate kan herproduseer en tot watter akkuraatheid. Vir modelle om geldig te wees moet daar ook goeie ooreenkomste wees tussen teoretiese resultate en bekende waarneembare eienskappe. In hierdie projek word sommige eieskappe van neutron sterre, soos die toestandandsvergelyking, versadigingsdigtheid, bindingsenergie, simmetrie-energie, gradiënt en onsaampersbaarheids parameters van die simmetrie-energie bestudeer. Die oorgangsdigtheid word dan gevind deur gebruik te maak van die dinamiese metode. Resultate van die vyftien Skyrme interaksie parameter stelle wys goeie ooreenstemming met die gepubliseerde waardes van die eienskappe van neutron sterre en is konsistent met hulle empiriese waardes afgelei van kern laboratorium data, wat die geldigheid van Skyrme interaksies vir die beskrywing van kernmaterie bevestig.

Thesis (MSc (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Due to the minimal length scale induced by non-commuting co-ordinates, it is not clear a priori what is meant by a position measurement on a non-commutative space. It was shown recently in a paper by Scholtz et al. that it is indeed possible to recover the notion of quantum mechanical position measurements consistently on the non-commutative plane. To do this, it is necessary to introduce weak (non-projective) measurements, formulated in terms of Positive Operator-Valued Measures (POVMs). In this thesis we shall demonstrate, however, that a measurement of position alone in non-commutative space cannot yield complete information about the quantum state of a particle. Indeed, the aforementioned formalism entails a description that is non-local in that it requires knowledge of all orders of positional derivatives through the star product that is used ubiquitously to map operator multiplication onto function multiplication in non-commutative systems. It will be shown that there exist several equivalent local descriptions, which are arrived at via the introduction of additional degrees of freedom. Consequently non-commutative quantum mechanical position measurements necessarily confront us with some additional structure which is necessary (in addition to position) to specify quantum states completely. The remainder of the thesis, based in part on a recent publication (\Noncommutative quantum mechanics { a perspective on structure and spatial extent", C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) will involve investigations into the physical interpretation of these additional degrees of freedom. For one particular local formulation, the corresponding classical theory will be used to demonstrate that the concept of extended, structured objects emerges quite naturally and unavoidably there. This description will be shown to be equivalent to one describing a two-charge harmonically interacting composite in a strong magnetic eld found by Susskind. It will be argued through various applications that these notions also extend naturally to the quantum level, and constraints will be shown to arise there. A further local formulation will be introduced, where the natural interpretation is that of objects located at a point with a certain angular momentum about that point. This again enforces the idea of particles that are not point-like. Both local descriptions are convenient, in that they make explicit the additional structure which is encoded more subtly in the non-local description. Lastly we shall argue that the additional degrees of freedom introduced by local descriptions may also be thought of as gauge degrees of freedom in a gauge-invariant formulation of the theory. / AFRIKAANSE OPSOMMING: As gevolg van die minimum lengteskaal wat deur nie-kommuterende ko ordinate ge nduseer word is dit nie a priori duidelik wat met 'n posisiemeting op 'n nie-kommutatiewe ruimte bedoel word nie. Dit is onlangs in 'n artikel deur Scholtz et al. getoon dat dit wel op 'n nie-kommutatiewe vlak moontlik is om die begrip van kwantummeganiese posisiemetings te herwin. Vir hierdie doel benodig ons die konsep van swak (nie-projektiewe) metings wat in terme van 'n positief operator-waardige maat geformuleer word. In hierdie tesis sal ons egter toon dat 'n meting van slegs die posisie nie volledige inligting oor die kwantumtoestand van 'n deeltjie in 'n niekommutatiewe ruimte lewer nie. Ons formalisme behels 'n nie-lokale beskrywing waarbinne kennis oor alle ordes van posisieafgeleides in die sogenaamde sterproduk bevat word. Die sterproduk is 'n welbekende konstruksie waardeur operatorvermenigvuldiging op funksievermenigvuldiging afgebeeld kan word. Ons sal toon dat verskeie ekwivalente lokale beskrywings bestaan wat volg uit die invoer van bykomende vryheidsgrade. Dit beteken dat nie-kommutatiewe posisiemetings op 'n natuurlike wyse die nodigheid van bykomende strukture uitwys wat noodsaaklik is om die kwantumtoestand van 'n sisteem volledig te beskryf. Die res van die tesis, wat gedeeltelik op 'n onlangse publikasie (\Noncommutative quantum mechanics { a perspective on structure and spatial extent", C.M. Rohwer, K.G. Zloshchastiev, L. Gouba and F.G. Scholtz, J. Phys. A: Math. Theor. 43 (2010) 345302) gebaseer is, behels 'n ondersoek na die siese interpretasie van hierdie bykomende strukture. Ons sal toon dat vir 'n spesi eke lokale formulering die beeld van objekte met struktuur op 'n natuurlike wyse in die ooreenstemmende klassieke teorie na vore kom. Hierdie beskrywing is inderdaad ekwivalent aan die van Susskind wat twee gelaaide deeltjies, gekoppel deur 'n harmoniese interaksie, in 'n sterk magneetveld behels. Met behulp van verskeie toepassings sal ons toon dat hierdie interpretasie op 'n natuurlike wyse na die kwantummeganiese konteks vertaal waar sekere dwangvoorwaardes na vore kom. 'n Tweede lokale beskrywing in terme van objekte wat by 'n sekere punt met 'n vaste hoekmomentum gelokaliseer is sal ook ondersoek word. Binne hierdie konteks sal ons weer deur die begrip van addisionele struktuur gekonfronteer word. Beide lokale beskrywings is gerie ik omdat hulle hierdie bykomende strukture eksplisiet maak, terwyl dit in die nie-lokale beskrywing deur die sterproduk versteek word. Laastens sal ons toon dat die bykomende vryheidsgrade in lokale beskrywings ook as ykvryheidsgrade van 'n ykinvariante formulering van die teorie beskou kan word.

Thesis (PhD (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: The aim of this thesis is threefold. Firstly, I give an overview of non-commutative quan- tum mechanics and build up a description of non-commutative piecewise constant poten- tial wells in this context. Secondly, I look at some of the stationary properties of a finite non-commutative well using the mathematical tools laid out in the first part. Lastly, I in- vestigate how non-commutativity affects the tunneling rate through a barrier. Throughout this work I give the normal commutative descriptions and results for comparsion. / AFRIKAANSE OPSOMMING: Die doel van hierdie tesis is drievoudig. Eerstens gee ek ’n oorsig van niekommutatiewe kwantummeganika en bou daarmee ’n beskrywing van niekommutatiewe deelswyskon- stante potensiaal putte op. Tweedens kyk ek na ’n paar van die stasionˆere eienskappe van ’n eindige niekommutatiewe potensiaal put deur die wiskunde te gebruik wat in die eerste deel uiteengesit is. Laastens ondersoek ek hoe niekommutatiwiteit die spoed van tonneling deur ’n potensiaal wal be¨ınvloed. Dwarsdeur die hierdie hele tesis gee ek die normale kommutatiewe beskrywings en resultate vir maklike vergelyking.

Berry curvature appears in the semi-classical theory of Bloch electrons already to first order in electromagnetic fields, resulting in profound modification of the carrier velocity and phase space density of states. Here we derive the equations of motion for the physical position and crystal momentum to second order in the fields. The dynamics still has a Hamiltonian structure, albeit with noncanonical Poisson brackets between the physical variables. We are able to expand both the carrier energy and the Poisson brackets to second order in the fields with terms of clear physical meaning. To demonstrate the utility of our theory, we obtain with much ease the electromagnetic response and orbital magnetic susceptibility. / text

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