Estudamos a Mecânica Quântica não-relativística de sistemas físicos caracterizados pela presença de um grau de liberdade extra, que não comuta com a coordenada temporal. Na linguagem da Geometria Não-Comutativa, tratamos de sistemas descritos por uma álgebra da forma F(Q) X \"A IND.\"teta\"\"(R X \"S POT.1\"), onde F(Q) é a álgebra de funções sobre o espaço de configurações usual \"Q\" e \"A IND.\"teta\"\"(R X \"S POT.1\") é uma deformação de F(R X \"S POT.1\"), conhecida como cilindro não-comutativo. Do ponto de vista geométrico, os geradores do cilindro não-comutativo correspondem à coordenada temporal e a uma coordenada espacial (extra) compacta, em analogia com o caso das teorias do tipo Kaluza-Klein. Mostramos que, como resultado da não-comutatividade entre o tempo e a dimensão extra, a evolução temporal dos sistemas descritos por F(Q) X \"A_t(R X S 1) é discretizada. Ao desenvolver a teoria de espalhamento para sistemas definidos nesse espaço-tempo, verificamos o aparecimento de um efeito inexistente no caso usual: transições entre um estado \"in\" com energia \"E IND.\"alfa\"\" e um estado \"out\" com energia \"E IND.\"beta\"\" (diferente de \"E IND.\"alfa\"\") passam a ser possíveis. Mais especificamente, transições serão possíveis sempre que \"E IND.\"beta\" -\" E IND.\"alfa\" = 2\"pi\"/\"teta\"n, com n \'PERTENCE A\' aos inteiros. As conseqüências desse fato são investigadas de maneira qualitativa, no caso específico de uma barreira uni-dimensional do tipo delta. Essa análise é baseada na aproximação de Born para a matriz de transição / We study the nonrelativistic Quantum Mechanics of physical systems characterized F(Q) X \"A IND.\"teta\"\"(R X \"S POT.1\"), by the presence of an extra degree of freedom which does not commute with the time coordinate. In the language of Noncommutative Geometry, we deal with systems described by an algebra of the form F(Q) X \"A IND.\"teta\"\"(R X \"S POT.1\"),, where F(Q) is the algebra of functions over the usual con¯guration space \"Q\" e \"A IND.\"teta\"\"(R X\"S POT.1\") is a deformation of F(R X \"S POT.1\"), known as noncommutative cylinder. From a geometric viewpoint, the generators of the noncommutative cylinder correspond to the time coordinate and to an extra compact spatial coordinate, just like in Kaluza-Klein theories. We show that because of the noncommutativity between the time coordinate and the extra degree of freedom, the time evolution of systems described by F(Q) X \"A_t(R X S 1) is discretized. We develop the scattering theory for such systems, and verify the presence of a new e®ect: transitions from an in state with energy \"E IND.\"alfa\"\" and an out state with energy \"E IND.\"beta\"\" (diferente de \"E IND.\"alfa\"\") are now allowed, in contrast to the usual case. In fact, transitions take place whenever \"E IND.\"beta\" -\" E IND.\"alfa\" = 2\"pi\"/\"teta\"n,, with n \'PERTENCE A\'. The consequences of this result are investigated in the case of a one-dimensional delta barrier. Our analysis is based on the Born approximation for the transition matrix.

In the presented thesis we study quantum reaction dynamics of H2O- using semiclassical method. Using ab initio quantum potential evaluated on a fine grid we derive analytical formula for potential energy surface, which is used for solving classical equations of motion. A reaction path is analyzed using contour plots with a focus on a saddle point area. Reaction dynamics analysis is focused on properties of interaction probability and cross section depending on impact parameter, collision energy and initial vibrational state of interacting molecule. Final results are compared with experimental data.

Les systèmes composés d'atomes et d'ions ultrafroids ont étés un sujet d'intérêt pour les physiciens atomiques et, plus récemment, pour la communauté des ions froids (simulation et calcul quantique avec des ions piégés). Ils sont considéré la possibilité d'utiliser un gaz d'atomes ultrafroids pour refroidir sympathiquement les ions car la modulation intrinsèque du mouvement, le micromouvement, représente une source de décohérence dans les applications des ions froids. L'intérêt envers ce système mixte est aussi motivé par l'étude de la physique d'impuretés et par une meilleure compréhension des réactions entre espèces ioniques et neutre ayant pour but la création d'ions moléculaires. Cette thèse a pour objectif d'étudier les effets du micromouvement dans les collisions atome-ion. Nous traitons au préalable les collisions à 1D d'une particule dans un piège harmonique (un ion) et d'un particule libre (une atome) en utilisant différentes approches numériques. Ce système est intéressant en soi en raison de la dimensionnalité mixte 0D-1D. Le potentiel atome-ion est modélisé par une interaction à portée nulle tout au cours de ce travail. Par la suite, nous traitons un problème similaire mais dans le cas d'une particule dans un piège harmonique décrivant un piège de Paul. Enfin, nous généralisons l'étude du micromouvement à un système modèle 3D avec un ion dans un piège de Paul sphérique 3D et un atome lourd au centre du piège. Nous discutons de l'influence du micromouvement en vue d'applications potentielles de ce système telle que la porte logique de phase. / Ultracold atom-ion systems have been a topic of interest for atomic physicists studying chemical reactions and since recently, the cold ion community (ion trap quantum computation and simulation). They have been looking at the possibility of using an ultracold atom gas to sympathetically cool ions since intrinsic motional modulation i.e micromotion is an inherent cause of decoherence in coherent applications of cold ions. Interest is also piqued by the possibility of using this hybrid system for studying impurity physics and to better understand ion-neutral reactions aimed at creation of molecular ions. In this thesis, we aim to study the effect of ion micromotion in atom-ion collision. As a prelude, we treat the 1D collision of a particle in a harmonic trap (ion) and a free particle (atom) using different numerical schemes. This system is of interest in its own right due to the mixed 0D-1D dimensionality. Atom-ion potential is simplified to a zero range potential all through out the work. Next we deal with a similar problem but with the trapped particle in a time dependent harmonic trap identical to an ion Paul trap. Finally we extend the study of micromotion to a model system in 3D with an ion in a 3D spherical Paul trap and a heavy atom at the trap centre. We discuss the effect micromotion has on potential applications of such a system, like a quantum phase gate.

Cette thèse est faite de deux parties. La première partie est un article rédigé conjointementavec Martin Puchol et Jialin Zhu. La deuxième partie est une série de résultats obtenus par moi-même liés au théorème de Riemann-Roch-Grothendieck pour les fibrés vectoriels plats. Dans la première partie, nous donnons une preuve analytique d'un résultat décrivant le comportement de la torsion analytique en théorie de de Rham lorsque la variété considérée est séparée en deux par une hypersurface. Plus précisément, nous donnons une formule liant la torsion analytique de la variété entière aux torsions analytiques associées aux variétés à bord avec des conditions limites relative ou absolue le long de l'hypersurface. Dans la deuxième partie de cette thèse, nous raffinons les résultats de Bismut-Lott pour les images directes des fibrés vectoriels plats au cas où le fibré vectoriel plat en question est lui-même la cohomologie holomorphe d'un fibré vectoriel le long d'une fibration plate à fibres complexes. Dans ce contexte, nous donnons une formule de Riemann-Roch-Grothendieck dans laquelle la classe de Todd du fibré tangent relatif apparaît explicitement. En remplaçant les classes de cohomologie par des formes explicites qui les représentent en théorie de Chern-Weil, nous généralisons ainsi des constructions de Bismut-Lott. / This thesis consists of two parts. The first part is an article written jointly with Martin Puchol and Jialin Zhu, the second part is a series of results obtained by myself in connection with the Riemann-Roch-Grothendieck theorem for flat vector bundles. In the first part, we give an analytic approach to the behavior of classical Ray-Singer analytic torsion in de Rham theory when a manifold is separated along a hypersurface. More precisely, we give a formula relating the analytic torsion of the full manifold, and the analytic torsion associated with relative or absolute boundary conditions along the hypersurface. In the second part of this thesis, we refine the results of Bismut-Lott on direct images of flat vector bundles to the case where the considered flat vector bundle is itself the fiberwise holomorphic cohomology of a vector bundle along a flat fibration by complex manifolds. In this context, we give a formula of Riemann-Roch-Grothendieck in which the Todd class of the relative holomorphic tangent bundle appears explicitly. By replacing cohomology classes by explicit differential forms in Chern-Weil theory, we extend the constructions of Bismut-Lott in this context.

Hazards associated with air pollution motivate the search for technologies capable of monitoring individual exposure to gaseous pollutants and particulate matter (PM). A Low-cost Optical Particle Counter (OPC), costing less than 50 USD, is an example of such technologies. Currently, OPCs are widely used to measure the concentration of particle matter in ambient air. While these low-cost air quality sensors are widely available, the accuracy and precision of these devices is highly uncertain. Consequently, the purpose of this thesis is to present an analytical model of two generic, low-cost OPCs based on the Laws of Conservation of Mass, Momentum, and Energy. These models utilize Mie scattering theory and Computational Fluid Dynamics models to quantify uncertainty and accuracy in low-cost OPCs based first principles. Modeling results indicate that the measurement of forward-scattered light may dramatically increase the accuracy of low-cost OPCs. These results also indicate that careful attention must be placed on the design of sensor flow passages so as to most efficiently transport particles to the scattering volume where they may be detected. A combination of careful attention to photodetector placement in the forward scattering regime as well as efficient transport to the scattering volume may increase low-cost OPC accuracy by magnitudes of order.

Cette thèse traite de plusieurs problèmes de théorie de la diffusion dans la limite semi-classique, c’est à dire des propriétés des fonctions propres généralisées d’un opérateur de Schrödinger à haute fréquence. Les fonctions propres généralisées d’un opérateur de Schrödinger sur l’espace euclidien, pour un potentiel lisse à support compact, peuvent toujours se décomposer comme la somme d’une partie entrante et d’une partie sortante, plus un terme négligeable à l’infini. La matrice de diffusion relie alors la partie entrante et la partie sortante de la fonction propre. Une première partie de ce travail concerne le spectre de la matrice de diffusion. On montre un résultat d’équidistribution des valeurs propres de la matrice de diffusion, sous l’hypothèse sans doute générique que les ensembles de points fixes de certaines applications définies à partir de la dynamique classique sont de mesure de Lebesgue nulle. Ce résultat était connu précédemment, sous l’hypothèse additionnelle que la dynamique classique est sans ensemble capté.Une seconde partie du travail concerne les ondes planes tordues, qui sont une famille particulière de fonctions propres généralisées d’un opérateur de Schrödinger, pouvant s'écrire comme la somme d'une onde plane et d'une partie purement sortante. Nous faisons l’hypothèse que la dynamique classique sous-jacente possède un ensemble capté hyperbolique, et qu’une certaine pression topologique est négative. Sous ces hypothèses, on obtient dans la limite semi-classique une description précise des ondes planes tordues comme une somme convergente d’états lagrangiens. On peut en particulier en déduire la mesure semi-classique associée aux ondes planes tordues. Si la variété est de courbure négative, et que le potentiel est nul, ces états lagrangiens sont associés à des lagrangiennes se projetant sans caustiques sur la variété de base. On peut alors en déduire des résultats sur les normes C^l et les ensembles nodaux des ondes planes tordues. Nous obtenons aussiune borne inférieure sur le nombre de domaine nodaux de la somme de deux ondes planes tordues de directions incidentes proches, pour une petite perturbation générique d’une métrique de courbure négative vérifiant la condition de pression topologique. / This thesis deals with several problems of scattering theory in the semi-classical limit, that is to say, with properties of the generalised eigenfunctions of a Schrödinger operator at high frequencies. The generalised eigenfunctions of a Schrödinger operator on the Euclidean space, with a compactly supported smooth potential, may always be written as the sum of an incoming wave and an outgoing wave, plus a term which is negligible at infinity. The scattering matrix relates the incoming part with the outgoing part. The first part of this work deals with the spectrum of the scattering matrix. We show an equidistribution result for the eigenvalues of the scattering matrix, under the hypothesis that the sets of fixed points of some maps defined from the classical dynamics has measure zero. This result was previously known under the additional assumption that the classical dynamics has an empty trapped set.A second part of this work deals with the distorted plane waves, which are a particular family of generalized eigenfunctions of a Schrödinger operator, which can be written as the sum of a plane wave and a purely outgoing part. We make the hypothesis that the underlying classical dynamics has a hyperbolic trapped set, and that a certain topological pressure is negative. Under these assumptions, we obtain in the semiclassical limit a precise description of distorted plane waves as a convergent sum of Lagrangian states. In particular, we can deduce from this the semiclassical measure associated to distorted plane waves. If we furthermore assume that the manifold has non-positive curvature, and that the potential is zero, these Lagrangian states project on the base manifold without caustics. We deduce from this results on the C^l norms and on the nodal sets of distorted plane waves. We also obtain a lower bound on the number of nodal domains of the sum of two distorted plane waves with close enough incoming directions , for a small generic perturbation of a metric of negative curvature satisfying the topological pressure assumption.

The impact parameter (IP) approximation is a semiclassical model in quantum scattering theory wherein N large masses interact with one small mass. We study this model in one spatial dimension using the tools of time-dependent scattering theory, considering a system of two large-mass particles and one small-mass particle. We demonstrate that the model's predictive power becomes arbitrarily good as the masses of the two heavy particles are made larger by studying the S-matrix for a particular scattering channel. We also show that the IP wave functions can be made arbitrarily close to the full three-body solution, uniformly in time, provided one of the large masses is fixed in place, and that such a result probably will not hold if we allow all the masses to move. / Ph. D.

No description available.

<p dir="ltr">Developing non-invasive and non-destructive testing in complex media continues to be a rich field of study (see e.g.[22, 28, 36, 76, 89] ). These types of tests have applications in medical imaging, geophysical exploration, and engineering where one would like to detect an interior region or estimate a model parameter. With the current rapid development of this enabling technology, there is a growing demand for new mathematical theory and computational algorithms for inverse problems in partial differential equations. Here the physical models are given by a boundary value problem stemming from Electrical Impedance Tomography (EIT), Diffuse Optical Tomography (DOT), as well as acoustic scattering problems. Important mathematical questions arise regarding existence, uniqueness, and continuity with respect to measured surface data. Rather than determining the solution of a given boundary value problem, we are concerned with using surface data in order to develop and implement numerical algorithms to recover unknown subregions within a known domain. A unifying theme of this thesis is to develop Qualitative Methods to solve inverse shape problems using measured surface data. These methods require very few a priori assumptions on the regions of interest, boundary conditions, and model parameter estimation. The counterpart to qualitative methods, iterative methods, typically require a priori information that may not be readily available and can be more computationally expensive. Qualitative Methods usually require more data.</p><p dir="ltr">This thesis expands the library of Qualitative Methods for elliptic problems coming from tomography and scattering theory. We consider inverse shape problems where our goal is to recover extended and small volume regions. For extended regions, we consider applying a modified version of the well-known Factorization Method [73]. Whereas for the small volume regions, we develop a Multiple Signal Classification (MUSIC)-type algorithm (see for e.g. [3, 5]). In all of our problems, we derive an imaging functional that will effectively recover the region of interest. The results of this thesis form part of the theoretical forefront of physical applications. Furthermore, it extends the mathematical theory at the intersection of mathematics, physics and engineering. Lastly, it also advances knowledge and understanding of imaging techniques for non-invasive and non-destructive testing.</p>

Thesis (MSc (Physics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: Non-commutative quantum mechanics is a generalisation of quantum mechanics which incorporates the notion of a fundamental shortest length scale by introducing non-commuting position coordinates. Various theories of quantum gravity indicate the existence of such a shortest length scale in nature. It has furthermore been realised that certain condensed matter systems allow effective descriptions in terms of non-commuting coordinates. As a result, non-commutative quantum mechanics has received increasing attention recently. A consistent formulation and interpretation of non-commutative quantum mechanics, which unambiguously defines position measurement within the existing framework of quantum mechanics, was recently presented by Scholtz et al. This thesis builds on the latter formalism, extends it to many-particle systems and links it up with non-commutative quantum field theory via second quantisation. It is shown that interactions of particles, among themselves and with external potentials, are altered as a result of the fuzziness induced by non-commutativity. For potential scattering, generic increases are found for the differential and total scattering cross sections. Furthermore, the recovery of a scattering potential from scattering data is shown to involve a suppression of high energy contributions, disallowing divergent interaction forces. Likewise, the effective statistical interaction among fermions and bosons is modified, leading to an apparent violation of Pauli’s exclusion principle and foretelling implications for thermodynamics at high densities. / AFRIKAANSE OPSOMMING: Nie-kommutatiewe kwantummeganika is ’n veralgemening van kwantummeganika wat die idee van ’n fundamentele kortste lengteskaal invoer d.m.v. nie-kommuterende ko¨ordinate. Verskeie teorie¨e van kwantum-grawitasie dui op die bestaan van so ’n kortste lengteskaal in die natuur. Dit is verder uitgewys dat sekere gekondenseerde materie sisteme effektiewe beskrywings in terme van nie-kommuterende koordinate toelaat. Gevolglik het die veld van nie-kommutatiewe kwantummeganika onlangs toenemende aandag geniet. ’n Konsistente formulering en interpretasie van nie-kommutatiewe kwantummeganika, wat posisiemetings eenduidig binne bestaande kwantummeganika raamwerke defineer, is onlangs voorgestel deur Scholtz et al. Hierdie tesis brei uit op hierdie formalisme, veralgemeen dit tot veeldeeltjiesisteme en koppel dit aan nie-kommutatiewe kwantumveldeteorie d.m.v. tweede kwantisering. Daar word gewys dat interaksies tussen deeltjies en met eksterne potensiale verander word as gevolg van nie-kommutatiwiteit. Vir potensiale verstrooi ¨ıng verskyn generiese toenames vir die differensi¨ele and totale verstroi¨ıngskanvlak. Verder word gewys dat die herkonstruksie van ’n verstrooi¨ıngspotensiaal vanaf verstrooi¨ıngsdata ’n onderdrukking van ho¨e-energiebydrae behels, wat divergente interaksiekragte verbied. Soortgelyk word die effektiewe statistiese interaksie tussen fermione en bosone verander, wat ly tot ’n skynbare verbreking van Pauli se uitsluitingsbeginsel en dui op verdere gevolge vir termodinamika by ho¨e digthede.