Thesis (Ph. D.)--University of Texas at Austin, 2008. / Vita. Includes bibliographical references.

Apresentamos neste trabalho um método desenvolvido com o objetivo de resolver as equações que regem o comportamento de sistemas quânticos com a utilização de Redes Neurais Artificiais. Detalhamos duas possíveis abordagens da física quântica: i) a descrição em termos da função de onda, ou representação de Schrödinger; ii) a descrição em termos da densidade eletrônica, desenvolvida a partir do modelo de Thomas-Fermi e da teoria do Funcional Densidade. Uma rede neural do tipo multicamada unidirecional com três camadas (de entrada, oculta e de saída) é utilizada para representar a função de onda ou a densidade eletrônica do sistema. Treinamos essa rede através do Algoritmo Genético, minimizando um funcional adequado a cada abordagem quântica. Esta metodologia foi aplicada à equação de Schrödinger para os seguintes sistemas de uma partícula: oscilador harmônico simples, oscilador duplo, potencial de Morse e átomo de hidrogênio. Em todos os casos, a energia do estado fundamental foi obtida com erro absoluto menor que 0,1% em relação aos valores exatos. Também resolvemos a equação de Thomas-Fermi e as equações auto-consistentes de Kohn-Sham para o átomo de Hooke e átomos de hélio, lítio e berílio. Nossos resultados foram comparados com resultados analíticos, quando disponíveis, ou com resultados obtidos por outros métodos numéricos. Para o átomo de Hooke, o erro absoluto entre o valor da energia encontrado pela rede e o resultado analítico foi de 0,6%.

The behaviour of electrons in nanostructures such as quantum wells is of interest for the design of new electronic and electro-optic devices, and also for exploration of basic many-body physics. This thesis develops and tests improved methods for describing such electronic behaviour. The system used for this work was the parabolic quantum well (PQW), an important special system which has recently attracted much experimental and theoretical attention. We firstly report self-consistent nonlinear groundstate solutions of the Poisson equation together with the Thomas-Fermi (TF) hydrostatic equations. In contrast to most previous solutions, all the electron density profiles were inhomogeneous and continuous. We also added a von Weizsacker term with and without the exchange/exchange-correlation to the above treatment, using a novel numerical approach allowing for wider electron gases than previously possible. We also report for the first time the effects of spatially varying effective mass and dielectric function in theories of this type. To investigate infrared response of these systems, we apply new hydrodynamic theories recently proposed by Dobson. By using this type of theory, we simultaneously satisfy the Harmonic Potential Theorem (extended generalized Kohn theorem) and obtain the correct 2D plasmon dispersion, as well as obtaining the correct spacing of standing plasmons. Other inhomogeneous hydrodynamic theories do not achieve this. We also showed analytically an exact solution for a plasmon mode at the Kohn frequency in addition to one found in the Harmonic Potential Theorem. An open hydrodynamic theory was then developed based on this type of mode. Numerical application of Kohn Frequency Theorem theory was shown and the results were compared with other existing hydrodynamic theories.

Dans ce travail, nous nous intéressons à l'analyse numérique de problèmes aux valeurs propres non linéaires, comme on peut en trouver en chimie quantique ou en mécanique. La résolution de ces problèmes étant très coûteuse, l'idée est de proposer de nouvelles méthodes permettant de simplifier la résolution de ce type de problèmes et ainsi diminuer le coût de calcul. L'analyse numérique est nécessaire pour comprendre si l'impact positif sur le coût de calcul total n'a pas de mauvaise conséquence sur la précision des résultats. On propose un complément aux travaux existants sur les estimations d'erreur a priori, afin d'obtenir des résultats équivalents à ceux connus dans le cas de problèmes aux valeurs propres linéaires. Ces résultats ont été utilisés pour la mise en oeuvre et l'analyse numérique de nouveaux schémas à deux grilles pour l'approximation de problèmes aux valeurs propres non linéaires. Ensuite, on propose d'adapter ce type de méthode de sous-grilles, pour une utilisation associée à la méthode des bases réduites.

Les propriétés microscopiques des plasmas chauds et denses --- plasmas couplés --- constituent un domaine d'étude essentiellement exploré par les théories de physique classique telles que le plasma à une composante, théorie basée sur un certain nombre de paramètres ajustables, en particulier l'ionisation.<br /><br />Nous nous proposons, dans ce travail de thèse, d'aborder cette thématique par une approche sans paramètre basée sur le couplage cohérent de la dynamique moléculaire classique des noyaux et de la théorie de la fonctionnelle de la densité sans orbitale pour les électrons. La composante électronique est ainsi représentée par une énergie libre semi-classique dont la seule variable pertinente est la densité locale.<br /><br />Ce modèle a été validé par comparaison avec une méthode ab initio, la dynamique moléculaire quantique, qui décrit également le fluide électronique par une énergie libre mais exprimée au moyen d'une théorie quantique de particules indépendantes. Suite à cette validation, la dynamique moléculaire sans orbitale a été mise à profit pour évaluer l'équation d'état, à l'équilibre thermodynamique, de plasmas de bore et de fer à très haute température et densité. De plus, des comparaisons avec les modèles classiques ont été entreprises sur les propriétés structurales et dynamiques. Enfin, les lois de mélange d'équations d'état ou de coefficient de transport ont été vérifiées par simulation directe d'un plasma constitué de deutérium et de cuivre.

The two-dimensional electronic systems (2DESs) have kept surprising physicists for the last few decades. Examples include the integer and fractional quantum Hall effects, cuprate superconductivity, and graphene. This thesis is intended to develop suitable theoretical tools which can be generalized to study new types of 2DESs with strong correlation feature. The first part of this thesis describes the investigation of heterostructures made by Mott insulators. This work is mostly motivated by the significant improvement of techniques for layer-by-layer growth of transition metal oxides in the last few years. We construct a toy model based on generalized Hubbard model complemented with long-ranged Coulomb interaction, and we study it by Hartree-Fock theory, dynamical mean-field theory, and Thomas-Fermi theory. We argue that interesting 2D strongly correlated electronic systems can be created in such heterostructures under several conditions. Since these 2D systems are formed entirely due to the gap generated by electron-electron interaction, they are not addiabatically connected to a noninteracting electron states. This feature makes these 2D systems distinguish from the ones created in semiconductor heterostructures, and they may be potential systems having non-Fermi liquid behaviors. The second part of this thesis is devoted to the study of collective excitations in high-temperature superconductors. One important achievement in this work is to develop a time-dependent mean-field theory for t-U-J-V model, an effective low energy model for cuprates. The time-dependent mean-field theory is proven to be identical to the generalized random-phase approximation (GRPA) which includes both the bubble and ladder diagrams. We propose that the famous 41 meV magnetic resonance mode observed in the inelastic neutron scattering measurements is a collective mode arising from a conjugation relation, which has been overlooked in previous work, between the antiferromagnetic fluctuation and the phase fluctuation of the d-wave superconducting order parameter near momentum ([pi, pi]). Furthermore, we find that this collective mode signals the strength of the antiferromagnetic fluctuations which are responsible for the suppression of the superfluid density in the underdoped cuprates even at zero temperature. Finally, we perform a complete analysis on an effective model with parameters fitted by experimental data of Bi2212 within the GRPA scheme and conclude that the short-range antiferromagnetic interactions which are a remnant of the parent Mott-insulator are more likely the pairing mechanism of the High-T[subscript c] cuprates. / text

In this thesis, the properties of the Bose-Einstein condensation (BEC) in low dimensions are reviewed. Three dimensional weakly interacting Bose systems are examined by the variational method. The effects of both the attractive and the repulsive interatomic forces are studied. Thomas-Fermi approximation is applied to find the ground state energy and the chemical potential. The occurrence of the BEC in low dimensional systems, is studied for ideal gases confined by both harmonic and power-law potentials. The properties of BEC in highly anisotropic trap are investigated and the conditions for reduced dimensionality are derived.