Nesta dissertação é feito um estudo numérico da evolução temporal das soluções da equação de Schrödinger não linear unidimensional discreta, considerando os efeitos de um potencial aperiódico (ou desordenado) e a influência de um campo elétrico de externo. A análise feita tem como foco principal a caracterizando das soluções como sendo estendidas ou localizadas, de acordo com a intensidade da não-linearidade ou correlação (U), da desordem (ε) e do campo elétrico (F), sendo que estas são dadas em unidades do hopping (V), sendo este o termo associado com a probabilidade da partícula pular a sítios vizinhos. Além disso, consideramos a influência de duas condições iniciais especificas: somente o sítio central da rede populado (distribuição delta), e uma distribuição gaussiana centrada no sítio central da rede com desvio padrão σ= 5 (distribuição gaussiana). A equação de Schrödinger estudada, descrita pela aproximação tight-binding, e resolvida numericamente através do algoritmo conhecido como método de Crank-Nicholson, que fornece a evolução temporal das amplitudes da função de onda (amplitudes de Wannier) em cada sítio da rede, mantendo a normalização da função de onda total, fornecendo assim a evolução dinâmica da probabilidade de encontrar a partícula em cada sítio. Utilizando as amplitudes de Wannier, algumas funções auxiliares locais e globais são calculadas a fim de obter informações importantes sobre a distribuição do pacote na rede ao longo do tempo, sendo elas a entropia de Shannon, o número de participação de Wegner, a função de Anderson e o centróide da distribuição. A análise dos resultados e feita através da análise gráfica do perfil do pacote de ondas na rede e da evolução temporal das funções auxiliares. Baseando-se nesta análise, pode-se perceber que tanto o aumento da intensidade da correlação como da desordem tendem a localizar o pacote de ondas, sendo que, para distribuições iniciais específicas, existem regiões de parâmetros onde o aumento da localização e acentuado e abrupto, permitindo-nos, em alguns casos, definir limiares de transição bem claros entre regiões de estados estendidos e localizados. Com a inserção do campo elétrico externo, pode-se observar um comportamento oscilatório do pacote de ondas, cuja forma depende das condições iniciais, com um período dependente do inverso do módulo do campo elétrico (F), caracterizando assim um efeito conhecido como oscilação de Bloch. A consideração destes três efeitos, não apenas isoladamente, mas associados conjuntamente, apresenta interessantes padr6es de localização dinâmica, principalmente nos casos com campo elétrico, onde o incremento da desordem e da correlação destroem as oscilações de Bloch e acabam localizando o pacote de ondas, de maneiras diferentes. / This dissertation presents a numerical study of the one-dimensional discrete non-linear Schrödinger equation considering the effects of an aperiodic (or disordered) potential and the influence of a do external electric field. The analysis is focused on the characterization of the solutions to be extended or localized, according to the intensity of non-linearity or correlation (U), disorder (ε) and electric field (F), and these are given in units of the hopping (V), the term associated with the probability of the particle to hop to nearest sites. We also consider the influence of two specific initial conditions: only the central site of the lattice populated (delta distribution), and a Gaussian distribution centered on the central lattice site with a standard deviation σ = 5 (gaussian distribution). The Schrödinger equation studied, described by the tight-binding approximation, is solved numerically using the algorithm known as the Crank-Nicholson method, which provides the temporal evolution of the amplitudes of the wave function (Wannier amplitudes) at each lattice site, keeping the normalization of the total wave function, thereby providing the dynamic evolution of the probability of finding the particle at each site. Using the Wannier amplitudes, some auxiliary local and global functions are calculated to obtain important information about the distribution of the packet on the lattice, these being the Shannon entropy, the number of participation of Wegner, the function of Anderson and the centroid of the distribution. The analysis of the results is done through the graphical analysis of the profile of the wave packet in the lattice and the temporal evolution of the auxiliary functions. Based on this analysis, one can see that both the increased intensity of correlation and the disorder tend to localize the wave packet, and, for specific initial distributions, there are regions of parameters where the increase in localization is sharp and abrupt allowing us, in some cases, to set clear transition thresholds between regions of extended and localized states. With the introduction of the external electric field, one can observe an oscillatory behavior of the wave packet, whose form depends on the initial conditions, with a period dependent on the inverse of the module of the electric field (F), thus demonstrating an effect known as Bloch oscillation. Considering these three effects, not only individually, but linked together, it presents interesting patterns of dynamical localization, especially in the case with the electric field, where the increase of disorder and correlation destroy the Bloch oscillations and end up localizing the wave packet, in different ways.

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Orientadores : Stefano de Leo, Pietro Rotelli / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-01T02:48:01Z (GMT). No. of bitstreams: 1 Ducati_GiseleCristina_D.pdf: 1064826 bytes, checksum: e206d2ad21fbc3926474a864bc60a0c7 (MD5) Previous issue date: 2002 / Doutorado / Doutor em Matemática Aplicada

A system of nonrelativistic charged particles and radiation is canonically quantized in the Coulomb gauge and Maxwell's equations in quantum electrodynamics are derived. By requiring form invariance of the Schrodinger equation under a space and time dependent unitary transformation, operator gauge transformations on the quantized electromagnetic potentials and state vectors are introduced. These gauge transformed potentials have the same form as gauge transformations in non-Abelian gauge field theories. A gauge-invariant method for solving the time-dependent Schrodinger equation in quantum electrodynamics is given. Maxwell's equations are written in a form which holds in all gauges and which has formal similarity to the equations of motion of non-Abelian gauge fields. A gauge-invariant derivation of conservation of energy in quantum electrodynamics is given. An operator gauge transformation is made to the multipolar gauge in which the potentials are expressed in terms of the electromagnetic fields. The multipolar Hamiltonian is shown to be the minimally coupled Hamiltonian with the electromagnetic potentials in the multipolar gauge. The model of a charged harmonic oscillator in a single-mode electromagnetic field is considered as an example. The gauge-invariant procedure for solving the time-dependent Schrodinger equation is used to obtain the gauge-invariant probabilities that the oscillator is in an energy eigenstate For comparison, the conventional approach is also used to solve the harmonic oscillator problem and is shown to give gauge-dependent amplitudes.

The nonlinear Schrödinger equation appears in different fields of physics, for example in the theory of Bose-Einstein condensates or in wave propagation models. From a mathematical point of view, the study of this equation is interesting and delicate, notably because it can have a very rich set of solutions with various behaviours.<p><p>In this thesis, we have been interested in standing waves, which satisfy an elliptic partial differential equation. When this equation is seen as a singularly perturbed problem, its solutions concentrate, in the sense that they converge uniformly to zero outside some concentration set, while they remain positive on this set.<p><p>We have obtained three kind of new results. Firstly, under symmetry assumptions, we have found solutions concentrating on a sphere. Secondly, we have obtained the same type of solutions for the Schrödinger-Poisson system. The method consists in applying the mountain pass theorem to a penalized problem. Thirdly, we have proved the existence of solutions of the nonlinear Schrödinger equation concentrating at a local maximum of the potential. These solutions are found by a more general minimax principle. Our results are characterized by very weak assumptions on the potential./<p><p>L'équation de Schrödinger non-linéaire apparaît dans différents domaines de la physique, par exemple dans la théorie des condensats de Bose-Einstein ou dans des modèles de propagation d'ondes. D'un point de vue mathématique, l'étude de cette équation est intéressante et délicate, notamment parce qu'elle peut posséder un ensemble très riche de solutions avec des comportements variés. <p><p>Dans cette thèse ,nous nous sommes intéressés aux ondes stationnaires, qui satisfont une équation aux dérivées partielles elliptique. Lorsque cette équation est vue comme un problème de perturbations singulières, ses solutions se concentrent, dans le sens où elles tendent uniformément vers zéro en dehors d'un certain ensemble de concentration, tout en restant positives sur cet ensemble. <p><p>Nous avons obtenu trois types de résultats nouveaux. Premièrement, sous des hypothèses de symétrie, nous avons trouvé des solutions qui se concentrent sur une sphère. Deuxièmement, nous avons obtenu le même type de solutions pour le système de Schrödinger-Poisson. La méthode consiste à appliquer le théorème du col à un problème pénalisé. Troisièmement, nous avons démontré l'existence de solutions de l'équation de Schrödinger non-linéaire qui se concentrent en un maximum local du potentiel. Ces solutions sont obtenues par un principe de minimax plus général. Nos résultats se caractérisent par des hypothèses très faibles sur le potentiel. / Doctorat en sciences, Spécialisation mathématiques / info:eu-repo/semantics/nonPublished

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Dans cette thèse, nous aborderons l'approximation de champ moyen pour des particules bosoniques. Pour un certain nombre d'états quantiques, la dérivation de la limite de champ moyen est connue, et il semble naturel d'étendre ces travaux à un cadre général d'états quantiques quelconques. L'approximation de champ moyen consiste à remplacer le problème à N corps quantique par un problème non linéaire, dit de Hartree, quand le nombre de particules est grand. Nous prouverons un résultat général pour un système de particules, confinées ou non, interagissant au travers d'un potentiel singulier. La méthode utilisée repose sur les mesures de Wigner. Notre contribution consiste en l'extension de la méthode des caractéristiques au cadre de champ de vitesse singulier associé à l'équation de Hartree. Cela complète les travaux d'Ammari et Nier et permet de prouver des résultats pour des potentiels critiques pour les équations de Hartree. En particulier, on s'intéressera à un système de bosons interagissant au travers d'un potentiel à plusieurs corps et nous démontrerons l'approximation de champ moyen sous une hypothèse de compacité forte sur ce dernier. Les résultats s’appuient en grande partie sur la flexibilité des mesures de Wigner, ce qui permet également de proposer une preuve alternative à l'approximation de champ moyen dans un cadre variationnel. / In this thesis, we justify the mean field approximation in a general framework for bosonic systems. The derivation of mean field dynamics is known for some specific quantum states. Therefore it is natural to expect the extension of these results for a general family of normal states. The mean field approximation for bosons consists in replacing the many-body quantum problem by a non linear one, so-called Hartree problem, when the number of particles tends to infinity. We establish a general result for bosons confined or not, interacting through a singular potential. The method used is based on Wigner measures. Our contribution consists in extending the characteristics method when the velocity field associated to the Hartree equation is subcritical or critical. It complements the work of Ammari and Nier and provides a result for critical potential for the Hartree equation. We also focus on bosonic systems interacting through a multi-body potential and we prove the mean field approximation under a strong assumption on this potential. All these results essentially rely on the flexibility of Wigner measures and we can give an alternative proof of the variational mean field approximation.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Meteorology, 1980. / Microfiche copy available in Archives and Science. / Vita. / Includes bibliographical references. / by William Allan Perrie. / Ph.D.

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Civil Engineering, 1980. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Vita. / Bibliography: leaves 198-203. / by Dick Kau-Ping Yue. / Sc.D.

Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Civil Engineering, 1981. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Bibliography: leaves 91-92. / by François-Marc Turpin. / M.S.