Orientador: Marcus Aloizio Martinez de Aguiar / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-24T22:36:15Z (GMT). No. of bitstreams: 1 Barreto_WendellPereira_M.pdf: 1811207 bytes, checksum: 0f8c73a79029cd1792710999b0f258a0 (MD5) Previous issue date: 2014 / Resumo: A evolução temporal de estados quânticos é estudada do ponto de vista semiclássico usando o propagador na representação de estados coerentes. No limite semiclássico o propagador pode ser calculado em termos de soluções complexas das equações de Hamilton que devem satisfazer condições de contorno apropriadas. No entanto, nem todas as soluções podem ser utilizadas no cálculo do propagador. Certas trajetórias, denominadas não contribuintes devem ser descartadas, pois dão contribuições incorretas ao propagador. Aqui, exploramos a questão das trajetórias não contribuintes, que é um dos problemas mais sérios na aplicação sistemática das expressões semiclássicas envolvendo órbitas complexas. Para isso consideramos uma classe de problemas unidimensionais não-lineares onde as soluções clássicas e quânticas poder ser obtidas analiticamente. Dessa forma, o propagador semiclássico pode ser escrito de forma explícita, o que permite uma análise detalhada da contribuição de cada trajetória. Definimos então um critério mais preciso para a exclusão de soluções espúrias e, enfim, melhorar o cálculo semiclássico. O sistema foco neste estudo foi o oscilador harmônico ao quadrado, cuja dinâmica tem solução analítica e está presente em problemas de óptica não linear / Abstract: The time evolution of quantum states is studied in the semiclassical limit using the semiclassical propagator in the coherent-state representation. In the semiclassical limit the quantum propagator can be calculated with complex solutions of Hamilton's equations satisfying appropriate boundary conditions. However, not all these solutions can be used in the expression for the propagator. Some trajectories, called non contributing trajectories, give incorrect contributions to the propagator and should be excluded. In this work the issue of non-contributing trajectories, which is one of the most serious problems in the systematic application of semiclassical expression involving complex orbits, is studied. We explore a class of nonlinear one-dimensional problems for which classical and quantum solutions can be analytically obtained. For these problems, the semiclassical propagator can be written explicitty allowing a detailed analisys of the contribution of each trajectory. In this work we will focus on the ''squared harmonic oscillator'', it can be solved analytically and it is present in problems of nonlinear optics / Mestrado / Física / Mestre em Física

Orientador: Marcus Aloizio Martinez de Aguiar / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica Gleb Wataghin / Made available in DSpace on 2018-08-10T00:40:17Z (GMT). No. of bitstreams: 1 Santos_LuisCoelhodos_D.pdf: 950495 bytes, checksum: 6d6e6d4fadee89455b54a57206af4e76 (MD5) Previous issue date: 2008 / Resumo: A supercompleteza da base de estados coerentes gera uma multiplicidade de representações da integral de trajetória de Feynman. Estas diferentes representações, embora equivalentes quanticamente, levam a diferentes limites semiclássicos. Baranger et al calcularam o limite semiclássico de duas formas para a integral de trajetória, sugeridas por Klauder e Skagerstam. Cada uma destas fórmulas envolve trajetórias governadas por uma diferente representação clássica do operador Hamiltoniano: a representação P em um caso e a representação Q no outro. Nesta tese, nós construímos outras duas representações da integral de trajetória, cujos limites semiclássicos envolvem diretamente a representação de Weyl do operador Hamiltoniano, isto é, a própria Hamiltoniana classica. Mostramos que, no limite semiclássico, a dinâmica na representação de Weyl é independente da largura dos estados coerentes e o propagador é também livre das correções de fase encontradas em todos os outros casos. Além disto, fornecemos uma conexão explícita entre as representações quânticas de Weyl e de Husimi no espaço de fases / Abstract: The overcompleteness of the coherent states basis gives rise to a multiplicity of representations of Feynman¿s path integral. These different representations, although equivalent quantum mechanically, lead to different semiclassical limits. Baranger et al derived the semiclassical limit of two path integral forms suggested by Klauder and Skagerstam. Each of these formulas involve trajectories governed by a different classical representation of the Hamiltonian operator: the P representation in one case and the Q representation in the other one. In this thesis we construct two other representations of the path integral whose semiclassical limit involves directly the Weyl representation of the Hamiltonian operator, i.e., the classical Hamiltonian itself. We show that, in the semiclassical limit, the dynamics in the Weyl representation is independent of the coherent states width and that the propagator is also free from the phase corrections found in all the other cases. Besides, we obtain an explicit connection between the Weyl and the Husimi phase space representations of quantum mechanics / Doutorado / Física Clássica e Física Quântica : Mecânica e Campos / Doutor em Ciências

Orientador: Marcus Aloizio Martinez de Aguiar / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-19T16:43:20Z (GMT). No. of bitstreams: 1 Veronez_Matheus_M.pdf: 3220654 bytes, checksum: 8a9b1ab2e2d0f8f82a8747e9a1f7c2fb (MD5) Previous issue date: 2011 / Resumo: Nas duas últimas décadas do séc. XX os estados coerentes entraram em cena como uma poderosa representação sobre a qual pode-se apoiar a mecânica quântica, possibilitando a extensão do cálculo de integrais de trajetória a uma classe de estados mais abrangente, da qual os autoestados de posição e momento são membros. Cálculos semiclássicos revelaram que as contribuições mais importantes ao propagador quântico provém de domínios centrados em trajetórias complexas no espaço de fase. O método dos estados coerentes acoplados emprega estados dinâmicos para desenvolver um esquema exato para resolver a equação de Schrödinger dependente do tempo, dinâmica esta que emprega trajetórias reais. O regime semiclássico deste método exato conduz a um resultado similar ao obtido a partir das integrais de trajetória, porém empregando trajetórias reais. Neste trabalho o interesse é desenvolver a teoria dos estados coerentes acoplados empregando as trajetórias complexas naturais à aproximação semiclássica e estudar a viabilidade deste método / Abstract: By the end of the last century the harmonic oscillator coherent states were extensively studied as a powerful representation for doing quantum mechanics on the phase space. They were employed in the development of a more general class of path integrals which has the usual Feynman path integral as a particular case. The semiclassical limit of these path integrals involves contributions of functions evaluated on complex trajectories on the phase space. The coupled coherent states (CCS), an exact method devised for solving Schrödinger\'s equation employing a set of path guided states driven by real trajectories, has its semiclassical limit in accordance with that provided by the path integral method, respecting the differences among the trajectories each one employs. In this work we extend the range of the CCS using complex trajectory guided states and we study the complex CCS theory thus obtained / Mestrado / Física / Mestre em Física

Orientador: Marcus Aloizio Martinez de Aguiar / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Fisica / Made available in DSpace on 2018-08-04T08:21:52Z (GMT). No. of bitstreams: 1 ParisioFilho_FernandoRobertodeLuna_D.pdf: 1316597 bytes, checksum: 7f79cd7aefdbaf70bddcdb50acb8e154 (MD5) Previous issue date: 2005 / Resumo: Esta tese aborda diversos aspectos da propagação semiclássica de estados coerentes. Determinamos uma expressão bastante geral para o propagador entre tais estados que, ao contrário das fórmulas existentes na literatura, é válida para pacotes de larguras quaisquer. O resultado, obtido via integração funcional, depende de trajetórias clássicas num espaço de fase complexificado. Aproximações baseadas em órbitas reais são também analisadas e demonstra-se a origem comum dos propagadores gaussianos de Heller e BAKKS. Em seguida, é feito um estudo bastante completo da propagação semiclássica de estados coerentes na representação de posição. Os resultados formais obtidos são aplicados explicitamente para o caso de um pacote gaussiano sob a influência de um potencial repulsivo suave. Para este sistema, a solução das equações de Hamilton e a própria função de onda semiclássica podem ser determinadas analiticamente. O problema das soluções não contribuintes, que se origina da aplicação do método do expoente estacionário, é resolvido através de imposições de consistência física. Os efeitos das cáusticas no espaço de fase, pontos onde a aproximação semiclássica de ordem quadrática diverge, são controlados através de correções envolvendo funções de Airy / Abstract: This thesis addresses di®erent aspects of the semiclassical propagation of coherent states. We have derived a general expression for the propagator connecting these states which, di®erently from previous formulae in the literature, is valid for packets of arbitrary widths. The result, obtained via functional integration, depends on classical trajectories in a complex phase space. Approximations based on real orbits are also analyzed and it is demonstrated that the Heller and BAKKS Gaussian propagators belong to the same category. Next we make a detailed study of the semiclassical propagation of coherent states in the position representation. The obtained formal results are applied to the case of a Gaussian packet under the influence of a smooth repulsive potential. For this system the solution of Hamilton's equations and the semiclassical wave function can be expressed analytically. The problem of non-contributing solutions, which originates from the application of the stationary exponent method, is solved by the introduction of some criteria of physical consistency. The e®ects of caustics in phase space, points where the lowest order semiclassical approximation diverges, are controlled by introducing corrections involving Airy functions / Doutorado / Física / Doutor em Ciências

Dans ce travail de thèse , je présente comment extraire les géométries discrètes de l'espace-temps de la formulation covariante de la gravitaté quantique à boucles, qui est appelé le formalisme de la mousse de spin. LQG est une théorie quantique de la gravité qui non-perturbativement quantifie la relativité générale indépendante d'un fond fixe. Il prédit que la géométrie de l'espace est quantifiée, dans lequel l'aire et le volume ne peuvent prendre que la valeur discrète. L'espace de Hilbert cinématique est engendré par les fonctions du réseau de spin. L'excitation de la géométrie peut être parfaitement visualisée comme des polyèdres floue qui collées à travers leurs facettes. La mousse de spin définit la dynamique de la LQG par une amplitude de la mousse de spin sur un complexe cellulaire avec un état du réseau de spin comme la frontiére. Cette thèse présente deux résultats principaux. Premièrement, la limite semi-classique de l'amplitude de la mousse de spin sur un complexe simplicial arbitraire avec une frontière est complètement étudiée. La géométrie discrète classique de l'espace-temps est reconstruite et classée par les configurations critiques de l'amplitude de la mousse de spin. Deuxièmement, la fonction de trois-point de LQG est calculé. Il coïncide avec le résultat de la gravité discrète. Troisièmement, la description des géométries discrètes de hypersurfaces nulles est explorée dans le cadre de la LQG. En particulier, la géométrie nulle est décrit par une structure singulière euclidienne sur la surface de type espace à deux dimensions définie par un feuilletage de l'espace-temps par hypersurfaces nulles. / In this thesis, I will present how to extract discrete geometries of space-time fromthe covariant formulation of loop quantum gravity (LQG), which is called the spinfoam formalism. LQG is a quantum theory of gravity that non-perturbative quantizesgeneral relativity independent from a fix background. It predicts that the geometryof space is quantized, in which area and volume can only take discrete value. Thekinematical Hilbert space is spanned by Penrose's spin network functions. The excita-tion of geometry can be neatly visualized as fuzzy polyhedra that glued through theirfacets. The spin foam defines the dynamics of LQG by a spin foam amplitude on acellular complex, bounded by the spin network states. There are three main results inthis thesis. First, the semiclassical limit of the spin foam amplitude on an arbitrarysimplicial cellular complex with boundary is studied completely. The classical discretegeometry of space-time is reconstructed and classified by the critical configurations ofthe spin foam amplitude. Second, the three-point function from LQG is calculated.It coincides with the results from discrete gravity. Third, the description of discretegeometries of null hypersurfaces is explored in the context of LQG. In particular, thenull geometry is described by a Euclidean singular structure on the two-dimensionalspacelike surface defined by a foliation of space-time by null hypersurfaces. Its quan-tization is U(1) spin network states which are embedded nontrivially in the unitaryirreducible representations of the Lorentz group.

La méthode de diffusion inverse est la plus efficace dans la théorie des systèmes intégrables. Introduite dans les années soixantes, d'importants résultats ont été obtenus pour les problèmes de dimension 1+1 et notamment sur l'interaction de solitons. Depuis quelques années, l'intérêt est porté sur des problèmes de dimensions supérieures comme les équations de Davey-Sterwartson, une généralisation de l'équation intégrable de Schrödinger cubique non linéaire en dimension 1+1. Des études numériques en limite semi-classique de l'équation de Davey-Stewartson II (DSII) défocalisant, font apparaître des points communs avec le cas réduit unidimensionnel, par exemple sur l'existence d'ondes de choc dispersives : des conditions initiales lisses mènent à une région d'oscillations rapides et modulées dans le voisinage des chocs des solutions des équations non dispersives dotées des mêmes conditions initiales.Cette thèse donne les premières étapes pour l'étude analytique de ce problème basée sur la méthode de la transformée de diffusion inverse. Les deux types de méthodes, directe et inverse, pour l'équation de DSII permettent de réécrire le problème sous la forme des équations D-bar. On considère la transformée spectrale directe pour l'équation DSII avec des conditions initiales lisses en limite semi-classique. La transformée spectrale directe mène à un système de Dirac elliptique singulièrement perturbé en deux dimensions. On introduit une méthode de type BKW pour ce problème et on montre qu'il est bien défini pour des paramètres spectraux k ∈ ℂ dont les modules sont suffisamment grands en controllant la solution d'une équation eikonale non linéaire. Aussi cette méthode donne des résultats numériques précis pour de tels k en limite semi-classique. Ces résultats reposent sur la solution numérique du système de Dirac singulièrement perturbé et la solution numérique du problème eikonal.On résout le problème eikonal de manière explicite pout tout k dans le cas d'un potentiel particulier. Ces calculs donnent une explication sur le fait que l'on ne puisse pas appliquer la méthode BKW pour des valeurs de |k| plus petites. On présente une nouvelle méthode numérique pour calculer la solution du problème eikonal avec des valeurs de |k| suffisamment grandes.Les calculs numériques de la transformée spectrale directe offrent une manière d'analyser le système de Dirac singulièrement perturbé pour des valeurs de |k| si petites qu'il n'y a pas de solution globale au problème eikonal. On donne une analyse semi-classique rigoureuse sur la solution pour des potentiels radiaux en k = 0, ce qui donne une expression asymptotique du coefficient de réflexion pour k = 0 et suggère une structure annulaire pour la solution, ce qui peut être utilisé quand |k| ≠ 0 est petit. L'étude numérique suggère aussi que pour certains potentiels, le coefficient de réflexion converge simplement, quand ε ↓ 0, vers une fonction limite définie pour des valeurs de k pour lesquelles le problème eikonal n'a pas de solution globale. On propose que les singularités de la fonction eikonale jouent un rôle aussi similaire que les points tournants de la théorie unidimensionelle. / Inverse scattering is the most powerful tool in theory of integrable systems. Starting in the late sixties resounding great progress was made in (1+1) dimensional problems with many break-through results as on soliton interactions. Naturally the attention in recent years turns towards higher dimensional problems as the Davey-Stewartson equations, an integrable generalisation of the (1+1)-dimensionalcubic nonlinear Schrödinger equation. The defocusing Davey-Stewartson II equation, in its semi-classical limit has been shown in numerical experiments to exhibit behavior that qualitatively resembles that of its one-dimensional reduction, namely the generation of a dispersive shock wave: smooth initial data develop a zone rapid modulated oscillations in the vicinity of shocks of solutions for the corresponding dispersionless equations for the same initial data. The present thesis provides a first step to study this problem analytically using the inverse scattering transform method. Both the direct and inverse scattering transform for DSII can be expressed as D-bar equations. We consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semi-classical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem and prove that it is well defined for sufficiently large modulus of the spectral parameter k ∈ ℂ by controlling the solution of an associated nonlinear eikonal problem. Further, we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. We present a new method for the numerical solution of the eikonal problem valid for sufficiently large |k|. For a particular potential we are able to solve the eikonal problem in a closed form for all k, acalculation that yields some insight into the failure of the WKB method for smaller values of |k|. The numerical calculations of the direct spectral transform indicate how to study the singularly perturbed Dirac system for values of |k| so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k = 0 and suggests an annular structure for the solution that may be exploited when |k| ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges point-wise as ε ↓ 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. We suggest that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory.

The celebrated Schrödinger equation is the key to understanding the dynamics of quantum mechanical particles and comes in a variety of forms. Its numerical solution poses numerous challenges, some of which are addressed in this work. Arguably the most important problem in quantum mechanics is the so-called harmonic oscillator due to its good approximation properties for trapping potentials. In Chapter 2, an algebraic correspondence-technique is introduced and applied to construct efficient splitting algorithms, based solely on fast Fourier transforms, which solve quadratic potentials in any number of dimensions exactly - including the important case of rotating particles and non-autonomous trappings after averaging by Magnus expansions. The results are shown to transfer smoothly to the Gross-Pitaevskii equation in Chapter 3. Additionally, the notion of modified nonlinear potentials is introduced and it is shown how to efficiently compute them using Fourier transforms. It is shown how to apply complex coefficient splittings to this nonlinear equation and numerical results corroborate the findings. In the semiclassical limit, the evolution operator becomes highly oscillatory and standard splitting methods suffer from exponentially increasing complexity when raising the order of the method. Algorithms with only quadratic order-dependence of the computational cost are found using the Zassenhaus algorithm. In contrast to classical splittings, special commutators are allowed to appear in the exponents. By construction, they are rapidly decreasing in size with the semiclassical parameter and can be exponentiated using only a few Lanczos iterations. For completeness, an alternative technique based on Hagedorn wavepackets is revisited and interpreted in the light of Magnus expansions and minor improvements are suggested. In the presence of explicit time-dependencies in the semiclassical Hamiltonian, the Zassenhaus algorithm requires a special initiation step. Distinguishing the case of smooth and fast frequencies, it is shown how to adapt the mechanism to obtain an efficiently computable decomposition of an effective Hamiltonian that has been obtained after Magnus expansion, without having to resolve the oscillations by taking a prohibitively small time-step. Chapter 5 considers the Schrödinger eigenvalue problem which can be formulated as an initial value problem after a Wick-rotating the Schrödinger equation to imaginary time. The elliptic nature of the evolution operator restricts standard splittings to low order, ¿ < 3, because of the unavoidable appearance of negative fractional timesteps that correspond to the ill-posed integration backwards in time. The inclusion of modified potentials lifts the order barrier up to ¿ < 5. Both restrictions can be circumvented using complex fractional time-steps with positive real part and sixthorder methods optimized for near-integrable Hamiltonians are presented. Conclusions and pointers to further research are detailed in Chapter 6, with a special focus on optimal quantum control. / Bader, PK. (2014). Geometric Integrators for Schrödinger Equations [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/38716 / Premios Extraordinarios de tesis doctorales