Orientador: Marcus Aloizio Martinez de Aguiar / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-22T04:47:13Z (GMT). No. of bitstreams: 1 Viscondi_ThiagodeFreitas_D.pdf: 5908171 bytes, checksum: 62e83e5e2d7f988db884e3964fd40971 (MD5) Previous issue date: 2013 / Resumo: A propagação semiclássica consiste na elaboração e aplicação de métodos para a resolução aproximada da equação de Schrödinger dependente do tempo, assumindo como hipótese que a ação clássica do sistema possui valor bastante superior à constante de Planck. Dentro deste contexto, o propagador quântico representa um elemento de interesse central, uma vez que esta grandeza corresponde à amplitude de probabilidade para a transição entre dois estados do sistema físico. Em um estágio preliminar de nosso trabalho, empregamos o conceito generalizado de estados coerentes para reformular o propagador quântico em termos de uma integral de caminho. Em seguida, com a utilização do método do ponto de sela, realizamos uma dedução detalhada para a aproximação semiclássica do propagador correspondente a uma ampla classe de grupos dinâmicos. A aplicação deste resultado formal está subordinada à resolução de equações clássicas de movimento sob condições de contorno, considerando um espaço de fase com dimensão duplicada. De maneira geral, a busca por trajetórias clássicas sujeitas a valores de contorno demonstra elevado custo computacional e complexidade técnica. Por esta razão, desenvolvemos três diferentes aproximações semiclássicas determinadas exclusivamente por condições iniciais. Em uma primeira situação, elaboramos um método de propagação constituído por uma integral sobre soluções clássicas no espaço de fase duplicado. No segundo caso, com a formulação do operador semiclássico de evolução temporal, eliminamos a necessidade pela duplicação dos graus de liberdade clássicos. A terceira abordagem, designada por propagador clássico corrigido, está definida pela contribuição de uma única trajetória. Com o propósito de avaliar a precisão e eficiência das expressões semiclássicas adquiridas, exemplificamos a aplicação destas ferramentas teóricas para os estados coerentes de SU(2) e SU(3). Por fim, apresentamos uma extensa discussão sobre as vantagens introduzidas pelo espaço de fase duplicado na implementação de uma aproximação semiclássica. Deste modo, verificamos que soluções clássicas tunelantes possuem uma importante participação na descrição precisa da penetração parcial de um pacote de onda em uma barreira de potencial finita. Além disto, mostramos que o fenômeno quântico de reflexão por um potencial atrativo está diretamente associado à ocorrência de trajetórias com comportamento não-clássico. / Abstract: The semiclassical propagation comprises the development and application of methods for obtaining approximate solutions to the time-dependent Schrödinger equation, assuming the hypothesis that the classical action of the system is much greater than the Planck constant. In this context, the quantum propagator represents an element of central interest, since this quantity corresponds to the probability amplitude for the transition between two states of thephysical system. In a preliminary stage of our work, we employ the generalized concept of coherent states to reformulate the quantum propagator in terms of a path integral. Then, with use of the saddlepoint method, we conduct a detailed derivation of the semiclassical approximation for the propagator corresponding to a wide class of dynamical groups. The application of this formal result depends on the resolution of classical equations of motion under boundary conditions, considering a phase space with doubled dimension. Generally, the search for classical trajectories subject to boundary values demonstrates high computational cost and technical complexity. For this reason, we have developed three distinct semiclassical approximations exclusively determined by initial conditions. In a first situation, we elaborate a propagation method composed of an integral over classical solutions in the doubled phase space. In the second case, with the formulation of the semiclassical time-evolution operator, we eliminate the need for the duplication of the classical degrees of freedom. The third approach, designated as corrected classical propagator, is defined by the contribution of a single trajectory. In order to evaluate the accuracy and efficiency of the obtained semiclassical expressions, we exemplify the application of these theoretical tools for the coherent states of SU(2) and SU(3). At last, we present an extensive discussion on the advantages introduced by the doubled phase space in implementing a semiclassical approximation. In this way, we find that tunneling classical solutions have an important participation in the accurate description of the partial penetration of a wave packet in a finite potential barrier. Furthermore, we show that the quantum phenomenon of reflection by an attractive potential is directly associated to the occurrence of trajectories with non-classical behavior. / Doutorado / Física / Doutor em Ciências

Orientador: Silvio Antonio Sachetto Vitiello / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin / Made available in DSpace on 2018-08-24T13:18:32Z (GMT). No. of bitstreams: 1 Abreu_BrunoRicardide_M.pdf: 1928743 bytes, checksum: 32226a9b6b2fe6d0ce77dbb9efc50309 (MD5) Previous issue date: 2014 / Resumo: A matriz densidade é um objeto fundamental na mecânica estatística de sistemas de muitos corpos quânticos. Através dela pode ser encontrado o valor esperado de qualquer observável do sistema de interesse. Neste trabalho calculamos a matriz densidade a baixas temperaturas para sistemas de muitos corpos que interagem via um potencial de pares através de convolucões da matriz densidade a altas temperaturas, onde é possível utilizar aproximações semi-clássicas / Abstract: The density matrix is a fundamental object in statistical mechanics of quantum many-body systems. Through it the observed value of any observable of a quantum mechanical system of interest can be found. In this work we calculate the density matrix at low temperatures of manybody systems that interact through pairwise potentials using a convolution procedure of the density matrix at high temperatures, where is possible to apply semi-classical approximations / Mestrado / Física / Mestre em Física

The topic of this thesis is a mathematically rigorous derivation of formulae for the magnetic force which is exerted on a part of a bounded magnetized body by its surrounding. Firstly, the magnetic force is considered within a continuous system based on macroscopic magnetostatics. The force formula in this setting is called Brown's force formula referring to W. F. Brown, who gave a mainly physically motivated discussion of it. This formula contains a surface integral which shows a nonlinear dependence on the normal. Brown assumes the existence of an additional term in the surface force which cancels the nonlinearity to allow an application of Cauchy's theorem in continuum mechanics to a magnetoelastic material. The proof of Brown's formula which is given in this work involves a suitable regularization of a hypersingular kernel and uses singular integral methods. Secondly, we consider a discrete, periodic setting of magnetic dipoles and formulate the force between a part of a bounded set and its surrounding. In order to pass to the continuum limit we start from the usual force formula for interacting magnetic dipoles. It turns out that the limit of the discrete force is different from Brown's force formula. One obtains an additional nonlinear surface term which allows one to regard Brown's assumption on the surface force as a consequence of the atomistic approach. Due to short range effects one obtains moreover an additional linear surface term in the continuum limit of the discrete force. This term contains a certain lattice sum which depends on a hypersingular kernel and the underlying lattice structure.

The topic of this thesis is a mathematically rigorous derivation of formulae for the magnetic force which is exerted on a part of a bounded magnetized body by its surrounding. Firstly, the magnetic force is considered within a continuous system based on macroscopic magnetostatics. The force formula in this setting is called Brown''s force formula referring to W. F. Brown, who gave a mainly physically motivated discussion of it. This formula contains a surface integral which shows a nonlinear dependence on the normal. Brown assumes the existence of an additional term in the surface force which cancels the nonlinearity to allow an application of Cauchy''s theorem in continuum mechanics to a magnetoelastic material. The proof of Brown''s formula which is given in this work involves a suitable regularization of a hypersingular kernel and uses singular integral methods. Secondly, we consider a discrete, periodic setting of magnetic dipoles and formulate the force between a part of a bounded set and its surrounding. In order to pass to the continuum limit we start from the usual force formula for interacting magnetic dipoles. It turns out that the limit of the discrete force is different from Brown''s force formula. One obtains an additional nonlinear surface term which allows one to regard Brown''s assumption on the surface force as a consequence of the atomistic approach. Due to short range effects one obtains moreover an additional linear surface term in the continuum limit of the discrete force. This term contains a certain lattice sum which depends on a hypersingular kernel and the underlying lattice structure. / Das Thema dieser Arbeit ist eine mathematisch strenge Herleitung von Formeln für die magnetische Kraft, die auf einen Teil eines beschränkten, magnetischen Körpers durch seine Umgebung ausgeübt wird. Zunächst wird die magnetische Kraft in einem kontinuierlichen System auf Grundlage der makroskopischen Magnetostatik betrachtet. Mit Bezug auf W. F. Brown, der eine vor allem physikalisch motivierte Herleitung der Kraftformel gegeben hat, wird diese auch Brownsche Kraftformel genannt. Das Oberflächenintegral in dieser Formel zeigt eine nichtlineare Abhängigkeit von der Normalen. Um Cauchys Theorem aus der Kontinuumsmechanik in einem magnetoelastischen Material anwenden zu können, nimmt Brown an, dass die Oberflächenkraft einen zusäatzlichen Term enthält, der den nichtlinearen Ausdruck aufhebt. Der Beweis der Brownschen Kraftformel in dieser Arbeit beruht auf einer geeigneten Regularisierung eines hypersingulären Kerns und benutzt Methoden für singuläre Integrale. Danach gehen wir von einem diskreten, periodischen System von magnetischen Dipolen aus und betrachten die Kraft zwischen einem Teil einer beschränkten Menge und der Umgebung. Um zum Kontinuumslimes überzugehen, starten wir von der üblichen Kraftformel für wechselwirkende magnetische Dipole. Es zeigt sich, dass sich der Limes der diskreten Kraft von der Brownschen Kraftformel unterscheidet. Man erhält einen zusätzlichen nichtlinearen Oberflächenterm, der es ermöglicht, Browns Annahme als Konsequenz des atomistischen Zugangs zu sehen. Kurzreichweitige Effekte führen zudem zu einem linearen Oberflächenterm im Kontinuumlimes der diskreten Kraft. Dieser Zusatzterm enthält eine gewisse Gittersumme, die von einem hypersingulären Kern und der Struktur des zugrundeliegenden Gitters abhängt.

This report aims to give an insight to the path integral formalism in quantum mechanics. After explaining the kernel's construction, some of its properties and ways to compute it, we see how it relates to the Schrödinger picture. Moreover, we see how its representation can change if it is defined in the space, momentum, time or energy space. Finally, we derive Born's expansion with the kernel showing how this formalism helps to understand perturbation theory and thus scattering. The path integral formalism is then used in quantum field theory with proofs and examples of simple correlation functions. Furthermore, supersymmetry in zero and one dimension are studied with use of the localization principle and the Witten index.

The focus of this thesis is to introduce the path integral and some of its applications. One interpretation of quantum mechanics is that a microscopic system which moves from an initial- to a final state moves through each possible intermediate state. The path integral uses the principle of least action to sum over all such intermediate states to find the evolution of a quantum mechanical system. We compare the path integral approach to that of the Schrödinger equation and show that the two give an equivalent description of quantum mechanics. To demonstrate the usefulness of the path integral, we introduce low-dimensional quantum field theory (QFT). In particular, we discuss Feynman diagrams. The idea behind Feynman diagrams is to sum over all possible weak interactions between fields to evaluate the properties of a system through the path integral. We also carry out a computation of a low energy effective action in a 0-dimensional model. The result of the computation shows that there is free energy also in a vacuum. Finally, we briefly generalize some of the previous discussion to 1-dimensional QFT. To give an example of a practical application, we give a qualitative discussion of how the path integral can be applied to statistical mechanics to predict the behaviour of superfluids. / Målet med den här rapporten är att introducera konceptet vägintegral och några av dess applikationer. En tolkning av kvantmekanik är att ett mikroskopiskt system som går från ett initialt- till ett slutgiltigt tillstånd kommer att passera genom alla möjliga mellanliggande tillstånd. Vägintegralen använder sig av principen om minsta verkan för att summera över alla sådana mellanliggande tillstånd för att hitta utvecklingen hos ett system. Vi kommer att jämföra vägintegralen med Schrödingers ekvation och visa att de två ger en ekvivalent beskrivning av kvantmekaniken. För att demonstrera vägintegralens användbarhet introducerar vi lågdimensionell kvantfältteori. Vi diskuterar speciellt Feynmandiagram. Idén bakom Feynmandiagram är att summera över alla möjliga svaga interaktioner mellan fält för att utvärdera fysikaliska egenskaper hos system med hjälp av vägintegraler. Vi kommer också att utvärdera en effektiv verkan i 0-dimensionell kvantfältteori. Resultatet visar att det finns fri energi även i ett vakuum. Slutligen generaliserar vi delar av vår tidigare diskussion till 1-dimensionell kvantfältteori. Som ett exempel på praktiska applikationer för vi en kvalitativ diskussion kring hur vägintegraler kan användas inom statistisk mekanik för att förutsäga egenskaper hos superfluider.

A real-space time-domain calculation of the frequency-dependent dielectric constant of nonmetallic crystals is outlined and the integrals required for this calculation are computed. The outline is based on time dependent current density functional theory and is partially implemented in the ab initio density functional theory FIREBALL program. The addition of a vector potential to the Hamiltonian of the system is discussed as well as the need to include the current density in addition to the particle density. The derivation of gradient integrals within a localized atomic-like orbital basis is presented for use in constructing the current density. Due to the generality of the derivation we also give the derivation of the kinetic energy, dipole, and overlap interactions.

Gemit Solutions is a company that develops web based solutions for geographical analysis on pollution. One of the solutions is called EnvoMap, which is a map based tool for analyzing pollution on the water and sewage system. The assignment that this project will be working on is to implement a new functionality to EnvoMap. The idea is that it should be possible for the user to choose an area on the map and get the underlaying data for that area. To solve this an iterative method for development will be used. The result will be a prototype that meets the desired functionality. This assignment also provides the opportunity to study an interesting topic about the usage of geographical data in software development. Methods that will be used for the study is literature study, implementation, documentation and analysis. The given result includes theories, experiences and recommendations from working on the project and will try to answer the thesis question about usage of geographical data in software development. / Gemit Solutions är ett företag som utvecklar webbaserade lösningar för bland annat geografisk analys av föroreningsbelastning. En av lösningarna heter EnvoMap och är ett kartbaserat system för analys av föroreningsbelastning på VA-nätet (vatten och avlopp). Uppgiften det här projekt skall lösa är att Gemit vill ha en ny funktionalitet på EnvoMap. Tanken är att det skall gå att välja ett område på kartan och få ut underliggande data för det området. För att lösa detta kommer en iterativ utvecklings metod att användas. Resultatet av detta är tänkt att ge en prototyp/lösningsförslag som möter de funktionella kraven. Den här uppgiften medför också möjligheten till en intressant frågeställning om användning av geografisk data vid mjukvaruutveckling, som rapporten kommer undersöka. Metoder som litteraturstudie, implementation, dokumentation och analys kommer användas för undersökningen. Resultat som ges av undersökningen innefattar teorier, erfarenheter och rekommendationer från arbetet och kommer att försöka besvara frågeställningen om användning av geografisk data vid mjukvaruutveckling.

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This dissertation introduces advances in the Wiener path integral (WPI) technique for determining efficiently and accurately the stochastic response of diverse nonlinear dynamical systems. First, a novel, general, formalism of the WPI technique is developed to account, in a direct manner, also for systems with non-Markovian response processes. Specifically, the probability of a path and the associated transition probability density function (PDF) corresponding to the Wiener excitation process are considered. Next, a functional change of variables is employed, in conjunction with the governing stochastic differential equation, for deriving the system response joint transition PDF as a functional integral over the space of possible paths connecting the initial and final states of the response vector. In comparison to alternative derivations in the literature, the herein-developed formalism does not require the Markovian assumption for the system response process. Overall, the veracity and mathematical legitimacy of the WPI technique to treat also non-Markovian system response processes are demonstrated. In this regard, nonlinear systems with a history-dependent state, such as hysteretic structures or oscillators endowed with fractional derivative elements, can be accounted for in a direct manner—that is, without resorting to any ad hoc modifications of the WPI technique pertaining, typically, to employing additional auxiliary filter equations and state variables. Next, a reduced-order WPI formulation is introduced for efficiently determining the stochastic response of diverse nonlinear systems with fractional derivative elements. This formulation can be also construed as a dimension reduction approach that renders the associated computational cost independent of the total number of stochastic dimensions of the problem. In fact, the proposed technique can determine, directly, any lower-dimensional joint response PDF corresponding to a subset only of the response vector components. This is accomplished by utilizing an appropriate combination of fixed and free boundary conditions in the related variational, functional minimization, problem. Notably, the reduced-order WPI formulation is particularly advantageous for problems where the interest lies in, few only, specific degrees-of-freedom whose stochastic response is critical for the design and optimization of the overall system. Further, an extrapolation approach within the WPI technique is developed that significantly enhances the computational efficiency of the technique without, practically, affecting the associated degree of accuracy. Overall, the WPI technique treats the system response joint transition PDF as a functional integral over the space of all possible paths connecting the initial and the final states of the response vector. Next, the functional integral is evaluated, ordinarily, by considering the contribution only of the most probable path. This corresponds to an extremum of the functional integrand, and is determined by solving a functional minimization problem that takes the form of a deterministic boundary value problem (BVP). This BVP corresponds to a specific grid point of the response PDF domain. Remarkably, the BVPs corresponding to two neighboring grid points not only share the same equations, but also the boundary conditions differ only slightly. This unique aspect of the technique is exploited, and it is shown that solution of a BVP and determination of the response PDF value at a specific grid point can be used for extrapolating and estimating efficiently and accurately the PDF values at neighboring points without the need for considering additional BVPs. Last, a joint time-space extrapolation approach within WPI technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of the above space-domain extrapolation scheme to account also for the temporal dimension. Specifically, it is shown that information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs. In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time-space domain at practically zero additional computational cost. Compared to an alternative brute-force implementation of the WPI technique, and to a standard Monte Carlo simulation (MCS) solution treatment, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude. Several representative numerical examples are considered to demonstrate the reliability of the developed techniques. Juxtapositions with pertinent MCS data are included as well.