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The path integral formulation of quantum mechanics generalizes the action principle of classical mechanics. The Feynman path integral is, roughly speaking, a sum over all possible paths that a particle can take between fixed endpoints, where each path contributes to the sum by a phase factor involving the action for the path. The resulting sum gives the probability amplitude of propagation between the two endpoints, a quantity called the propagator. Solutions of the Feynman path integral formula exist, however, only for a small number of simple systems, and modifications need to be made when dealing with more complicated systems involving singular potentials, including the Coulomb potential. We derive a generalized path integral formula, that can be used in these cases, for a quantity called the pseudo-propagator from which we obtain the fixed-energy amplitude, related to the propagator by a Fourier transform. The new path integral formula is then successfully solved for the Hydrogen atom in two and three dimensions, and we obtain integral representations for the fixed-energy amplitude. / Vägintegral-formuleringen av kvantmekanik generaliserar minsta-verkanprincipen från klassisk mekanik. Feynmans vägintegral kan ses som en summa över alla möjliga vägar en partikel kan ta mellan två givna ändpunkter A och B, där varje väg bidrar till summan med en fasfaktor innehållande den klassiska verkan för vägen. Den resulterande summan ger propagatorn, sannolikhetsamplituden att partikeln går från A till B. Feynmans vägintegral är dock bara lösbar för ett fåtal simpla system, och modifikationer behöver göras när det gäller mer komplexa system vars potentialer innehåller singulariteter, såsom Coulomb--potentialen. Vi härleder en generaliserad vägintegral-formel som kan användas i dessa fall, för en pseudo-propagator, från vilken vi erhåller fix-energi-amplituden som är relaterad till propagatorn via en Fourier-transform. Den nya vägintegral-formeln löses sedan med framgång för väteatomen i två och tre dimensioner, och vi erhåller integral-representationer för fix-energi-amplituden.

Relations between propagators and Green functions of Hamiltonians which are SUSY partners have been obtained. New exact propagators for the family of multi-well, time-dependent and non-hermitian potentials have been calculated. Non-conservative SUSY transformation has been studied in the case of multichannel Schrodinger equation with different thresholds. Spectrum (bound/virtual states and resonances) of the non-conservative SUSY partner of zero potential has been founded. Exactly solvable model of the magnetic induced Feshbach resonance has been constructed. This model was tested in the case of Rb$^{85}$. Conservative SUSY transformations of the first and the second order have been studied in the case of multichannel Schrodinger equation with equal thresholds. Transformations which introduce non-trivial coupling between scattering channels have been founded. The first order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter has been founded in the case of two channel scattering with partial waves of different parities. In the case of coinciding parities we have found the second order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter. Phenomenological two channel $^3S_1-^3D_1$ neutron-proton potential has been obtained by using single channel, phase equivalent and coupling SUSY transformations applied to zero potential.

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

Flow propagators have been frequently used in characterisation of porous media and the study of fluid transport behaviour. Previous work considered the shape of measured flow propagators using Nuclear Magnetic Resonance (NMR) discussed the influence of pore geometry, dispersion, relaxation and internal gradients. In addition, numerically simulated flow propagators were also reported. However, a uantitative numerical analysis of local contributions to flow propagators has not been considered in the literature, yet may provide significant new insights into the flow behaviour through complex porous media. In this work we use two types of beads to realize a dual-scale bead pack consisting of micro- and macropore regions for the NMR experiments. A low-field NMR system (2 MHz) was used to measure flow propagators for this sample. We further generated a dual-scale Gaussian Random Field (GRF) image based on porosity, beads diameters and volume fraction of each type of bead for numerical simulations. A Lattice Boltzmann Method (LBM) and Random Walk (RW) technique were combined to derive the simulated flow propagators and validated against experiments. We carry out a local analysis of the flow propagators showing a significant difference in bandwidth of displacements in micro- and macro-pore regions. In addition, the local flow propagators indicate a linear relationship between mixing (the fluid exchange on regions' boundaries) and flow velocities as well as a non-linear correlation between mixing and evolution times.

The successful prediction of spacecraft motion is often heavily based upon assumptions used to simplify the problem without compromising solution accuracy. For many analyses, a primary assumption used is the decoupling of trajectory and attitude dynamics when calculating trajectories. In cases where spacecraft or objects have high area to mass ratios, non-conservative effects such as atmospheric drag and solar radiation pressure can greatly perturb spacecraft translational motion based on rotational state. A modular, six degree of freedom (6DOF) simulation with coupled orbit and attitude dynamics has been developed to model spacecraft and orbits of arbitrary geometries. First, the basis for the modular rotational and translational equations of motion are introduced. Next, formulations are provided for the gravity gradient torque, solar radiation pressure, aerodynamic, and non-spherical gravity potential sources of perturbations, and the Marshall Engineering Thermosphere atmospheric model used is described. A first test case is performed using the 6DOF simulation to simulate the deorbit of the spacecraft Lightsail 1, which flew in 2015. Next, predictive cases are demonstrated using the simulation for a theoretical sail-boom-rocket combination representative of a debris removal scenario, and for the Aerodynamic Deorbit Experiement, which will demonstrate a passively stable drag sail technology and characterize its effectiveness on orbit. All simulation cases have had aerodynamic perturbation formulations compared against high fidelity Direct Simulation Monte Carlo runs, and suggestions have been made for the future development of the simulation tool.

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Nous proposons deux méthodes pour calculer le spectre des ondes de cisaillement dans les cristaux phononiques (CP) 1D et 2D. Commençant notre étude par les CP 1D, nous développons la méthode des impédances scalaires pour la couche sur le substrat 1D.Le focus principal de ce travail est sur les CP 2D : en particulier, on considère la couche sur le substrat 2D, la plaque à conditions libres 2D et la couche entre les deux substrats 2D. Comme la matrice propagateur M à travers la cellule unitaire obtenue via l’expansion des ondes planes dans une coordonnée peut avoir des composants très grandes, notre approche consiste à la substituer par sa résolvante R= (zI−M)−1qui est numériquement stable (où z est un nombre complexe hors des pecM). Deux autres outils centraux définis par la résolvante, le projecteur spectral P de t propagateur Md pour les ondes évanescentes, entrent en jeu pour le cas des CP avec un substrat. La méthode de la résolvante, fournissant les équations de dispersion et du champ d’ondes en termes de R,P de tMd, a de multiples avantages. Elle est d’une bonne précision grâce à la solution exacte dans une coordonnée, efficace grâce à la réduction du problème à une seule cellule unitaire, même pour un substrat semi-infini, et polyvalente, puisque applicable pour les structures uniformes ou périodiques à 1D ou 2D. De plus, la méthode peut être généralisée aux CP à 3D et aux ondes vectorielles.Dans les exemples numériques, nous calculons les bandes d’arrêt de basse fréquence et les comparons avec les profils de symétrie axiale et les profils perturbées. / We propose two methods for calculating the shear waves spectra in 1D and 2D phononiccrystal (PC) wave guides. Starting this study with 1D PC, we consider the 1D-periodic coated substrate. Here we develop scalar impedance method providing efficient means for analysis and calculation of dispersion spectrum. The main focus of our work in on the 2D PC’s: the 2D PC layer on a substrate, the freePC plate and the PC plate sandwiched between two substrates.Since the propagator Mover a unit cell approximated by Fourier harmonics in one coordinate can have very large components, we introduce its resolvent R= (zI−M)−1(z is a complex number outside of specM) as a numerically stable substitute. Another two key tools given in terms of there solvent, a spectral projector Pd and propagator Md for the decreasing modes, come intoplay in the case of a wave guide with a substrate. The resolvent method providing simple dispersion and wave field equations in termsof R,Pd and Md has several advantages. It is of a good precision due to the exact solution in one direction, computationally cheap due to the reduction of the problem to one unitcell even in a semi-infinite substrate, and versatile since it is applicable to uniform, 1D- or 2D-periodic structures. More over, it is extendible to P/SVwaves and 3D PC.In numerical examples, we model low-frequency band gaps and compare them for the mirror-symmetric and perturbed profiles.

<p> Although the electronic pairing hypothesis of Bardeen, Cooper and Schrieffer is verified by experiment, the lattice polarisation mechanism for over-screening the repulsion between mates in a pair need not be the sole mechanism. </p> <p> Alternatives to the lattice polarisation mechanism are discussed and a new approach to electronic polarisation is developed. Propagator formalism is reworked using a charge density propagator to avoid the Dyson equation for the screened potential. The fermi liquid approach to electrons in a metal is avoided, too, because it misses the correlations which give a solid its atomicity. An atomic picture is contrived instead and used to predict what observable features of a superconductor are influenced by atomic correlations. </p> / Thesis / Doctor of Philosophy (PhD)

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