Padrões de difração de elétrons com e sem efeito de fase Aharonov-Bohm e a divergência em sua forma assintótica. - Favarato, Cássio Cecato (2019)
It is common during undergraduate courses to study the phenomenon of diffraction of electromagnetic waves by single and double slit systems via scalar diffraction theory and the Kirchoff integral. Alternatively to this, in this thesis, we analyzed the electron diffraction and interference patterns, via Feynman path integrals. For this, we start from the conceptual model, as originally proposed by RPFeynman in Quantum Mechanics and Path Integrals but which, over time, has been treated more rigorously by other authors, and with greater sophistication and clarity by M.Beau in Feynman path integral approach to electron diffraction for one and two slits: analytical results. At first, we preserve all the analysis given by this author to the problem, with some notes about probable misconceptions and construct the free propagators and wave functions for each stage of the movement for the different systems studied. H. Yabuki, in Feynman Path Integrals in the Young Double-Slit Experiment, makes it clear that in the employed representation, despite being able to extract information about the most diverse possible trajectories, the probabilistic weight of events as loops are relatively negligible, validating the superposition principle as the sum of the emerging wave functions of each slot region. From this we obtain the expression that takes the electronic distributions on the screen. By means of the variation of parameters, such as the geometric and Fresnel number contained in it, the different regimes of the optics appear: Fraunhouffer, Intermediate and Fresnel. Therefore, it was realized that these regimes could be recovered through special conditions conferred the asymptotic forms of Fresnel functions. We extend these arguments to the expression that leads to the electronic distribution with Aharonov-Bohm (AB) phase effect. Throughout a succession of approximations, we reached an expression with physical meaning and the existence of other mathematical expressions that carried divergences in their domains, corroborating the inexistence of an analogous expression to that typical of the regime of Fraunhoffer with mixed phase, that manifests the asymmetries AB. Finally, we demonstrate the emergence of the Berry phase and its relation to the AB phase.