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dc.contributor.advisorVillanueva, Alfredo
dc.contributor.authorIntroíni Isbardo, Mario
dc.date.accessioned2018-05-16T17:18:39Z
dc.date.available2018-05-16T17:18:39Z
dc.date.issued2012-05
dc.identifier.urihttps://hdl.handle.net/20.500.11801/658
dc.description.abstractEn Mecánica Cuántica, la descripción del estado de una partícula en un cierto dominio es un problema del mayor interés para aquellos que quieran explorar los entresijos de la materia a nivel molecular, atómico o subatómico. La Ecuación de Schrӧdinger es la ecuación principal de la Mecánica Cuántica, y debido a su importancia ha sido extensivamente estudiada en espacios Euclidianos. En este trabajo exploraremos nuevos espacios en donde resolver la Ecuación de Schrӧdinger Independiente del Tiempo, empleando el método de Separación de Variables. Concretamente, en variedades bidimensionales (superficies) de curvatura Gaussiana no constante. Con este fin, consideraremos primero su resolución en el espacio de Darboux D1, conocida en la literatura. Partiendo de D1, el cual es una superficie de revolución, buscamos nuevas superficies en donde separar la ecuación. Las superficies quedarán definidas por su métrica. Presentamos tres métricas y dos familias uniparamétricas de métricas Riemannianas en las que la ecuación se resuelve empleando la técnica de separación de variables. El método utilizado para obtener las nuevas superficies es de una singular importancia porque permite encontrar nuevos dominios de solubilidad de una ecuación diferencial, o encontrar nuevas variables en función de las cuales se pueda lograr dicho objetivo. Realizamos luego el estudio geométrico del espacio D1 y de las nuevas superficies obtenidas, calculando la curvatura Gaussiana y resolviendo las ecuaciones de las geodésicas. Finalmente, presentamos algunas parametrizaciones y sus correspondientes gráficas.
dc.description.abstractIn Quantum Mechanics, the description of the state of a particle in a certain domain is a problem of the utmost interest for those who wish to explore the details of matter at a molecular, atomic or subatomic level. Schrödinger’s Equation is the main equation of Quantum Mechanics, and due to its importance it has been extensively studied in Euclidean Space. In this work we will explore new spaces in which we can solve the Time Independent Schrödinger’s Equation, employing the method of Separation of Variables. More specifically, in two-dimensional manifolds (surfaces) of non constant Gaussian curvature. To this end, we first consider its solution in Darboux D1 space, which is known from the literature. Starting from D1, which is a surface of revolution, we look for new surfaces where we can separate the equation. The surfaces will be defined by their metric. We present three metrics and two families of Riemannian metrics depending on a single parameter where the equation is solved by separation of variables. The method employed to obtain the new surfaces is quite important, the reason being that it enables us to find new domains where we can solve a particular differential equation, or to find new variables in terms of which this goal can be achieved. In a later stage we conduct a geometric study of the space D1 and of the new surfaces obtained, calculating the Gaussian curvature and solving the equations of the geodesics. We conclude this work presenting some parameterizations and their corresponding graphics.
dc.language.isoesen_US
dc.subjectSchrӧdinger equationen_US
dc.subjectVariablesen_US
dc.subjectGaussian curvatureen_US
dc.subject.lcshDarboux transformationen_US
dc.subject.lcshGeodesics (Mathematics)en_US
dc.subject.lcshQuantum theory -- Mathematicsen_US
dc.subject.lcshSchrӧdinger equationen_US
dc.subject.lcshSeparation of variablesen_US
dc.subject.lcshManifolds (Mathematics)en_US
dc.subject.lcshCurvatureen_US
dc.titleSolución de la ecuación estacionaria de Schrӧdinger en variaciones bidimensionales de curvatura no constanteen_US
dc.title.alternativeSolutions of Schrӧdinger stationary equation on two dimensional manifolds of non constant curvatureen_US
dc.typeThesisen_US
dc.rights.licenseAll rights reserveden_US
dc.rights.holder(c)2012 Mario Introíni Isbardoen_US
dc.contributor.committeeBarety, Julio E.
dc.contributor.committeeRozga, Krzysztof
dc.contributor.representativePagán, Omell
thesis.degree.levelM.S.en_US
thesis.degree.disciplineApplied Mathematicsen_US
dc.contributor.collegeCollege of Arts and Sciences - Sciencesen_US
dc.contributor.departmentDepartment of Mathematicsen_US
dc.description.graduationSemesterSpringen_US
dc.description.graduationYear2012en_US


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