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Solutions of the Schrödinger equation with fractal boundary conditions

Authors: Robert Heidrich

Ref.: Bachelor thesis, Martin-Luther University of Halle-Wittenberg (2019)

Abstract: In general I want to find solutions for the time independent Schrödinger equation for a single particle which is captured in a fractal structure. This investigation could be useful because we run across fractals everywhere in nature and technology. Most organs build up fractal structures to maximize there surface area and today we can find fractal antennas in cellphones to cover a wide spectrum of frequencies. There has been research how electrons behave in non integer dimensions so it is a current topic in science. In addition it is known that the spectrum of electrons can create fractal structures such as the Hofstadter butterfly if they are exposed to a magnetic field so it is worth to have a look at the spectrum when fermions are already placed in some fractal structure. To compute the solutions I will use the finite difference method. I will try different sets of stencil rules (i.e. an arrangement of mesh points) and adjusting the grid to achieve accurate solutions. For benchmarking the code it is necessary to derive analytic results for one and two dimensional problems to compare these solutions with the numeric ones the program will generate. In addition to the previous tasks I am going to explain the abilities of the used fractal geometry and derive the fractal dimension. In this thesis I will only use fractals which can be produced with iterative methods. After finishing the calculations I will try to find a link between the correlation of the eigenvalues and the fractal dimension for different iteration steps. If there is a connection it can be shown we can predict features directly out of the boundary conditions.