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A Comparative Analysis of Photon and Electron Wave Functions in Spherically Symmetric Potentials

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Michael Bush

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2016

We derived the wave function for light in a finite spherical well potential by drawing an analogy to an electron in an analogous potential energy well. The theory behind vector calculus on a spherical basis was examined. The wave equation for light, derived from Maxwellâ€™s equations, and the Pauli equation equation for electrons were combine into a unified form that was solved separation of variables, infinite series solutions, and numerical methods. The potential well for light was established by considering an environment with a constant index of refraction inside a spherical boundary and a different, but still constant, index of refraction outside the boundary. It was determined that the radial part of the wave function oscillated more inside the boundary as radial quantum number increased. The distance between the origin and the first peak amplitude inside the boundary increased as the angular momentum quantum number increased. For light, the wave number was found to be complex, therefore the temporal part of the wave function was a dampened oscillator. The time constant of the temporal part of the wave function increased as angular momentum quantum increased and decreased when the radial quantum number increased.