| Semiclassical methods for solving problems in quantum theory have long been used. The most common technique, referred to as the Wentzel-Kramers-Brillouin (WKB) approximation, has a number of shortcomings; for example, it requires piecewise solutions and connection formulas to avoid divergences in the solution at the classical turning points. Other techniques, aside from WKB, exist which also generate approximate solutions but avoid these difficulties. One important but less well known semiclassical technique is the method of comparison equations. Using a modified method of comparison equations, inspired by supersymmetric quantum mechanics, approximate solutions to the Dirac equation in curved spacetime will be discussed.; Fermions, most notably neutrinos which constitute the second most abundant particle in the universe, play an important role in cosmology. Solving the Dirac equation in cosmological spacetimes is useful to understand quantum effects. Comparison methods are used to solve the Dirac equation in Friedmann-Robertson-Walker (FRW) models. These solutions are used to examine current conservation and neutrino oscillations in the early universe.; Next, the technique is applied to a different class of spacetimes, namely those relating to black holes. Motivation for examining this situation is the possible application of using neutrinos to examine black hole formation during stellar collapse. Using the comparison equation method, solutions to the Dirac equation in the Kerr metric are generated and applied to examining the lack of fermion superradiance in the Kerr metric, as well as other scattering phenomena. |
