On Nelson's stochastic mechanics for a semiclassical parabolic state. - Angharad Williams (2012)
This thesis presents an analysis of a stochastic process characterising a parabolic motion with small random perturbations. This process arises from considerations of the Bohr correspondence limit of the atomic elliptic state. It represents the semiclassical behaviour of a particle, describing a parabolic orbit under a Coulomb potential. By first considering the analogous clfussical mechanical system, we investigate the difference between the classical and semiclassical systems. Chapter 1 begins by introducing Nelson's stochastic mechanics as a reformulation of Schrodinger's wave mechanics. Comparisons are drawn between the classical and quantum Kepler problems. In Chapter 2, we consider earlier results of Durran, Neate and Truman, together with a derivation of the parabolic state by considering the limit of the eccentricity of the semiclassical elliptic diffusion. We proceed to analyse the resulting stochastic differential equation, proving the existence of a solution in the weak sense. A complete analysis of the trajectory and time- dependence of the corresponding classical system is also provided. Chapter 3 focuses on asymptotic series solutions to more general stochastic differential equations in both one and two dimensions. Methods considered are used to find the first order quantum correction to the parabolic orbit in terms of time-ordered products. We conclude in Chapter 4 by applying the Levi-Civita transformation to the semiclassical orbit, yielding first order quantum corrections to both its Cartesian coordinates and areal velocity.