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On Nelson's stochastic mechanics for a semiclassical parabolic state. / Angharad Williams

Swansea University Author: Angharad Williams

Abstract

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 par...

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Published: 2012
Institution: Swansea University
Degree level: Doctoral
Degree name: Ph.D
URI: https://cronfa.swan.ac.uk/Record/cronfa42359
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spelling 2018-08-02T16:24:28.9633834 v2 42359 2018-08-02 On Nelson's stochastic mechanics for a semiclassical parabolic state. d2ec327326604268405d852548ee9250 NULL Angharad Williams Angharad Williams true true 2018-08-02 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. E-Thesis Mathematics. 31 12 2012 2012-12-31 COLLEGE NANME Mathematics COLLEGE CODE Swansea University Doctoral Ph.D 2018-08-02T16:24:28.9633834 2018-08-02T16:24:28.9633834 Faculty of Science and Engineering School of Mathematics and Computer Science - Mathematics Angharad Williams NULL 1 0042359-02082018162448.pdf 10798067.pdf 2018-08-02T16:24:48.2470000 Output 2912140 application/pdf E-Thesis true 2018-08-02T16:24:48.2470000 false
title On Nelson's stochastic mechanics for a semiclassical parabolic state.
spellingShingle On Nelson's stochastic mechanics for a semiclassical parabolic state.
Angharad Williams
title_short On Nelson's stochastic mechanics for a semiclassical parabolic state.
title_full On Nelson's stochastic mechanics for a semiclassical parabolic state.
title_fullStr On Nelson's stochastic mechanics for a semiclassical parabolic state.
title_full_unstemmed On Nelson's stochastic mechanics for a semiclassical parabolic state.
title_sort On Nelson's stochastic mechanics for a semiclassical parabolic state.
author_id_str_mv d2ec327326604268405d852548ee9250
author_id_fullname_str_mv d2ec327326604268405d852548ee9250_***_Angharad Williams
author Angharad Williams
author2 Angharad Williams
format E-Thesis
publishDate 2012
institution Swansea University
college_str Faculty of Science and Engineering
hierarchytype
hierarchy_top_id facultyofscienceandengineering
hierarchy_top_title Faculty of Science and Engineering
hierarchy_parent_id facultyofscienceandengineering
hierarchy_parent_title Faculty of Science and Engineering
department_str School of Mathematics and Computer Science - Mathematics{{{_:::_}}}Faculty of Science and Engineering{{{_:::_}}}School of Mathematics and Computer Science - Mathematics
document_store_str 1
active_str 0
description 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.
published_date 2012-12-31T03:52:48Z
_version_ 1763752614532481024
score 11.013776

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