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Title: Excursions in quantum and stochastic mechanics
Author: Yu, K. Y.
Awarding Body: University College of Swansea
Current Institution: Swansea University
Date of Award: 1993
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This thesis uses Melson's stochastic mechanics to obtain some new expressions for Poisson-Léy excursion measures for a variety of physically interesting quantum mechanical systems. The main results of this thesis are contained in Chap. (II)-(IV) where we deal exclusively with one-dimensional time-homogeneous diffusion processes. In Chap. (I) we provide the necessary preliminaries for the study of Nelson diffusion processes. Chap. (II) introduces some results of Truman and Williams necessary for our analysis in the following chapter for one-dimensional time-homogeneous diffusions with inaccessible boundaries, namely quantum mechanical expressions for the transition density, the distribution of first hitting times and the Poisson-Léy excursion measures. For the latter we have been able to derive the first few terms of a corresponding asymptotic series introduced by Truman and Truman and Williams. By considering the above we determine explicit excursion measures for the spherical square well ground-state and the Ornstein-Uhlenbeck process. In Chap. (III) we determine the spectrum and where applicable the distribution of first hitting times and excursion measures for the ground-state Nelson diffusion process for some potentials which fall into the general classical categories described by Titchmarsh. By considering a potential which falls outside the description of the classical categories, we see that this gives rise to some interesting spectral properties. In Chap. (IV) we treat diffusions on (0,∞) with an accessible boundary at the origin 0. Following Mandl's analysis, for a sticky-type boundary at 0, we show that the invariant measure obtained in Chap. (II) contains an atom at the origin. Using the Ornstein-Uhlenbeck process as well as the Wrong-signed Bessel process as examples, we determine explicit expressions for the transition densities, where we have given a complete description of the corresponding inverse Laplace transforms.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available