Deterministic chaos and the de Broglie-Bohm causal interpretation of quantum mechanics
| dc.contributor.advisor | Parmenter, Robert H. | en_US |
| dc.contributor.author | Valentine, Robert Warren, 1964- | |
| dc.creator | Valentine, Robert Warren, 1964- | en_US |
| dc.date.accessioned | 2013-04-18T09:30:49Z | |
| dc.date.available | 2013-04-18T09:30:49Z | |
| dc.date.issued | 1996 | en_US |
| dc.identifier.uri | http://hdl.handle.net/10150/282109 | |
| dc.description.abstract | In this thesis, properties of particle trajectories associated with the de Broglie-Bohm causal interpretation of quantum mechanics are studied. These trajectories are shown to exhibit deterministic chaos and adiabatic invariance under certain conditions. The very basic elements of the causal interpretation are presented in the first chapter. These include the equations of motion for the particle and the quantum potential. A brief discussion of the philosophically agreeable features of the theory is also included. In Chapter 2, properties of chaotic systems are studied. We define deterministic chaos for a flow and present methods for calculating the maximum Lyapunov exponent. The properties of the different types of systems and the conditions that lead to chaos in these systems are analyzed. We study in detail the specific example of the two-dimensional harmonic oscillator in Chapter 3. We find that different types of trajectories include those which are periodic and chaotic. The necessary conditions for obtaining chaos are determined for a superposition of stationary states. Systems which are qualitatively similar to the harmonic oscillator are covered in Chapter 4. These include the two-dimensional infinite well, an infinite well bisected by a finite barrier, and a Rydberg atom in an external electromagnetic field. In Chapter 5, the effect of a spin 1/2 wavefunction is considered. The causal equations of motion for a spin 1/2 particle are introduced. We find that chaotic trajectories are easily obtained. The causal analogue of the geometric phase is defined in Chapter 6. This phase is shown to be an adiabatic invariant for periodic trajectories. We define the geometric frequency for both periodic and aperiodic trajectories. Finally, in Chapter 7 we examine trajectories associated with stationary states. We define necessary conditions for chaos to arise in the trajectories. The properties of entangled boson and fermion systems are analyzed. | |
| dc.language.iso | en_US | en_US |
| dc.publisher | The University of Arizona. | en_US |
| dc.rights | Copyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. | en_US |
| dc.subject | Physics, General. | en_US |
| dc.subject | Physics, Elementary Particles and High Energy. | en_US |
| dc.title | Deterministic chaos and the de Broglie-Bohm causal interpretation of quantum mechanics | en_US |
| dc.type | text | en_US |
| dc.type | Dissertation-Reproduction (electronic) | en_US |
| thesis.degree.grantor | University of Arizona | en_US |
| thesis.degree.level | doctoral | en_US |
| dc.identifier.proquest | 9706144 | en_US |
| thesis.degree.discipline | Graduate College | en_US |
| thesis.degree.discipline | Physics | en_US |
| thesis.degree.name | Ph.D. | en_US |
| dc.description.note | This item was digitized from a paper original and/or a microfilm copy. If you need higher-resolution images for any content in this item, please contact us at repository@u.library.arizona.edu. | |
| dc.identifier.bibrecord | .b34261187 | en_US |
| dc.description.admin-note | Original file replaced with corrected file October 2023. | |
| refterms.dateFOA | 2018-07-01T17:01:24Z | |
| html.description.abstract | In this thesis, properties of particle trajectories associated with the de Broglie-Bohm causal interpretation of quantum mechanics are studied. These trajectories are shown to exhibit deterministic chaos and adiabatic invariance under certain conditions. The very basic elements of the causal interpretation are presented in the first chapter. These include the equations of motion for the particle and the quantum potential. A brief discussion of the philosophically agreeable features of the theory is also included. In Chapter 2, properties of chaotic systems are studied. We define deterministic chaos for a flow and present methods for calculating the maximum Lyapunov exponent. The properties of the different types of systems and the conditions that lead to chaos in these systems are analyzed. We study in detail the specific example of the two-dimensional harmonic oscillator in Chapter 3. We find that different types of trajectories include those which are periodic and chaotic. The necessary conditions for obtaining chaos are determined for a superposition of stationary states. Systems which are qualitatively similar to the harmonic oscillator are covered in Chapter 4. These include the two-dimensional infinite well, an infinite well bisected by a finite barrier, and a Rydberg atom in an external electromagnetic field. In Chapter 5, the effect of a spin 1/2 wavefunction is considered. The causal equations of motion for a spin 1/2 particle are introduced. We find that chaotic trajectories are easily obtained. The causal analogue of the geometric phase is defined in Chapter 6. This phase is shown to be an adiabatic invariant for periodic trajectories. We define the geometric frequency for both periodic and aperiodic trajectories. Finally, in Chapter 7 we examine trajectories associated with stationary states. We define necessary conditions for chaos to arise in the trajectories. The properties of entangled boson and fermion systems are analyzed. |
