Doctoral Dissertations
Keywords and Phrases
Decoupling; Foldy-Wouthuysen; Pseudo-Hermitian; Ultrarelativistic
Abstract
"In this dissertation, three questions, concerning approximation methods for the eigenvalues of quantum mechanical systems, are investigated: (i) What is a pseudo-Hermitian Hamiltonian, and how can its eigenvalues be approximated via numerical calculations? This is a fairly broad topic, and the scope of the investigation is narrowed by focusing on a subgroup of pseudo-Hermitian operators, namely, PT-symmetric operators. Within a numerical approach, one projects a PT-symmetric Hamiltonian onto an appropriate basis, and uses a straightforward two-step algorithm to diagonalize the resulting matrix, leading to numerically approximated eigenvalues. (ii) Within an analytic ansatz, how can a relativistic Dirac Hamiltonian be decoupled into particle and antiparticle degrees of freedom, in appropriate kinematic limits? One possible answer is the Foldy-Wouthuysen transform; however, there are alternative methods which seem to have some advantages over the time-tested approach. One such method is investigated by applying both the traditional Foldy-Wouthuysen transform and the "chiral" Foldy-Wouthuysen transform to a number of Dirac Hamiltonians, including the central-field Hamiltonian for a gravitationally bound system; namely, the Dirac-(Einstein-)Schwarzschild Hamiltonian, which requires the formalism of general relativity. (iii) Are there are pseudo-Hermitian variants of Dirac Hamiltonians that can be approximated using a decoupling transformation? The tachyonic Dirac Hamiltonian, which describes faster-than-light spin-1/2 particles, is γ5-Hermitian, i.e., pseudo-Hermitian. Superluminal particles remain faster than light upon a Lorentz transformation, and hence, the Foldy-Wouthuysen program is unsuited for this case. Thus, inspired by the Foldy-Wouthuysen program, a decoupling transform in the ultrarelativistic limit is proposed, which is applicable to both sub- and superluminal particles"--Abstract, page iii.
Advisor(s)
Jentschura, Ulrich D.
Committee Member(s)
Parris, Paul Ernest, 1954-
Hale, Barbara N.
Madison, Don H.
Mohr, Peter J.
Department(s)
Physics
Degree Name
Ph. D. in Physics
Sponsor(s)
National Science Foundation (U.S.)
Publisher
Missouri University of Science and Technology
Publication Date
Fall 2015
Pagination
xv, 295 pages
Note about bibliography
Includes bibliographic references (pages 284-292).
Rights
© 2015 Jonathan Howard Noble, All rights reserved.
Document Type
Dissertation - Open Access
File Type
text
Language
English
Subject Headings
Perturbation (Mathematics)
Approximation theory
Dirac equation
Thesis Number
T 10832
Electronic OCLC #
936208916
Recommended Citation
Noble, Jonathan Howard, "Approximation methods in relativistic eigenvalue perturbation theory" (2015). Doctoral Dissertations. 2455.
https://scholarsmine.mst.edu/doctoral_dissertations/2455
Comments
This research has been supported, both in the summer months and also during a number of research semesters, by the National Science Foundation, within grants PHY-8555454, PHY-1068547 and PHY-1403973.