Keywords and Phrases
Decoupling; Foldy-Wouthuysen; Pseudo-Hermitian; Ultrarelativistic
"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.
Jentschura, Ulrich D.
Parris, Paul Ernest, 1954-
Hale, Barbara N.
Madison, Don H.
Mohr, Peter J.
Ph. D. in Physics
National Science Foundation (U.S.)
Missouri University of Science and Technology
xv, 295 pages
© 2015 Jonathan Howard Noble, All rights reserved.
Dissertation - Open Access
Electronic OCLC #
Noble, Jonathan Howard, "Approximation methods in relativistic eigenvalue perturbation theory" (2015). Doctoral Dissertations. 2455.