Publication

Analytic representation and Dirac contour representation for quantum systems

Mohammed, Aisha F.A.
Publication Date
End of Embargo
Supervisor
Vourdas, Apostolos
Lei, Ci
Keywords
Finite dimensional Hilbert space, SU(2) coherent states, Extended complex plane, Analytic representations, Dirac contour representation, Hilbert spaces H(2j+1)+ and H(2j+1)−, Quantum systems
Rights
Creative Commons License
The University of Bradford theses are licenced under a Creative Commons Licence.
Peer-Reviewed
Open Access status
Accepted for publication
Institution
University of Bradford
Department
Department of Computer Science. Faculty of Engineering and Informatics
Awarded
2023
Embargo end date
Collections
Theses
Show all metadata
Files
PhD Thesis
Adobe PDF, 4.76 MB
Additional title
Abstract
This research explores analytic and Dirac contour representation systems with finite d-dimensional Hilbert space, where the dual variables, position x, and momentum p, take values in Z(d)(the integer modulo d). It introduces the SU(2) formalism, defining angular momentum states and operators, and investigates the SU(2) formalism in the harmonic oscillator context and SU(2) coherent states and their resolution of identity. Analytical representations based on SU(2) coherent states are developed and numerically implemented in the extended complex plane. The investigation explores time evolution in periodic and non-periodic systems, examining special cases like joining two paths and the behavior of zeros of the analytic representation function approaching zero and ±∞. Furthermore, the research explores Dirac contour representation in a finite dimensional Hilbert space in the extended complex plane. It defines the scalar product of states and proves the relationship between Dirac ket and Dirac bra representations. Time evolution in periodic and non-periodic systems is investigated with numerical implementations and visualizations using MATLAB. Additionally, this study applies Dirac contour representation to finite-dimensional Hilbert spaces at positive and negative temperatures, denoted as H(2j+1)+ and H(2j+1)−, respectively. It defines operators within these spaces and introduces the scalar product of two states in each system. The investigation explores L operators capable of mapping states from H(2j+1)+ to H(2j+1)− and vice versa, examining their Dirac contour representation and obtaining intriguing numerical results visualized using MATLAB.
URI
https://bradscholars.brad.ac.uk/handle/10454/20428
Version
Citation
Link to publisher’s version
Link to published version
Link to Version of Record
Type
Thesis
Qualification name
PhD
Notes