Analytic representation and Dirac contour representation for quantum systems
Aisha F.A. Mohammed
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.
•Jan 1, 2023•0 citations