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Analytic representation and Dirac contour representation for quantum systems

Aisha F.A. Mohammed

Published Jan 1, 2023• 0 citations
OpenAlex

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.

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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.
The Dirac contour representation functions fk(z) and fb(z) are employed to represent theket states |f ⟩ and bra states ⟨f |, respectively, in quantum systems with a finite-dimensional Hilbertspace H_2j+1. The scalar product within these quantum systems is defined using a contour integral.Moreover, a numerical approach is utilized to examine the time evolution of both periodic and non-periodic systems, utilizing several Hamiltonian matrices. Furthermore, the stability of periodic systemsis investigated. In addition to these aspects, we study the most significant application of the Dirac con-tour representation, which is its capability to handle an extended Hilbert space, suitable for describingquantum physics at negative temperatures.
Analytic representations of finite quantum systems based on SU(2)SU(2)SU(2) coherent states in C∪{∞}C\cup\{\infty\}C∪{∞} and potential quantum applications to energy works
This study considers the finite-dimensional Hilbert space with the quantum systems. TheSU(2)SU(2)SU(2)coherent states in the extended complex plane, which is stereo graphically equivalent to a sphere, serve as the foundation for analytical representation functions of the states in this space. The analytical representation function is practically implemented using the numerical style for both the periodic and non-periodic systems. This study shows in detail the zeroes trajectories of these analytic representation functions which apply the chronological evolution of the system and derive vital findings regarding to the behaviour of the quantum system. More potentially, there are several crucial applications of quantum mechanics to advance energy works for more in depth investigation in future pathways.
Various Euclidean, hyperbolic and elliptic analytic representations are introduced and relations among them are discussed. The Bargmann analytic representation in the complex plane is considered and its relation to other phase-space methods for the harmonic oscillator is reviewed. The general theory that relates the growth of analytic functions with the density of their zeros is applied to Bargmann functions and it leads to theorems on the completeness of sequences of Glauber coherent states. Two hyperbolic analytic representations in the unit disc, based on SU (1, 1) coherent states and also on phase states are introduced. A third analytic representation in the complex plane based on Barut–Girardello states is also considered and transformations which relate it to the other ones are studied. In the case of systems with finite-dimensional Hilbert space, an elliptic analytic representation in the extended complex plane and also another analytic representation based on theta functions are introduced. The Berezin formalism in the Euclidean, hyperbolic and elliptic cases is discussed. Contour analytic representations in these three cases are also presented.
Finite quantum systems with d-dimension Hilbert space, where position x and \nmomentum p take values in Zd(the integers modulo d) are studied. An analytic \nrepresentation of finite quantum systems, using Theta function is considered. \nThe analytic function has exactly d zeros. The d paths of these zeros on the \ntorus describe the time evolution of the systems. The calculation of these \npaths of zeros, is studied. The concepts of path multiplicity, and path winding \nnumber, are introduced. Special cases where two paths join together, are also \nconsidered. A periodic system which has the displacement operator to real \npower t, as time evolution is also studied. \nThe Bargmann analytic representation for infinite dimension systems, with \nvariables in R, is also studied. Mittag-Leffler function are used as examples of \nBargmann function with arbitrary order of growth. The zeros of polynomial \napproximations of the Mittag-Leffler function are studied.
Quantum systems in a d-dimensional Hilbert space are considered, where \nthe phase spase is Z(d) x Z(d). An analytic representation in a cell S in the \ncomplex plane using Theta functions, is defined. The analytic functions have \nexactly d zeros in a cell S. The reproducing kernel plays a central role in \nthis formalism. Wigner and Weyl functions are also studied. \nQuantum systems with positions in a circle S and momenta in Z are also \nstudied. An analytic representation in a strip A in the complex plane is also \ndefined. Coherent states on a circle are studied. The reproducing kernel is \ngiven. Wigner and Weyl functions are considered.
Quantum systems with a finite Hilbert space, where position x and momen- \ntum p take values in Z(d) (integers modulo d), are studied. An analytic \nrepresentation of finite quantum systems is considered. Quantum states are \nrepresented by analytic functions on a torus. This function has exactly d \nzeros, which define uniquely the quantum state. The analytic function of a \nstate can be constructed using its zeros. As the system evolves in time, the \nd zeros follow d paths on the torus. Examples of the paths ³n(t) of the zeros, \nfor various Hamiltonians, are given. In addition, for given paths ³n(t) of the \nd zeros, the Hamiltonian is calculated. Furthermore, periodic finite quantum \nsystems are considered. Special cases where M of the zeros follow the same \npath are also studied, and general ideas are demonstrated with several ex- \namples. Examples of the path with multiplicity M = 1; 2; 3; 4; 5 are given. It \nis evidenced within the study that a small perturbation of the initial values \nof the zeros splits a path with multiplicity M into M different paths.