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Autonomous generation of all Wigner functions and marginal probability densities of Landau levels

Demircioglu, B, Abdullah Verçin

Published Jan 1, 2002• 5 citations
OpenAlex

Abstract

Generation of Wigner functions of Landau levels and determination of their symmetries and generic properties are achieved in the autonomous framework of deformation quantization. Transformation properties of diagonal Wigner functions under space inversion, time reversal and parity transformations are specified and their invariance under a four-parameter subgroup of symplectic transformations are established. A generating function for all Wigner functions is developed and this has been identified as the phase-space coherent state for Landau levels. Integrated forms of generating function are used in generating explicit expressions of marginal probability densities on all possible two dimensional phase-space planes. Phase-space realization of unitary similarity and gauge transformations as well as some general implications for the Wigner function theory are presented.

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Autonomous generation of all Wigner functions and marginal probability densities of Landau levels

B Demircioğlu, A Verçin

arXiv:quant-ph/0211115•Annals of Physics•Apr 30, 2003•10 citations
Generation of Wigner functions of Landau levels and determination of their symmetries and generic properties are achieved in the autonomous framework of deformation quantization. Transformation properties of diagonal Wigner functions under space inversion, time reversal and parity transformations are specified and their invariance under a four-parameter subgroup of symplectic transformations are established. A generating function for all Wigner functions is developed and this has been identified as the phase-space coherent state for Landau levels. Integrated forms of generating function are used in generating explicit expressions of marginal probability densities on all possible two dimensional phase-space planes. Phase-space realization of unitary similarity and gauge transformations as well as some general implications for the Wigner function theory are presented.
Following an approach based on generating function method phase space characteristics of Landau system are studied in the autonomous framework of deformation quantization. Coherent state property of generating functions is established and marginal probability densities along canonical coordinate lines are derived. Well defined analogs of inner product, Cauchy-Bunyakowsy-Schwarz inequality and state functional have been defined in phase space and they have been used in analyzing the uncertainty structures. The general form of the uncertainty relation for two real-valued functions is derived and uncertainty products are computed in states described by Wigner functions. Minimum uncertainty state property of the standard coherent states is presented and uncertainty structures in the case of phase space generalized coherent states are analyzed.
The Wigner phase-space distribution function provides the basis for Moyal's deformation quantization alternative to the more conventional Hilbert space and path integral quantizations. The general features of time-independent Wigner functions are explored here, including the functional (``star'') eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux (``supersymmetric'') isospectral potential recursions; and their canonical transformations. These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P\"oschl-Teller potential, and the Liouville potential.
This work presents a comprehensive study of the exotic Landau model in a two-dimensional noncommutative plane. Beginning with the classical formulation where two conserved quantities Pi\mathcal{P}_iPi​ and Ki\mathcal{K}_iKi​ are derived, we proceed to the quantum level where these lead to two independent oscillator representations generating bosonic Fock spaces ΓPΓ_{\mathcal{P}}ΓP​ and ΓKΓ_{\mathcal{K}}ΓK​. Coherent states satisfying all Klauder's criteria are explicitly constructed, and their physical properties including normalization, continuity, resolution of the identity, temporal stability, and action identity are rigorously proven. We further develop matrix vector coherent states and quaternionic vector coherent states, examining their mathematical structure and physical implications. Detailed calculations of the free particle propagator via path integrals, uncertainty relations, and time evolution of probability densities are provided.
We consider the time-independent Wigner functions of phase-space quantum mechanics (also known as deformation quantization) for a Morse potential. First, we find them by solving the *-eigenvalue equations, using a method that can be applied to potentials that are polynomial in an exponential. A Mellin transform converts the *-eigenvalue equations to difference equations, and factorized solutions are found directly for all values of the parameters. The symbols found this way are of both diagonal and off-diagonal density operator elements in the energy basis. The Wigner transforms of the density matrices built from the known wavefunctions are then shown to confirm the solutions.
The Wigner distribution function (WDF) is a quasi-probability distribution that provides a phase-space representation of quantum states. It offers insights into the state dynamics and contains statistical information of it, such as the marginal probabilities. This thesis investigates a protocol for estimating the WDF of quantum states using symplectic transformations, which are linear canonical transformations that preserve phase-space structure and can be implemented experimentally using the technique of Gaussian optics. These transformations are studied because the time evolution of the WDF can be described by means of these transformations in cases where the Hamiltonian is a second-order (quadratic) polynomial in the phase-space variables. Furthermore, it was found that the Fourier transform of the marginal probability (measured at the laboratory) is related to the characteristic function (Fourier transform of the Wigner function). This relationship is given by Equation 2.46, which states that, with each intensity data obtained by applying a given symplectic transformation, the value of the characteristic function (Fourier transform of the WDF) at a given point of the phase space can be obtained. Based on this, an optical setup was implemented to perform symplectic transformations, such as free-space propagation and passing through a lens. The study focuses on the WDF of a one-dimensional electric field of two Gaussian beams, separated by a distance 2d, where d = 1.388 ± 0.004mm. The two beams are passed through the transformation system (propagation-lens-propagation) and then the marginal probability (intensity) is measured with a CCD camera. After this, the Fourier transform was computationally applied to the intensity data and compared with the theoretical plots of the characteristic function (Equation 2.2). In doing so, it was found that it is necessary to multiply the data obtained by an additional scaling factor to improve them to better fit the theoretical results. Finally, the protocol to be followed to reconstruct the Wigner function of this state with the obtained data was described.