The Schrödinger and Pauli-Dirac Oscillators in Noncommutative Phase Space

We investigate the non-relativistic Schrödinger and Pauli-Dirac oscillators in noncommutative phase space using the five-dimensional Galilean covariant framework. The Schrödinger oscillator presented the correct energy spectrum whose non isotropy is caused by the noncommutativity with an expected similarity between this system and the particle in a magnetic field. A general Hamiltonian for the 3-dimensional Galilean covariant Pauli-Dirac oscillator was obtained and it presents the usual terms that appears in commutative space, like Zeeman effect and spin-orbit terms. We find that the Hamiltonian also possesses terms involving the noncommutative parameters that are related to a type of magnetic moment and an electric dipole moment.     

Non-commutative Geometry in Massless and Massive Particles

In this paper, we study the symmetries of massless and massive particles action. By considering the non-commutative space-time, we find appropriate non-commutative relation for relativistic particles which leaves invariant the non-commutative Minkowski space-time. We show that non-commutativity break the scale and conformal invariance in massless and massive action. So, in non-commutative space-time the massless and massive particles have same symmetry.     

Modified Form of Wigner Functions for Non-Hamiltonian Systems

Quantization of non-Hamiltonian systems （such as damped systems） often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasiprobability distribution functions. We show that a modified form of the Wigner functions satisfies a ＊-genvalue equation and can be derived from deformation quantization for such systems.     

Noncrossing Normal Ordering for Functions of Boson Operators

Normally ordered forms of functions of boson operators are important in many contexts in particular concerning Quantum Field Theory and Quantum Optics. Beginning with the seminal work of Katriel (Lett. Nuovo Cimento 10(13):565–567, 1974), in the last few years, normally ordered forms have been shown to have a rich combinatorial structure, mainly in virtue of a link with the theory of partitions. In this paper, we attempt to enrich this link. By considering linear representations of noncrossing partitions, we define the notion of noncrossing normal ordering. Given the growing interest in noncrossing partitions, because of their many unexpected connections (like, for example, with free probability), noncrossing normal ordering appears to be an intriguing notion. We explicitly give the noncrossing normally ordered form of the functions (a ^r (a ^† ) ^s ) ⁿ ) and (a ^r +(a ^† ) ^s ) ⁿ , plus various special cases. We are able to establish for the first time bijections between noncrossing contractions of these functions, k-ary trees and sets of lattice paths.     

Master Equation in Phase Space for a Uniaxial Spin System

A master equation, for the time evolution of the quasi-probability density function of spin orientations in the phase space representation of the polar and azimuthal angles is derived for a uniaxial spin system subject to a magnetic field parallel to the axis of symmetry. This equation is obtained from the reduced density matrix evolution equation (assuming that the spin-bath coupling is weak and that the correlation time of the bath is so short that the stochastic process resulting from it is Markovian) by expressing it in terms of the inverse Wigner-Stratonovich transformation and evaluating the various commutators via the properties of polarization operators and spherical harmonics. The properties of this phase space master equation, resembling the Fokker-Planck equation, are investigated, leading to a finite series (in terms of the spherical harmonics) for its stationary solution, which is the equilibrium quasi-probability density function of spin “orientations” corresponding to the canonical density matrix and which may be expressed in closed form for a given spin number. Moreover, in the large spin limit, the master equation transforms to the classical Fokker-Planck equation describing the magnetization dynamics of a uniaxial paramagnet.     

A Simple Scheme for Directly Measuring the Wigner Functions of Cavity Fields

We propose a scheme for directly measuring the Wigner functions of cavity fields.The scheme is based on the Raman interaction between atoms and cavity fields.We find a simple and direct relation between the Wigner function and the atomic population difference,By suitablychoosing the interaction time,we find that the wigner function is just two times of the atomic population difference.Thus,one can obtain the Wigner function by measuring the atomic populations and calculating the population difference.     

The Navier–Stokes Equations in a Space of Bounded Functions

We establish a blow-up rate of the Navier–Stokes equations subject to the non-slip boundary condition for a certain class of domains including bounded and exterior domains.     

Exact Solutions of the Klein-Gordon Equation for the Scarf-Type Potential via Nikiforov-Uvarov Method

In this paper we present the exact solutions of the one-dimensional Klein-Gordon equation for the Scarf-type potential with equal scalar and vector potentials. Exact solutions and corresponding energy eigenvalues equation are obtained using Nikiforov-Uvarov mathematical method for the s-wave bound state. The PT-symmetry and Hermiticity for this potential are also considered. It will be shown that the obtained results of the Scarf-type potential are reduced to the results of the well-known potentials in the special cases.     

A Note on Wigner Functions and ＊-Genvalue Equation

In deformation quantization, static Wigner functions obey functional ,-genvalue equation, which is equivalent to time-independent Schrodinger equation in Hilbert space operator formalism of quantum mechanics. This equivalence is proved mostly for Hamiltonian with form H^ = （1/2）p^2 ＋ V（x^） [D. Fairlie, Proc. Camb. Phil. Soc. 60 （1964） 581]. In this note we generalize this proof to a very general Hamiltonian H^（x^,p^） and give examples to support it.     

Comments on “Barut-Girardello Coherent States for the Parabolic Cylinder Functions”

In Chenaghlou and Faizy (Int. J. Theor. Phys. 2008), the authors claim that they have constructed the Barut-Girardello coherent states for the parabolic cylinder functions. However, we point out here that by introducing these coherent states, Schrödinger was able to put forth the idea of “coherent states of the quantum harmonic oscillator” over eighty years ago. These coherent states are derived not only from the Barut-Girardello eigenvalue equation, but also from the Schrödinger and the Klauder-Perelomov approaches. Thus, contrary to their claim, the authors have not introduced new coherent states. In particular, a wide range of the parabolic cylinder functions do not form an orthonormal basis.     

Early Universe with Variable Cosmological Constants in Higher Dimensional Space Time

The field equations with variable cosmological and gravitational constants are consider in the presence of perfect fluid for Kaluza-Klein type cosmological model. The exact solutions of the field equations are obtained by using the gamma law equation of state p=(γ−1)ρ in which the parameter γ depends on scale factor R. The functional form of γ(R) is used to analyze a wide range of cosmological solution at early universe for two phases in cosmic history: inflationary phase and radiation dominated phase. The corresponding physical interpretation of cosmological solution are also discussed in the framework of higher dimensional space time.     

Marginal Distribution of Wigner Function in Mesoscopic RLC Circuit at Finite Temperature and Its Application

By means of the Weyl correspondence and Wigner theorem the marginal distribution of Wigner function in mesoscopic RLC circuit at finite temperature was discussed. Here we endow the Wigner function with a new physical meaning, i.e., its marginal distributions’ statistical average for q ²/(2C) and p ²/(2L) are the temperature-related energy stored in capacity and in inductance of the mesoscopic RLC circuit, respectively.     

Exact Solution of the Pseudoharmonic Oscillator in the Space of Constant Positive Curvature

We present analytically the exact solution of the radial Schrödinger equation with the pseudoharmonic oscillator potential in constant positive curvature representation. Exact bound state eigenfunctions and eigenvalues obtained using factorization method. Finally, energy eigenvalues obtained here compared with the results of the theoretical methods in the limit of flat space.     

Algebraic Approach for Shape Invariant Potentials in?Klein-Gordon Equation

The Shape invariant method has the algebraic structure and its algebras are infinite-dimensional. These algebras are converted into finite-dimensional under conditions. Based on the property of this method we obtain the algebraic structure of some physical potentials in the s-wave Klein-Gordon equation with scalar and vector potential which satisfy the Shape invariant condition.     

Time Operators and Approximation of Continuous Functions

The main purpose of this paper is to study time operators associated with generalized shifts and determined by the Haar and Faber–Schauder bases on the space of continuous functions. It is given the characterization of the domains of the constructed time operators and their scalings. It is also shown how scalings of time operators affect the dynamics of associated semigroups of shift operators.     

Law of Large Numbers for Non-Local Functions on Probabilistic Cellular Automata

We prove a multi-dimensional version of the law of large numbers for invariant measures of a large class of probabilistic cellular automata, whose transition probabilities satisfy some inequalities, which are known to assure their ergodicity. In some non-ergodic cases analogous results have been obtained for local functions. We deal with a larger class of functions, which includes some non-local ones.     

Wigner Functions and Tomograms of?the?Klauder-Perelomov Coherent States for?the?Pseudoharmonic Oscillator

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, the Wigner functions of the Klauder-Perelomov coherent states (KP-CSs) for the pseudoharmonic oscillator (PHO) are obtained and the variations of the Wigner functions with the parameters k and z are discussed. Moreover, the tomograms of the KP-CSs for the PHO are calculated by virtue of intermediate coordinate-momentum representation in quantum optics. Project 10574060 supported by the National Natural Science Foundation of China and project X071049 supported by Science Foundation of Liaocheng University.     

Phase—Locking in Coupled Chaotic Oscillators

The transition from the phase-unlocking state to the phase-locking state is found at the desynchronization of synchronous chaos of coupled oscillators.In the phase-locking case,the motions of all oscillators are chaotic and desynchronized,however spatial ordering is identified in their phase distribution.     

Hyperscaling for Oriented Percolation in $$$$ Space–Time Dimensions

Consider nearest-neighbor oriented percolation in \(d+1\) space–time dimensions. Let \(\rho ,\eta ,\nu \) be the critical exponents for the survival probability up to time t, the expected number of vertices at time t connected from the space–time origin, and the gyration radius of those vertices, respectively. We prove that the hyperscaling inequality \(d\nu \ge \eta +2\rho \), which holds for all \(d\ge 1\) and is a strict inequality above the upper-critical dimension 4, becomes an equality for \(d=1\), i.e., \(\nu =\eta +2\rho \), provided existence of at least two among \(\rho ,\eta ,\nu \). The key to the proof is the recent result on the critical box-crossing property by Duminil-Copin et al. [6].     

Laser-based sensor for a coolant leak detection in a nuclear reactor

Currently, the nuclear industry needs strongly a reliable detection system to continuously monitor a coolant leak during a normal operation of reactors for the ensurance of nuclear safety. In this work, we propose a new device for the coolant leak detection based on tunable diode laser spectroscopy (TDLS) by using a compact diode laser. For the feasibility experiment, we established an experimental setup consisted of a near-IR diode laser with a wavelength of about 1392 nm, a home-made multi-pass cell and a sample injection system. The feasibility test was performed for the detection of the heavy water (D₂O) leaks which can happen in a pressurized heavy water reactor (PWHR). As a result, the device based on the TDLS is shown to be operated successfully in detecting a HDO molecule, which is generated from the leaked heavy water by an isotope exchange reaction between D₂O and H₂O. Additionally, it is suggested that the performance of the new device, such as sensitivity and stability, can be improved by adapting a cavity enhanced absorption spectroscopy and a compact DFB diode laser. We presume that this laser-based leak detector has several advantages over the conventional techniques currently employed in the nuclear power plant, such as radiation monitoring, humidity monitoring and FT-IR spectroscopy.