Wigner Functions for Non-Hamiltonian Systems on Noncommutative Space

We propose a modified form of Wigner functions for generic non-Hamiltonian systems on noncommutative space and prove that it satisfies the corresponding ＊-genvalue equation. In addition, as an example, we derive exact energy spectra and Wigner functions for a non-Hamiltonian toy model on the noncommutative space.     

Finite Phase Space,Wigner Functions,and Tomography for Two-Qubit States

We discuss the Wigner functions and tomographic probability distributions of two-qubit states. We give the kernel of the map, which provides the expression of the state tomogram in terms of the Wigner function of the two-qubit state, in an explicit form. Also we obtain the kernel of the inverse map and elucidate the connection of the constructed maps with the star-product scheme of quantization.     

带电线性谐振子在非对易空间中的Wigner函数

在利用Wigner函数性质的基础上, 考虑到空间变量的对易关系中包含了坐标 坐标的非对易性, 得到了带电线性谐振子在非对易空间中的Wigner函数。 Based on the property of wigner function, the Wigner function of charged Linear Harmonic Oscillator in non commutative space was obtained by considering the noncommutative of the coordinate coordinate in the relation of space variable.     

A Relation of the Noncommutative Parameters in Generalized Noncommutative Phase Space

We introduce the deformed boson operators which satisfy a deformed boson algebra in some special types of generalized noncommutative phase space.Based on the deformed boson algebra,we construct coherent state representations.We calculate the variances of the coordinate operators on the coherent states and investigate the corresponding Heisenberg uncertainty relations.It is found that there are some restriction relations of the noncommutative parameters in these special types of noncommutative phase space.     

Dirac Oscillator in Noncommutative Phase Space

We study the Dirac oscillators in a noncommutative phase space. The results show that the energy gap of Dirac oscillator was changed by noncommutative effect. In addition, we obtain the non-relativistic limit of the energy spectrum.     

On Quantum Mechanics on Noncommutative Quantum Phase Space

In this work, we develop a general framework in which Noncommutative Quantum Mechanics (NCQM), characterized by a space noncommutativity matrix parameter θ=ε_ij^kθ_k and a momentum noncommutativity matrix parameter β=ε_ij^kβ_k, is shown to be equivalent to Quantum Mechanics (QM) on a suitable transformed Quantum Phase Space (QPS). Imposing some constraintson this particular transformation, we firstly find that the product of the two parameters θ and β possesses alower bound in direct relation with Heisenberg incertitude relations, and secondly that the two parameters are equivalent but with opposite sign, up to a dimension factor depending on thephysical system under study. This means that noncommutativity is represented by a unique parameter which may play the role of a fundamental constant characterizing the whole NCQPS. Within our framework, we treat some physical systems on NCQPS : free particle, harmonic oscillator, system of two-charged particles, Hydrogen atom. Amongthe obtained results, we discover a new phenomenon which consists of a free particle on NCQPS viewed as equivalent to a harmonic oscillator with Larmor frequency depending on β,representing the same particle in presence of a magnetic field$vec{B}=q^{-1}vec{beta}$. For the other examples, additionalcorrection terms depending on β appear in the expression of the energy spectrum. Finally, in the two-particle system case, we emphasize the fact that for two opposite charges noncommutativity is effectively feeled with opposite sign.     

Relativistic Hydrogen-Like Atom on a Noncommutative Phase Space

The energy levels of hydrogen-like atom on a noncommutative phase space were studied in the framework of relativistic quantum mechanics. The leading order corrections to energy levels 2S _1/2, 2P _1/2 and 2P _3/2 were obtained by using the ?? and the \(\bar \theta \) modified Dirac Hamiltonian of hydrogen-like atom on a noncommutative phase space. The degeneracy of the energy levels 2P _1/2 and 2P _3/2 were removed completely by ??-correction. And the \(\bar \theta \)-correction shifts these energy levels.     

非对易相空间中角动量的分裂

非对易空间效应是一种在弦尺度下出现的物理效应. 本文首先介绍了在Schwinger表象中角动量的3个分量用产生--消灭算符的表示形式, 接着讨论了非对易相空间的量子力学代数; 然后用对易空间谐振子的产生-消灭算符表示出了在非对易情况下的角动量; 最后讨论了非对易相空间中角动量的分裂.     

Hydrogen Atom Spectrum in Noncommutative Phase Space

We study the energy levels of the hydrogen atom in the noncommutative phase space with simultaneous spacespace and momentum-momentum noncommutative relations, We find new terms compared to the case that only noncommutative space-space relations are assumed. We also present some comments on a previous paper [Alavi S A hep-th/0501215].     

Deformations of Fluid Dynamics and Friedmann Equation on Noncommutative Phase Space

Noncommutative phase space is one of the widely studied extensions of ordinary phase space, and has profound implications in cosmological physics. In this paper we study the dynamics of perfect fluid on noncommutative phase space, as well as deformations of the Friedmann equation. The Lagrangian formalism is used to take into account of the phase space noncommutativities. Then a map from canonical Lagrangian variables to Eulerian variables is employed to derive the equations of motion of the mass and current densities. We find that both these two equations receive noncommutative corrections that are linear in the noncommutative parameters. However, we also find that in the approximation of vanishing comoving velocity the leading order noncommutative correction due to momentum noncommutativity on the Friedmann equation is zero.     

DKP Oscillator with Spin-0 in Three-dimensional Noncommutative Phase Space

The DKP equation with Dirac oscillator potential for spin-0 particles has been studied when both space-space noncommutativity and momentum-momentum noncommutativity are considered. The exact wave functions and corresponding energy levels have been found. Due to the noncommutative effect, the energy spectrum is not degenerate.     

非对易相空间中各向同性谐振子的能级分裂

非对易空间的效应是出现在弦尺度下的一种物理效应. 本文介绍了量子力学非对易空间的代数关系; 讨论了非对易相空间中服从玻色-爱因斯坦统计的粒子的连续性条件, 最后给出了非对易相平面和非对易相空间中的线性谐振子的能级分裂.     

Theta Functions on Noncommutative Tori

Ordinary theta functions can be considered as holomorphic sections of line bundles over tori. We show that one can define generalized theta functions as holomorphic elements of projective modules over noncommutative tori (theta vectors). The theory of these new objects is not only more general, but also much simpler than the theory of ordinary theta-functions. It seems that the theory of theta vectors should be closely related to Manin's theory of quantized theta functions, but we don't analyze this relation.     

Generalized Wigner Operator and Bivariate Normal Distributionin p-q Phase Space

We introduce a kind of generalized Wigner operator, whose normally ordered form can lead to the bivariate normal distribution in p-q phase space. While this bivariate normal distribution corresponds to the pure vacuum state in the generalized Wigner function phase space, it corresponds to a mixed state in the usual Wigner function phase space.     

电磁场中带电粒子在非对易相空间的能级

非对易空间效应是出现在弦的尺度下的一种物理效应。 首先扼要介绍了非对易相空间中的量子力学代数、 Moyal Weyl乘法和广义Bopp变换, 然后讨论了电磁场中带电粒子的Hamiltonian算符, 最后给出了其在非对易相空间中的能级情况。     

Klein-Gordon Oscillator in Noncommutative Phase Space Under a Uniform Magnetic Field

Using a direct substitution method, Klein-Gordon oscillator in a uniform magnetic field is researched in the noncommutative phase space (NCPS), the corresponding exact energy is obtained, and the analytic eigenfunction is presented in terms of the confluent hypergeometric. It is shown that the Klein-Gordon oscillator in uniform magnetic field in noncommutative phase space has the similar behaviors to the Landau problem in commutative space. In addition, the non-relativistic limit of the energy spectrum is obtained.     

Generalized Wigner Operator and Bivariate Normal Distribution in p-q Phase Space

We introduce a kind of generalized Wigner operator, whose normally ordered form can lead to the bivariate normal distribution in p-q phase space. While this bivariate normal distribution corresponds to the pure vacuum state in the generalized Wigner function phase space, it corresponds to a mixed state in the usual Wigner function phase space.     

Dirac Oscillator in Noncommutative Phase Space and (Anti)-Jaynes-Cummings Models

We study planar Dirac oscillator in noncommutative phase space. The model is solved exactly. The relation between this model and Jaynes-Cummings (JC) or anti-Jaynes-Cummings (AJC) models is investigated. We find that the behaviors of this model depend qualitatively on the signs of a dimensionless parameter κ. For a negative κ, we find that there is a map from this model to a model which contains only AJC terms. However, for a positive κ, there is a map from this model to a model which contains both AJC and JC terms simultaneously. Our investigation may afford a new way to study the noncommutative Dirac oscillator by means of quantum optics method, and vice verse.     

The Harmonic Oscillator Influenced by Gravitational Wave in Noncommutative Quantum Phase Space

Dynamical property of harmonic oscillator affected by linearized gravitational wave (LGW) is studied in a particular case of both position and momentum operators which are noncommutative to each other. By using the generalized Bopp’s shift, we, at first, derived the Hamiltonian in the noncommutative phase space (NPS) and, then, calculated the time evolution of coordinate and momentum operators in the Heisenberg representation. Tiny vibration of flat Minkowski space and effect of NPS let the Hamiltonian of harmonic oscillator, moving in the plain, get new extra terms from it’s original and noncommutative space partner. At the end, for simplicity, we take the general form of the LGW into gravitational plain wave, obtain the explicit expression of coordinate and momentum operators.