Two-Mode Wigner Operator in 〈ξ| Representation and 〈τ| Representation

We derive the two-mode Wigner operator in the 〈ξ| representation and 〈τ| representation, where |ξ〉 is common eigenvector of the mass-weighted relative coordinate and the mass-combinatorial momentum.and |τ〉 common eigenvector of the mass-weighted relative coordinate and the mass-combinatorial momentum. As an application,we calculate the Wigner function of some two-mode state.     

A new finite-dimensional pair coherent state studied by virtue of the entangled state representation and its statistical behavior

In this paper we construct a new type of finite-dimensional pair coherent states |ξ, q〉 as realizations of SU(2) Lie algebra. Using the technique of integration within an ordered product of operator, the nonorthogonality and completeness properties of the state |ξ, q〉 are investigated. Based on the Wigner operator in the entangled state |τ〉 representation, the Wigner function of |ξ, q〉 is obtained. The properties of |ξ, q〉 are discussed in terms of the negativity of its Wigner function. The tomogram of |ξ, q〉 is calculated with the aid of the Radon transform between the Wigner operator and the projection operator of the entangled state |η, κ₁, κ₂〉. In addition, using the entangled state |τ〉 representation of |ξ, q〉 to show that the states |ξ, q〉 are just a set of energy eigenstates of time-independent two coupled oscillators.     

Solution of the spin-one DKP oscillator under an external magnetic field in noncommutative space with minimal length

The spin-one Duffin-Kemmer-Petiau(DKP) oscillator under a magnetic field in the presence of the minimal length in the noncommutative coordinate space is studied by using the momentum space representation. The explicit form of energy eigenvalues is found, and the eigenfunctions are obtained in terms of the Jacobi polynomials. It shows that for the same azimuthal quantum number, the energy E increases monotonically with respect to the noncommutative parameter and the minimal length parameter. Additionally, we also report some special cases aiming to discuss the effect of the noncommutative coordinate space and the minimal length in the energy spectrum.     

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.     

Noncommutative coordinate picture of the quantum phase space

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit coordinates and discuss the physical and dynamical picture. The isomorphism is then used as a base for the translation of the differential symplectic geometry of the infinite dimensional manifolds onto the observable algebra as a noncommutative geometry. Hence, we obtain the latter from the physical theory itself. We have essentially an extended formalism of the Schr̎odinger versus Heisenberg picture which we describe mathematically as like a coordinate map from the phase space, for which we have presented argument to be seen as the quantum model of the physical space, to the noncommutative geometry coordinated by the six position and momentum operators. The observable algebra is taken essentially as an algebra of formal functions on the latter operators. The work formulates the intuitive idea that the noncommutative geometry can be seen as an alternative, noncommutative coordinate, picture of familiar quantum phase space, at least so long as the symplectic geometry is concerned.     

Operator Realization of Radial- and Azimuthal- Differentiations Obtained by Using the Entangled State Representation

We find Bose operator realization of radial- and azimuthal-differential operations in polar coordinate system by virtue of the entangled state |η〉 representation, which indicates that |η〉 representation just fits to describe the polar coordinate operators in quantum mechanics. The Bose operator corresponding to the Laplacian operation $\frac{\partial^{2}}{\partial r^{2} }+\frac{1}{r}\frac{\partial}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{2} }{\partial\varphi^{2}}$ for 2-dimensional system and its eigenvector are also obtained. Their new applications are partly presented.     

奇偶对相干态的维格纳函数和层析图函数

利用纠缠态η〉表象下的维格纳算符,重构了奇偶对相干态的维格纳函数.根据维格纳函数在相空间中随变量ρ和γ的变化规律,讨论了奇偶对相干态的非经典性质和量子干涉效应.研究发现,奇偶对相干态总呈现非经典性质,并且当q取奇数时,奇偶对相干态更容易出现非经典性质.奇偶对相干态的量子干涉效应的显著程度与q取值有关,但对于q的同一取值,奇对相干态的量子干涉效应更为显著.利用纠缠态η〉表象下的维格纳算符Δ1,2(ρ,γ)和纠缠态η,τ1,τ2〉的投影算符之间满足的拉东变换,获得了奇偶对相干态的量子层析图函数.     

Atomic coherent states as energy eigenstates of a Hamiltonian describing a two-dimensional anisotropic harmonic potential in a uniform magnetic field

In this paper we find that a set of energy eigenstates of a two-dimensional anisotropic harmonic potential in a uniform magnetic field is classified as the atomic coherent states |τ> in terms of the spin values of j in the Schwinger bosonic realization. The correctness of the above conclusions can be verified by virtue of the entangled state <η| representation of the state |τ>.     

On the spectra of noncommutative 2D harmonic oscillator

The spectra and wave functions of the 2-dimensional harmonic oscillator in a noncommutative plane are revised by using the path integral formulation in coordinate space and momentum space, respectively. We perform the path integral formulation in coordinate space first. Then we study this problem in momentum space. The propagator is computed both in coordinate space and in momentum space. The modification due to noncommutativity of eigenvalues and eigenfunctions is studied. Both the small and large noncommutative parameter limits are discussed. PACS 11.10.Ef     

Are vortex numbers preserved?

We study noncommutative vortex solutions that minimize the action functional of the Abelian Higgs model in 2-dimensional noncommutative Euclidean space. We first consider vortex solutions which are deformed from solutions defined on commutative Euclidean space to the noncommutative one. We construct solutions whose vortex numbers are unchanged under the noncommutative deformation. Another class of noncommutative vortex solutions via a Fock space representation is also studied.     

New Approach for Calculating Tomograms of Quantum States by Virtue of the IWOP Technique

We show that for tomographic approach there exist two mutual conjugate quantum states [p,σ, τ） and [x,μ, ν） （named the intermediate coordinate-momentum representation）, and the two Radon transforms of the Wigner operator are just the pure-state density matrices [p）σ1τσ1τ （p｜ and （p）λ,ν,λ,ν,（x｜ respectively. As a result, the tomogram of quantum states can be considered as the module-square of the states＇ wave function in these two representations. Throughout the paper we fully employ the technique of integration within an ordered product of operators. In this way we establish a new convenient formalism of quantum tomogram.     

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.     

Unitary Transformation in kq Representation

In this paper, we discuss the unitary transformation induced by Z₆ rotations of noncommutative space on the states |k,q,s>, which plays a key role in construction for noncommutative solitons T²/Z₆ by GHS method. As a result, we prove a well-known ``Gauss Sum' formula in the number theory through a concise way.     

Noncommutative Geometrical Structures of Multi-Qubit Entangled States

We study the noncommutative geometrical structures of quantum entangled states. We show that the space of a pure entangled state is a noncommutative space. In particular we show that by rewriting the coordinate ring of a conifold or the Segre variety we can get a q-deformed relation in noncommutative geometry. We generalized our construction into a multi-qubit state. We also in detail discuss the noncommutative geometrical structure of a three-qubit state.     

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.     

Recapitulating Quantum Phase Space Representation by Virtue of Normally Ordered Gaussian Form of Wigner Operator

Using the normally ordered Gaussian form of the Wigner operator we recapitulate the quantum phase space representation, we derive a new formula for searching for the classical correspondence of quantum mechanical operators; we also show that if there exists the eigenvector ｜q〉λ,v of linear combination of the coordinate and momentum operator, （λQ ＋ vP）, where λ,v are real numbers, and ｜q〉λv is complete, then the projector ｜q〉λ,vλ,v〈q｜ must be the Radon transform of Wigner operator. This approach seems concise and physical appealing.     

Optical Tomography for Single- and Two-Mode Squeezed Chaotic Fields

Different from the previous methods of obtaining optical tomography of quantum states, in this paper we use the Radon transform between the single- (two-)mode Wigner operator and the pure-state density operator |q〉 _{f,g f,g} 〈q| (|η,τ ₁,τ ₂〉〈η,τ ₁,τ ₂|) to analytically obtain optical tomograms for single- (two-)mode squeezed chaotic fields, where the states |q〉 _f,g (|η,τ ₁,τ ₂〉) refer to the newly introduced intermediate coordinate-momentum (entangled) states. More interestingly, we find that the tomograms of squeezed chaotic fields and those of coherent states are related by a integration. Thus, we establish a new convenient approach of obtaining tomograms for quantum states.     

Parseval Theory of Complex Wavelet Transform for Wavelet Family Including Rotational Parameters

A rotational parameter R_θ has been introduced to complex wavelet transform (CWT). The rotational CWT (RCWT) corresponds to a matrix element 〈ψ|U₂ (θ;μ;k)|F〉in the context of quantum mechanics, where U₂(θ;μ;k) is a two-mode rotational displacing-squeezing operator in the 〈η| representation. Based on this, the Parseval theorem and the inversion formula of RCWT have been proved. The concise proof not only manifestly shows the merit of Dirac's representation theory but also leads to a new orthogonal property of complex mother wavelets in parameter space.     

Coordinate-Momentum Intermediate Representation and Marginal Distributions of Quantum Mechanical Bivariate Normal Distribution

We introduce bivariate normal distribution operator for state vector [ψ） and find that its marginal distribution leads to one-dimensional normal distribution corresponding to the measurement probability ｜λ,v〈x｜.ψ〉｜^2, where ｜x〉λ,v is the coordinate-momentum intermediate representation. As a by-product, the one-dimensional normal distribution in statistics can be explained as a Radon transform of two-dimensional Gaussian function.     

Lie algebra type noncommutative phase spaces are Hopf algebroids

For a noncommutative configuration space whose coordinate algebra is the universal enveloping algebra of a finite-dimensional Lie algebra, it is known how to introduce an extension playing the role of the corresponding noncommutative phase space, namely by adding the commuting deformed derivatives in a consistent and nontrivial way; therefore, obtaining certain deformed Heisenberg algebra. This algebra has been studied in physical contexts, mainly in the case of the kappa-Minkowski space-time. Here, we equip the entire phase space algebra with a coproduct, so that it becomes an instance of a completed variant of a Hopf algebroid over a noncommutative base, where the base is the enveloping algebra.