Exact solution to two-dimensional isotropic charged harmonic oscillator in uniform magnetic field in non-commutative phase space

In this paper,the isotropic charged harmonic oscillator in uniform magnetic field is researched in the non-commutative phase space;the corresponding exact energy is obtained,and the analytic eigenfunction is presented in terms of the confluent hypergeometric function.It is shown that in the non-commutative space,the isotropic charged harmonic oscillator in uniform magnetic field has the similar behaviors to the Landau problem.     

The non-commutative oscillator, symmetry and the Landau problem

The isotropic oscillator on a plane is discussed where the coordinate and momentum space are both considered to be non-commutative. We also discuss the symmetry properties of the oscillator for three separate cases when the non-commutative parameters Θ and for x and p-space, respectively, satisfy specific relations. We compare the Landau problem with the isotropic oscillator on non-commutative space and obtain a relation between the two non-commutative parameters and the magnetic field of the Landau problem.     

Constraints in Noncommutative Chern-Simons Mechanics and Hall Effect

We study noncommutative Chern-Simons mechanics and noncommutative Hall effect by Dirac theory in this paper. The magnetic field is introduced by means of minimal coupling. We show that the constraint set will enlarge when a dimensionless parameter takes zero value. In order to illustrate our idea, we study two specific models. One is noncommutative Chern-Simons mechanics which describes a charged particle on a noncommutative plane interacting with a perpendicular uniform magnetic field. The other is a charged particle on a noncommutative plane with a background uniform electromagnetic field. We show that when the dimensionless parameter tends to zero, the particle will live in a lower dimensional space in both models. Noncommutative Chern-Simons mechanics will reduce to a harmonic oscillator and the classical equations of motion of a charged particle in the background of a uniform electromagnetic field are governed by classical Hall law when the dimensionless parameter tends to zero.     

Time-dependent harmonic oscillator in a magnetic field and an electric field on the non-commutative plane

In the non-commutative space, wave functions and geometric phases are derived for the time-dependent harmonic oscillator in external time-dependent magnetic and electric field. Explicit forms of the coherent states are also given, which are not the minimum uncertainty states for the coordinates and momenta.     

Non-commutative harmonic oscillator in magnetic field and continuous limit

The spectra of a charged harmonic oscillator minimally coupled to a perpendicular magnetic field in the non-commutative plane are studied by using the path integral formulation. We get the spectra in a mapping-independent way. Interestingly, we find that the spectra have no continuous limit when the dimensionless parameter tends to zero. In order to get a finite result, a truncation is inevitable. Finally, we give a reasonable explanation of truncation from the constrained theory point of view.     

均匀磁场中二维各向同性带电谐振子的守恒量与对称性研究

由牛顿第二定律得到二维各向同性带电谐振子在均匀磁场中运动的运动微分方程,通过对运动微分方程的直接积分得到系统的两个积分(守恒量).利用Legendre变换建立守恒量与Lagrange函数间的关系,从而求得系统的Lagrange函数,并讨论与守恒量相应的无限小变换的Noether对称性与Lie对称性,最后求得系统的运动学方程.     

Decoherence of quantum Brownian motion in noncommutative space

The decoherence of a harmonic oscillator under two-dimensional quantum Brownian motion on a noncommutative plane is investigated. The interaction with the environment is considered by two separate models so-called coupled and uncoupled. The two-dimensional master equation and its noncommutative counterpart are derived for both employed models. The rate of the linear entropy (predictability sieve) is chosen as a criterion to investigate the purity in the presence of the space noncommutativity. Besides, a two-dimensional charged harmonic oscillator on a plane which is imposed by a perpendicular magnetic field is introduced as a realization of our model. Therefore, our approach provides a formalism to investigate the influence of the magnetic field on the decoherence of the pure states. We show that in the high magnetic field limit the rate of the decoherence will be decreased.     

Generalized Spiked Harmonic Oscillator in Non-commutative Space

In this paper non-commutative Schrodinger equation is considered for generalized Spiked harmonic oscillator potential. The energy shift due to non-commutativeity is obtained via the perturbation theory. Furthermore we show that the degeneracy of the initial spectral line is broken in transition from commutative space to non-commutative space.     

非对易空间中耦合谐振子的能级分裂

非对易空间效应的出现引起了物理学界的广泛兴趣。 介绍了非对易空间中量子力学的代数关系,在所考虑的空间变量的对易关系中包含了坐标 坐标的非对易性, 并且把 Moyal-Weyl 乘法在非对易空间中通过一个Bopp变换转变成普通的乘法。 然后给出了非对易空间中耦合谐振子的能级分裂情况。 The effect of noncommutativity of space have caused the physical academic circles widespread interest. In this paper, the non commutative (NC) is introduced, which contain non commutative of coordinate coordinate, and find that the Moyal Weyl product in NC space can be replaced with a Bopp shift. Then, the energy splitting of the coupling harmonic oscillator in non commutative spaces are discussed.     

Forced Time-Dependent Harmonic Oscillators in Non-Commutative Space

For the time-dependent harmonic oscillator and generalized harmonic oscillator with or without external forces in non-commutative space, wave functions, and geometric phases are derived using the Lewis-Riesenfeld invariant. Coherent states are obtained as the ground state of the forced system. Quantum fluctuations are calculated too. It is seen that geometric phases and quantum fluctuations are greatly affected by the non-commutativity of the space.     

Lie algebraic approach of a charged particle in presence of a constant magnetic field via the quadratic invariant

In this paper we consider the problem of a charged harmonic oscillator under the influence of a constant magnetic field. The system is assumed to be isotropic and the magnetic field is applied along the z-axis. The canonical transformation is invoked to remove the interaction term and the system is reduced to a model containing the second harmonic generation. Two classes of the real and complex quadratic invariants (constants of motion) are obtained. We have employed the Lie algebraic technique to find the most general solution for the wave function for both real and complex invariants. Some discussions related to the advantage of using the quadratic invariants to solve the Cauchy problem instead of the direct use of the Hamiltonian itself are also given.     

非对易相空间三模坐标动量耦合谐振子的能谱

本文用不变本征算符方法研究非对易相空间中三模坐标动量耦合谐振子的能谱,分别得到了非耦合和坐标动量耦合两种情况下谐振子能谱的解析解,其中包括受非对易参数θ和φ影响的解λ0,1和λ1,1,以及不受非对易参数θ和φ影响的解λ0,2和λ1,2.然后,分析了两类耦合参数κ和η对三模坐标动量耦合谐振子能谱的影响.结果发现,耦合参数κ和η对λ1,1的影响是相同的,且当κ=η时,耦合系数κ和η对λ1,1是没有影响的.     

Quantum-mechanical oscillator in a uniform force field

The symmetry group of an isotropic oscillator explaining additional degeneracy of its energy levels was for the first time considered by Demkov [1–3]. He directed attention to the fact that harmonic oscillator Hamiltonian symmetry was beyond the scope of usual geometric concepts and could be expressed using canonical transformations relating coordinates and momenta. We show in this paper how the symmetry properties of the harmonic oscillator Hamiltonian can be used to solve the dynamic problem of an oscillator in a uniform force field.     

Propagator for the anisotropic three-dimensional charged harmonic oscillator in a constant magnetic field using the Schwinger action principle

The propagator for the anisotropic three-dimensional charged harmonic oscillator in the presence of a constant external magnetic field is calculated using the Schwinger action principle.     

外加磁场下抛物型量子线中的带电激子

在一维等效模型下采用有效差分法对抛物型量子阱线中带电激子的束缚能进行了计算,分析了约束势以及磁场对带电激子束缚能的影响,并对带正电激子(X⁺)和带负电激子(X^-)的情况进行了比较.结果表明:电子和空穴的振子强度对带电激子的稳定性有重要影响,X⁺的束缚能不总是比X^-的大,随着空穴振子强度的增加束缚能的函数曲线将会出现交叉,这同实验得到的结果符合;磁场的存在会增加粒子间的束缚,并且磁场对束缚能的影响同振子强度大小有关. 关键词： 带电激子 量子线 束缚能 磁场     

Classical mechanics in non-commutative phase space

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space.The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity.First,new Poisson brackets have been defined in non-commutative phase space.They contain corrections due to the non-commutativity of coordinates and momenta.On the basis of this new Poisson brackets,a new modified second law of Newton has been obtained.For two cases,the free particle and the harmonic oscillator,the equations of motion are derived on basis of the modified second law of Newton and the linear transformation (Phys.Rev.D,2005,72:025010).The consistency between both methods is demonstrated.It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space.but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.     

Classical mechanics in non-commutative phase space

In this paper the laws of motion of classical particles have been investigated in a non-commutative phase space. The corresponding non-commutative relations contain not only spatial non-commutativity but also momentum non-commutativity. First, new Poisson brackets have been defined in non-commutative phase space. They contain corrections due to the non-commutativity of coordinates and momenta. On the basis of this new Poisson brackets, a new modified second law of Newton has been obtained. For two cases, the free particle and the harmonic oscillator, the equations of motion are derived on basis of the modified second law of Newton and the linear transformation (Phys. Rev. D, 2005, 72: 025010). The consistency between both methods is demonstrated. It is shown that a free particle in commutative space is not a free particle with zero-acceleration in the non-commutative phase space, but it remains a free particle with zero-acceleration in non-commutative space if only the coordinates are non-commutative.     

Generation of squeezed states in a homogeneous magnetic field

It is shown that a homogeneous magnetic field acting on a charged harmonic oscillator in the ground state may produce a state with reduced fluctuation of one of the quadrature phase amplitudes. The amount of squeezing in the final state depends on the time during which the magnetic field was switched on.     

双外场对费米气体Joule-Thomson效应的调控简

研究匀强电场和谐振场构成的双外场对带电费米粒子系统化学势和Joule-Thomson系数的调控作用.在满足Thomas-Fermi近似的条件下,给出有限温度范围内匀强电场和谐振场中带电费米粒子系统的化学势和内能随温度变化的显式函数关系,分析双外场对带电费米粒子系统焦耳-汤姆逊系数的调控作用.研究结果表明,随着温度的升高,带电费米粒子系统的焦汤系数值逐渐由负变正,且临界温度Tc随外电场增强而降低.在适当温度下,外势场的存在会使带电费米粒子系统产生焦耳-汤姆逊正效应.     

双外场对费米气体Joule-Thomson效应的调控

研究匀强电场和谐振场构成的双外场对带电费米粒子系统化学势和Joule-Thomson系数的调控作用．在满足Thomas-Fermi近似的条件下,给出有限温度范围内匀强电场和谐振场中带电费米粒子系统的化学势和内能随温度变化的显式函数关系,分析双外场对带电费米粒子系统焦耳-汤姆逊系数的调控作用.研究结果表明：随着温度的升高,带电费米粒子系统的焦汤系数值逐渐由负变正,且临界温度Tc随外电场增强而降低;在适当温度下,外势场的存在会使带电费米粒子系统产生焦耳-汤姆逊正效应.