Landau problem in noncommutative quantum mechanics

The Landau problem in non-commutative quantum mechanics (NCQM) is studied. First by solving the Schrodinger equations on noncommutative (NC) space we obtain the Landau energy levels and the energy correction that is caused by space-space noncommutativity. Then we discuss the noncommutative phase space case, namely, space-space and momentum-momentum non-commutative case, and we get the explicit expression of the Hamiltonian as well as the corresponding eigenfunctions and eigenvalues.     

Heisenberg algebra for noncommutative Landau problem

The Landau problem on non-commutative quantum mechanics is studied, where the Heisenberg algebra and the Landau energy levels as well as the non-commutative angular momentum are constructed in detail in non-commutative space and non-commutative phase space respectively.     

Landau analog levels for dipoles in non-commutative space and phase space

We investigate the analog of Landau quantization, for a neutral polarized particle in the presence of homogeneous electric and magnetic external fields, in the context of non-commutative quantum mechanics. This particle, possessing electric and magnetic dipole moments, interacts with the fields via the Aharonov–Casher and He–McKellar–Wilkens effects. For this model we obtain the Landau energy spectrum and the radial eigenfunctions of the non-commutative space coordinates and non-commutative phase space coordinates. Also we show that the case of non-commutative phase space can be treated as a special case of the usual non-commutative space coordinates.     

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].     

非对易相空间中的狄拉克振子的能级和Wigner函数

非对易几何、弦论和圈量子引力理论的发展,使非对易空间受到越来越多的关注.非对易量子理论不同于平常的量子理论,它是弦尺度下的特殊的物理效应,处理非对易量子力学问题需要特殊方法.本文首先介绍了Moyal方程与Wigner函数,利用Moyal-Weyl乘法与Bopp变换将H(x,p)变换成^H(^x,^p),考虑坐标—坐标、动量—动量的非对易性,实现对非对易相空间中星乘本征方程的求解.并利用非对易相空间量子力学的代数关系,讨论了非对易相空间中狄拉克振子的Wigner函数和能级,研究结果发现非对易相空间中狄拉克振子的能级明显依赖于非对易参数.     

Klein-Gordon oscillators in noncommutative phase space

We study the Klein-Gordon oscillators in non-commutative (NC) phase space.We find that the Klein-Gordon oscillators in NC space and NC phase-space have a similar behaviour to the dynamics of a particle in commutative space moving in a uniform magnetic field.By solving the Klein-Gordon equation in NC phase space,we obtain the energy levels of the Klein-Gordon oscillators,where the additional terms related to the space-space and momentum-momentum non-commutativity are given explicitly.     

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.     

Klein-Gordon oscillators in noncommutative phase space

We study the Klein-Gordon oscillators in non-commutative (NC) phase space. We find that the Klein-Gordon oscillators in NC space and NC phase-space have a similar behaviour to the dynamics of a particle in commutative space moving in a uniform magnetic field. By solving the Klein-Gordon equation in NC phase space, we obtain the energy levels of the Klein-Gordon oscillators, where the additional terms related to the space-space and momentum-momentum non-commutativity are given explicitly.     

On atomic analogue of Landau quantization

We have studied the physics of atoms with permanent electric dipole moment and nonvanishing magnetic moment interacting with an electric field and inhomogeneous magnetic field. This system can be demonstrated as the atomic analogue of Landau quantization of charged particles in a uniform magnetic field. This Landau-like atomic problem is also studied with space-space noncommutative coordinates.     

Wigner Function for Klein--Gordon Landau Problem

First we calculate the Wigner phase-space distribution function for the Klein-Gordan Landau problem on a commmutative space. Then we study the modifications introduced by the coordinate-coordinate noncommuting and momentum-momentum noncommuting, namely, by using a generalized Bopp's shift method we construct the Wigner function for the Klein-Gordan Landau problem both on a noncommutative space (NCS) and a noncommutative phase space (NCPS).     

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.     

On non-commutative geodesic motion

In this work we study the geodesic motion on a noncommutative space–time. As a result we find a non-commutative geodesic equation and then we derive corrections of the deviation angle per revolution in terms of the non-commutative parameter when we specify the problem of Mercury’s perihelion. In this way, we estimate the noncommutative parameter based in experimental data.     

二维模糊球上的量子霍尔流体

在回顾了Haldane对量子Hall效应在二维球面S2上的描述后,本文构造了二维模糊球S2上的非对易代数及其Hilbert空间的Moyal结构.通过构造模糊球上不可压缩量子霍尔流体的非对易Chern-Simons理论,求解具有准粒子源的Gaussian约束,找出模糊球上的Calogero矩阵及最低Landau能级Laughlin波函数的完全集,此Laughlin波函数由旋量坐标推广的Jack多项式表示.     

Quantum Field Theory with a Minimal Length Induced from Noncommutative Space

From the inspection of noncommutative quantum mechanics, we obtain an approximate equivalent relation for the energy dependence of the Planck constant in the noncommutative space, which means a minimal length of the space. We find that this relation is reasonable and it can inherit the main properties of the noncommutative space. Based on this relation, we derive the modified Klein-Gordon equation and Dirac equation. We investigate the scalar field and φ4 model and then quantum electrodynamics in our theory, and derive the corresponding Feynman rules. These results may be considered as reasonable approximations to those of noncommutative quantum field theory. Our theory also shows a connection between the space with a minimal length and the noncommutative space.     

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.     

Thermodynamics of a Charged Particle in a Noncommutative Plane in a Background Magnetic Field

Landau system in noncommutative space has been considered. To take into account the issue of gauge invariance in noncommutative space, we incorporate the Seiberg-Witten map in our analysis. Generalised Bopp-shift transformation is then used to map the noncommutative system to its commutative equivalent system. In particular we have computed the partition function of the system and from this we obtained the susceptibility of the Landau system and found that the result gets modified by the spatial noncommutative parameter θ. We also investigate the de Hass–van Alphen effect in noncommutative space and observe that the oscillation of the magnetization and the susceptibility gets noncommutative corrections. Interestingly, the susceptibility in the noncommutative scenario is non-zero in the range of the magnetic field greater than the threshold value which is in contrast to its commutative counterpart. The results obtained are valid upto all orders in the noncommutative parameter θ.     

Noncommutative Quantum Hall Effect of Bilayer Systems

In this paper we study the bilayer quantum Hall (QH) effect on a noncommutative phase space (NCPS). By using perturbation theory, we calculate the energy spectrum, eigenfunction, Hall current, and Hall conductivity of the bilayer QH system, and express them in terms of noncommutative parameters θ and \bar{θ}, respectively. In our calculation, we assume that these parameters vary from layer to layer.     

Quantum dot with spin-orbit interaction in noncommutative phase space and analog Landau levels

We have studied a quantum dot with Rashba spin-orbit interaction in noncommutative phase space. The energy eigenvalues are analogous to Landau energy levels. It is shown that this system is related with a physically realizable model of a quantum dot with Rashba spin-orbit interaction in a magnetic field whereby a relation is derived among the noncommutative parameters, spin-orbit coupling strength and magnetic field.     

非对易相空间Dirac oscillator的研究

在极坐标系中研究了非对易相空间中的Dirac oscillator问题.研究显示:系统的波函数可以表示为合流超几何函数,而非对易相空间Dirac oscillator的量子行为类似于朗道问题.最后,对η=0和对易极限两种特殊情况进行了简单讨论.     

Dirac Oscillator in a Galilean Covariant Non-commutative Space

We study the Galilean Dirac oscillator in a non-commutative situation, with space-space and momentum-momentum non-commutativity. The wave equation is obtained via a ‘Galilean covariant’ approach, which consists in projecting the covariant equations from a (4,1)-dimensional manifold with light-cone coordinates, to a (3,1)-dimensional Galilean space-time. We obtain the exact wave functions and their energy levels for the plane and discuss the effects of non-commutativity.