矩形弹子球中的量子谱和经典周期轨道

利用SU(2)相干态的表示,我们构造了二维矩形弹子球中与经典周期轨道对应的波函数.经典周期轨道和量子波函数之间的关系可以通过物理图像清晰的表示出来.另外,利用周期轨道理论,我们计算了二维矩形弹子球体系的量子谱的傅立叶变换ρ(L).变换谱|ρN(L)|2对L图像中的峰可以和粒子在二维矩形腔中运动的经典轨迹的长度相比较.量子谱中的每一条峰正好对应一条经典周期轨道的长度,表明量子力学和经典力学的对应关系.     

谐振子系统的量子-经典轨道、Berry相及Hannay角

从量子-经典轨道和几何相两方面, 研究了二维旋转平移谐振子系统的量子-经典对应. 通过广义规范变换得到了Lissajous经典周期轨道和Hannay角. 另外, 使用含时规范变换解析推导了旋转平移谐振子系统Schrödinger方程的本征波函数和Berry相, 得出结论: 原规范中的非绝热Berry相是经典Hannay角的-n倍. 最后, 使用SU(2)自旋相干态叠加, 构造一稳态波函数, 其波函数的概率云很好地局域于经典轨道上, 满足几何相位和经典轨道同时对应.     

等腰直角三角形的二维量子谱和经典轨道

本文研究了等腰直角三角形中基于其波函数和能级结构的二维量子谱．虽然这个量子台球系统的本征态是无法分离的,但是关于两个变量的问题是完全可解的．通过对二维量子系统的波函数做相应的傅里叶变换,得到了系统的二维量子谱,把得到的结果和经典的二维量子台球轨道做相应性的对比发现：傅里叶变换的量子谱的峰值位置和经典轨道的长度之间存在着很好的对应关系,这说明经典计算的结果和量子计算的结果符合得非常好,从而进一步验证了周期轨道理论的正确性．     

二维高斯波束对多层球粒子电磁散射的解析解

基于矢量波函数在球和柱坐标系中表达式之间的转换关系,提出了一种求解球坐标系中二维高斯波束波形因子的方法,得到了二维高斯波束波形因子在球坐标系中的解析公式.结合广义米理论推导了在轴二维高斯波束入射多层球粒子的电磁散射的解析解,并对散射强度随散射角的分布进行了数值模拟,结果与平面波入射情况进行了比较. 关键词： 矢量波函数 波形因子 电磁散射 广义米理论     

单核子在具有八极形变的平均势场中的混沌运动(I)相干态在势场中传播的理论表示

单核子在具有八极以上形变的平均势场中发现有混沌运动.通过研究系统非定态波函数的时间演化特征,而使研究量子混沌与研究经典混沌的思路更趋一致,特别是能体现量子状态对初始条件的敏感性.给出了当初态选为二维不对称谐振子动力学对称条件下的、满足坐标动量最小测不准关系的相干态时,八极形变耦合作用下量子状态正则变量的期望值及测不准度随时间演化的理论表示.     

一维抛物势场中单电子量子比特的性质

根据-维抛物势场中单电子能量本征波函数的特点,以基态和第二激发态波函数构造一个量子比特,并研究量子比特的性质.数值计算表明,量子比特内电子的概率密度随空间坐标和时间的变化而变化,在平衡位置概率密度幅值最大,在其他位置相对较小;且各个空间点的概率密度均随时间做周期性振荡,振荡周期完全由抛物势场的性质决定.     

从量子谱到经典轨道：矩形腔中的弹子球

用体系的本征值和本征波函数定义一种新的量子谱函数．这种量子谱函数的傅里叶变换包含了体系从一个给定点到另一个给定点的许多经典轨道的信息．以二维矩形腔中的弹子球运动体系为例的初步研究验证了这一结论．　　 关键词： 经典量子对应 半经典物理     

复杂势场量子弹球中疤痕态的量子化条件

量子疤痕是波函数在经典不稳定周期轨道周围反常凝聚的一种量子或波动现象.人们对疤痕态的量子化条件进行了大量研究,对深入理解半经典量子化起到了一定的促进作用.之前大部分研究工作主要集中在硬墙量子弹球上,即给定边界形状的无穷深量子势阱系统.本文研究具有光滑复杂势场的二维量子弹球系统,考察疤痕态的量子化条件及其重复出现的规律,得到了与硬墙弹球不一样的结果,对理解这类现象是一个有益的补充.这些结果将有助于理解具有无规长程杂质分布的二维电子系统的态密度谱和输运行为.     

二维氢原子的双波描述

运用双波量子理论,给出了二维氢原子的双波函数描述,并讨论了其经典极限.结果表明,双波函数描述单个粒子的运动状态,并将通常的量子力学描述结果作为系综统计平均值包含在其中. 关键词：     

电子关联与二维晶格的不稳定性

本文从电子相互作用的屏蔽库仑势出发,构造了Wannier函数来计算二维不稳定晶格的能带及电子波函数,用相关基函数理论计算了二维非均匀体系的电子关联函数,由此研究了电子关联对二维晶格不稳定性的影响。结果表明,二维体系与一维体系不同,电子相互作用使晶格二聚化减弱。 关键词：     

Exact microscopic wave function for a topological quantum membrane

The higher dimensional quantum Hall liquid constructed recently supports stable topological membrane excitations. Here we introduce a microscopic interacting Hamiltonian and present its exact ground state wave function. We show that this microscopic ground state wave function describes a topological quantum membrane. We also construct variational wave functions for excited states using the noncommutative algebra on the four sphere. Our approach introduces a nonperturbative method to quantize topological membranes.     

Chirality and Dirac operator on noncommutative sphere

We give a derivation of the Dirac operator on the noncommutative 2-sphere within the framework of the bosonic fuzzy sphere and define Connes' triple. It turns out that there are two different types of spectra of the Dirac operator and correspondingly there are two classes of quantized algebras. As a result we obtain a new restriction on the Planck constant in Berezin's quantization. The map to the local frame in noncommutative geometry is also discussed.     

Competing topological orders in the ν=12/5 quantum Hall state

We provide numerical evidence that a p(x)-ip(y) paired Bonderson-Slingerland (BS) non-Abelian hierarchy state is a strong candidate for the observed ν=12/5 quantum Hall plateau. We confirm the existence of a gapped incompressible ν=12/5 quantum Hall state with shift S=2 on the sphere, matching that of the BS state. The exact ground state of the Coulomb interaction at S=2 is shown to have a large overlap with the BS trial wave function. Larger overlaps are obtained with BS-type wave functions that are hierarchical descendants of general p(x)-ip(y) weakly paired states at ν=5/2. We perform a finite-size scaling analysis of the ground-state energies for ν=12/5 states at shifts corresponding to the BS (S=2) and 3-clustered Read-Rezayi (S=-2) universality classes. This analysis reveals very tight competition between these two non-Abelian topological orders.     

The fuzzy supersphere

We introduce the fuzzy supersphere as sequence of finite-dimensional, noncommutative ₂-graded algebras tending in a suitable limit to a dense subalgebra of the ₂-graded algebra of ^∞-functions on the (2|2)-dimensional supersphere. Noncommutative analogues of the body map (to the (fuzzy) sphere) and the super-deRham complex are introduced. In particular we reproduce the equality of the super-deRham cohomology of the supersphere and the ordinary deRham cohomology of its body on the “fuzzy level”.     

A perturbative approach to fuzzifying field theories

We propose a procedure for computing noncommutative corrections to the metric tensor, and apply it to scalar field theory written on coordinate patches of smooth manifolds. The procedure involves finding maps to the noncommutative plane where differentiation and integration are easily defined, and introducing a star product. There are star product independent, as well as dependent, corrections. Applying the procedure for two different star products, we find the lowest order fuzzy corrections to scalar field theory on a sphere which is stereographically projected to the plane.     

Charged Particles in Monopole Background on Fuzzy Sphere

In this note we study the quantum mechanics of a charged particle on fuzzy sphere and in the presence of magnetic monopoles. We discuss the proper inclusion of the electromagnetic interaction in the Hamiltonian through the covariant form of the momentum operator. We consider two different kinds of monopoles. The first one is associated with projective modules and obtained from the corresponding projector. The second one we obtain by solving directly the noncommutative Maxwell equations over the fuzzy sphere. Among these, are the monopole connections for which the Hamiltonian operator can be diagonalized in an algebraic way.      

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.     

Extracting excitations from model state entanglement

We extend the concept of an entanglement spectrum from the geometrical to the particle bipartite partition. We apply this to several fractional quantum Hall wave functions on both sphere and torus geometries to show that this new type of entanglement spectra completely reveals the physics of bulk quasihole excitations. While this is easily understood when a local Hamiltonian for the model state exists, we show that the quasihole wave functions are encoded within the model state even when such a Hamiltonian is not known. As a nontrivial example, we look at Jain's composite fermion states and obtain their quasiholes directly from the model state wave function. We reach similar conclusions for wave functions described by Jack polynomials.     

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.     

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 θ^-, respectively. In our calculation, we assume that these parameters vary from laver to laver.