The generalized nonlinear Schrödinger hierarchy, its integrable coupling system and their bi-Hamiltonian structure

Based on a subalgebra G of Lie algebra A₂, a new Lie algebra G^∗ is constructed. By making use of the Tu scheme, the generalized nonlinear Schrödinger hierarchy and its integrable coupling are both obtained with the help of their corresponding special loop algebras. At last, by means of the quadratic-form identity, their bi-Hamiltonian structures of the generalized nonlinear Schrödinger hierarchy and its integrable coupling system are worked out respectively. The approach presented in this Letter can be used in other integrable hierarchies.     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’     

Aharonov-Bohm effect on AdS2 and nonlinear supersymmetry of reflectionless Pöschl-Teller system

We explain the origin and the nature of a special nonlinear supersymmetry of a reflectionless Pöschl-Teller system by the Aharonov-Bohm effect for a non-relativistic particle on the AdS₂. A key role in the supersymmetric structure appearing after reduction by a compact generator of the AdS₂ isometry is shown to be played by the discrete symmetries related to the space and time reflections in the ambient Minkowski space. We also observe that a correspondence between the two quantum non-relativistic systems is somewhat of the AdS/CFT holography nature.     

Unstable trajectories and the quantum mechanical uncertainty

There is still an ongoing discussion about various seemingly contradictory aspects of classical particle motion and its quantum mechanical counterpart. One of the best accepted viewpoints that intend to bridge the gap is the so-called Copenhagen Interpretation. A major issue there is to regard wave functions as probability amplitudes (usually for the position of a particle). However, the literature also reports on approaches that claim a trajectory for any quantum mechanical particle, Bohmian mechanics probably being the most prominent one among these ideas. We introduce a way to calculate trajectories as well, but our crucial ingredient is their well controlled local (thus also momentaneous) degree of instability. By construction, at every moment their unpredictability, i.e., their local separation rates of neighboring trajectories, is governed by the local value of the given modulus square of a wave function. We present extensive numerical simulations of the H and He atom, and for some velocity-related quantities, namely angular momentum and total energy, we inspect their agreement with the values appearing in wave mechanics. Further, we interpret the archetypal double slit interference experiment in the spirit of our findings. We also discuss many-particle problems far beyond He, which guides us to a variety of possible applications.     

The double slit experiment,the Aharonov-Bohm effect,and a new quantum potential theory: Towards a realistic interpretation of quantum mechanics

In this paper, a new proposal of the quantum potential theory is presented, along with a discussion of the two-slit interference experiment and the Aharonov-Bohm effect, which are explicitly calculated within the scheme adopted. The present formulation holds the notion of particle trajectory and follows the interpretation suggested by de Broglie and Bohm, the quantum potential being a manifestation of the quantum wave. This wave always satisfies the Schrödinger equation and is considered as areal and separate entity from the particle with which is associated. In our approach we try to circumvent some conceptual difficulties that prevent the de Broglie-Bohm's theory from fulfilling a complete objective program. New effects are predicted, specially for the Aharonov-Bohm experiment in the regime of high magnetic fields, and a setup for their detection is proposed.     

Connecting the quantum and classical worlds

By considering (non‐relativistic) quantum mechanics as it is done in practice in particular in condensed‐matter physics, it is argued that a deterministic, unitary time evolution within a chosen Hilbert space always has a limited scope, leaving a lot of room for embedding the quantum‐classical transition into our current theories without recurring to difficult‐to‐accept interpretations of quantum mechanics. Nonunitary projections to initial and final states, the breaking of time‐reversal symmetry, a change of Hilbert space, and the introduction of classical concepts such as external potentials or localized atomic nuclei are widespread in quantum mechanical calculations. Furthermore, quantum systems require classical environments that enable the symmetry breaking that is necessary for creating the atomic configurations of molecules and crystals. This paper argues that such classical environments are provided by finite‐temperature macroscopic systems in which the range of quantum correlations and entanglement is limited. This leads to classical behavior on larger scales, and to collapse‐like events in all dynamical processes that become coupled to the thermalized degrees of freedom.

     

The linear and nonlinear intersubband optical absorption coefficients and refractive index changes in a V-shaped quantum well under the applied electric and magnetic fields

In this study, the changes in the optical absorption coefficients and the refractive index in a V-shaped quantum well have investigated theoretically. Within the effective mass approximation, the electronic structure of the V-shaped quantum well is calculated by numerical methods from the Schrödinger equation. Optical properties are obtained using the compact density-matrix approach. In the present work, the linear, third-order nonlinear and total absorption and refractive index changes investigated as a function of the quantum well width, the incident optical intensity, strengths of the magnetic and electric fields. Our results show that the magnetic and electric fields strengths, the quantum well width and incident optical intensity have a great effect on the optical characteristics of these structures.     

Effect of magnetic fields on the linear and nonlinear intersubband optical absorption coefficients and refractive index changes in square and graded quantum wells

In this study, both the intersubband optical absorption coefficients and the refractive index changes as dependent on the magnetic field are calculated in square and graded quantum wells. Our results show that the position and the magnitude of the linear and total absorption coefficients and refractive index changes depend on the magnetic field strength and the shape of potential. The incident optical intensity has a great effect on the total absorption and refractive index changes.     

两自由度弱非线性耦合系统的一阶近似Lie对称性与近似守恒量

采用变劲度系数的耦合弹簧构建一实际的两自由度弱非线性耦合系统, 用近似Lie对称性理论研究系统的一阶近似Lie对称性与近似守恒量, 得到6个一阶近似Lie对称性和一阶近似守恒量, 其中1个一阶近似守恒量实为系统的精确守恒量, 4个一阶近似守恒量为平凡的一阶近似守恒量, 只有1个一阶近似守恒量为稳定的一阶近似守恒量. 关键词： 两自由度弱非线性耦合系统 近似Lie对称性 近似守恒量     

创立量子力学的睿智才思(续2)——纪念矩阵力学和波动力学诞生80～81周年

1922年前后玻尔-索末菲的旧量子论在原子结构理论上取得辉煌成就的同时，它的致命弱点也开始暴露出来．反常塞曼效应是其一，原子光谱的多重线结构是其二，更不要说氢分子离子H2^＋问题了．一时多少杰出的物理学家为这些问题绞尽脑汁，设计了各种物理模型，提出一些“代用（德文Er-satz）”理论，诸如海森伯和泡利的Zwang之类，到头来还只得愁眉以对，一筹莫展．无怪乎海森伯把反常塞曼效应叫做“光谱项的动物学和塞曼植物学（Term Zoology and Zeeman Botany）”，意即在这个领域里物理学家只能像生物学家那样记录现象和描绘事实，提不出像样的理论．到1925年下半年，人们手里有了矩阵力学，问题是否可以解决了呢？万事俱备，只欠“东风”，那就是“自旋”．     

半圆和四分之一圆二维弹子球的量子谱分析

采用近来提出的量子谱函数,我们把闭合轨道理论应用到半圆和四分之一圆弹子球系统,这种量子谱函数的傅利叶变换包含了连接任意两点的许多经典轨道的信息.计算表明量子谱的傅立叶变换和经典轨道的长度符合的很好.从这两个体系可以看出半经典理论为经典和量子力学提供了很好的桥梁作用.     

The structure of the quantum mechanical state space and induced superselection rules

The role of superselection rules for the derivation of classical probability within quantum mechanics is investigated and examples of superselection rules induced by the environment are discussed     

Revisiting novel symmetries in coupled ${ mathcal N }=2$ supersymmetric quantum systems: examples and supervariable approach

We revisit the novel symmetries in ${ mathcal N }=2$ supersymmetric quantum mechanical models by considering specific examples of coupled systems. Further, we extend our analysis to a general case and list out all the novel symmetries. In each case, we show the existence of two sets of discrete symmetries that correspond to the Hodge duality operator of differential geometry. Thus, we are able to provide a proof of the conjecture which points out the existence of more than one set of discrete symmetry transformations corresponding to the Hodge duality operator. Moreover, we derive on-shell nilpotent symmetries for a generalized superpotential within the framework of supervariable approach.     

Laplace-Beltrami operator on symmetric space X ≡ SOo(2, 3)/SO(2) × SO(3) of rank 2 and the related 2-body one dimensional quantum systems

The explicit form of the Laplace-Beltrami operator Δ_LB on symmetric space X  SO₀(2,3)/SO(2) xSO(3) of rank 2 is given. Using the representation of SO₀(2,3) the eigenfunctions of δ_LB, i.e. plane waves on X are constructed. The integral representation of the zonal spherical functions of the group SO_o(2, 3) is presented and the related 2-body one-dimensional quantum systems are considered.     

负值量子条件熵与双量子系统一类混合态纠缠量度

运用负值量子条件熵研究了双量子系统一类混合态的纠缠量度.给出了负值量子条件作为条件熵纠缠度的定义，证明了条件熵纠缠满足作为2×2系统一类混合纠缠态量度的四个基本条件.当双量子系统处于纯态时，条件熵纠缠度即为部分熵纠缠度.应用条件熵纠缠度研究了真空腔场中两全同二能级原子之间纯态和一类混合态纠缠的时间演化，比较了相同条件下两全同原子系统concurrence纠缠度的时间演化.结果表明，两纠缠度演化规律完全一致，验证了负值量子条件熵可以作为双量子系统纯态和一类混合态的纠缠量度. 关键词： 双量子系统 负值量子条件熵 条件熵纠缠度 混合态纠缠度     

Wetting layer and size effects on the nonlinear optical properties of semi oblate and prolate Si0.7Ge0.3/Si quantum dots

Semi oblate and semi prolate are among the most probable self-organized nanostructures shapes. The optoelectronic properties of such nanostructures are not just manipulated with the height and lateral size but also with the wetting layer element. The practical interest of derivatives of germanium and silicon has a great important role in optoelectronic devices. This study is a contribution to the analysis of linear and nonlinear optical properties of Si_0.7Ge_0.3/Si. In the framework of the effective mass approximation, we solve numerically the Schrödinger equation relative to one particle confined in Si_0.7Ge_0.3/Si semi prolate and semi oblate quantum dots by using the finite element method and by taking into consideration the effect of the wetting layer. The energy spectrum of the lowest states and the dipolar matrix for the fourth allowed transitions are determined and discussed. We also calculate the detailed optical properties, including absorption coefficients, refractive index changes, second and third harmonic generation as a function of the quantum dot sizes. We found that with the change in the size of prolate and oblate quantum dots, there is a shift in the resonance peaks for the absorption coefficient and refractive index. It is due to the modification in the energy levels with changing size. The study proves a redshift in the second harmonic generation and third harmonic generation coefficients with an increase in the height/radius of the oblate/prolate quantum dot, respectively. We also demonstrated the variation of wavefunction inside the quantum dot with the change in wetting layer thickness.     

The affine Weyl group symmetry of Desargues maps and of the non-commutative Hirota-Miwa system

We study recently introduced Desargues maps, which provide simple geometric interpretation of the non-commutative Hirota-Miwa system. We characterize them as maps of the A-type root lattice into a projective space such that images of vertices of any basic regular N-simplex are collinear. Such a characterization is manifestly invariant with respect to the corresponding affine Weyl group action, which leads to related symmetries of the Hirota-Miwa system.     

极化激元系统时间演化的量子涨落特性和非经典统计行为

将极化激元系统约化成模型单模光子-TO声子有效相互作用系统， 在此基础上以解析形式讨论了系统的力学量、压缩态、量子涨落特性以及亚泊松分布等非经典效应的动力学演化行为.结果表明，光子场与极化波量子场彼此交换能量过程随时间演化呈振荡性质，光子场和声子场都可以演化成压缩态，其二阶压缩度随时间演化成复杂周期振荡特性，这种非经典特性是非线性相互作用的结果并且以k₁项和k₂项同时存在并相互关联为前提.而此时光子和声子统计分布随时间演化呈现介于超Poison分布和亚Poison分布之间复杂周期振荡的新结果，非线性作用k₁项和k₂项对这种非经典统计行为都有贡献. 关键词： 极化激元系统动力学演化 单模光场-TO声子有效模型哈密顿量 量子涨落与压缩态 亚泊松分布