大约1935年3月：Schrödinger猫佯谬

奥地利理论物理学家Erwin Schroe-dinger提出的波动方程式对量子论的贡献深远,此方程式后来以他的名字命名。但很意外的是,他最有名的研究却是1935年所提出的一个思考性实验——Schroedinger猫的吊诡,此实验引起哲学家的兴趣,但也让爱猫人士感到毛骨悚然。     

薛定格猫态佯谬的双波解答

粒子在不可穿透器壁前的入射和反射运动，在动能大时为薛定格猫态，经典力学和双波量子力学对这体系的找述为完全描述，物理上下引起任何问题，普通量子测量假设引出佯谬，因为一个波函数在大能量条件下不描述单个粒子而描述系统，佯谬的根源是量子力学解释中不正确的前提和假设。     

相对论性Schrödinger方程的Green函数

使用量子场论中的Schrödinger波函数描述2-费密子系统. 导出了时间相关Green函数及其谱表示. 得到了Schrödinger方程与波函数的归一化条件, 它表明当位势的能量相关性可忽略时, 这种波函数就是几率振幅. 给出了另外几种形式的等时方程的Green函数, 这几种方程仍然有归一化的问题, 它们的位势是非厄密的.     

The generalized Schrödinger–Langevin equation

In this work, for a Brownian particle interacting with a heat bath, we derive a generalization of the so-called Schrödinger–Langevin or Kostin equation. This generalization is based on a nonlinear interaction model providing a state-dependent dissipation process exhibiting multiplicative noise. Two straightforward applications to the measurement process are then analyzed, continuous and weak measurements in terms of the quantum Bohmian trajectory formalism. Finally, it is also shown that the generalized uncertainty principle, which appears in some approaches to quantum gravity, can be expressed in terms of this generalized equation.     

非线性Schrödinger方程组的精确解组

运用Maple语言程序,在没有假设的条件下,得到了具有耦合特性的非线性Schr(o)dinger方程组的行波精确解组及其约束条件方程,它们的表达式涵盖了所有的耦合解组与非耦合解组,具有任意性.耦合解组的算例函数及其特性分析,解释了a螺旋蛋白质螺旋链运动模型的行波孤立子解的耦合效应,揭示了增加、稳定和控制蛋白质活性和功能的方向.文章的研究方法,为求解耦合的非线性微分方程组的行波精确解组探索了蹊径.     

薛定谔猫佯谬与重离子耗散反应激发函数的测量

 量子力学的建立与相对论的提出,是20世纪物理学最伟大的成就,它们构成了现代科学的理论基础,也是当代人类物质文明的理论基础。有趣的是,一方面量子理论已经为无数的实验事实所证实,到目前为止甚至还找不到一个与它的预言相悖的实际事例;另一方面,量子力学的基本概念又与人们对现实宏观世界的传统认知极不协调。这种不协调实质上是对量子理论的根基的质疑。所以,从量子力学诞生前后一直到现在的近百年间,对量子理论的争议一天也没有停止过。其中,最著名的议题之一就是所谓的薛定谔猫佯谬。     

The Schrödinger–Langevin equation with and without thermal fluctuations

The Schrödinger–Langevin equation (SLE) is considered as an effective open quantum system formalism suitable for phenomenological applications involving a quantum subsystem interacting with a thermal bath. We focus on two open issues relative to its solutions: the stationarity of the excited states of the non-interacting subsystem when one considers the dissipation only and the thermal relaxation toward asymptotic distributions with the additional stochastic term. We first show that a proper application of the Madelung/polar transformation of the wave function leads to a non zero damping of the excited states of the quantum subsystem. We then study analytically and numerically the SLE ability to bring a quantum subsystem to the thermal equilibrium of statistical mechanics. To do so, concepts about statistical mixed states and quantum noises are discussed and a detailed analysis is carried with two kinds of noise and potential. We show that within our assumptions the use of the SLE as an effective open quantum system formalism is possible and discuss some of its limitations.     

A Bohmian approach to the non-Markovian non-linear Schrödinger–Langevin equation

In this work, a non-Markovian non-linear Schrödinger–Langevin equation is derived from the system-plus-bath approach. After analyzing in detail previous Markovian cases, Bohmian mechanics is shown to be a powerful tool for obtaining the desired generalized equation.     

Painlevé Integrability of Nonlinear Schrödinger Equations with bothSpace- and Time-Dependent Coefficients

We investigate the Painlevé integrability of nonautonomous nonlinearSchrödinger (NLS) equations with both space- and time-dependent dispersion, nonlinearity, and external potentials. The Painlevé analysis is carried out without using the Kruskal's simplification, which results in more generalized form of inhomogeneous equations. The obtained equations are shown to be reducible to the standard NLS equation by using a point transformation. We also construct the corresponding Lax pair and carry out its Kundu-type reduction to the standard Lax pair. Special cases of equations from choosing limited form of coefficients coincide with the equations from the previous Painlevé analyses and/or become unknown new equations.     

光纤中变系数非线性Schrödinger方程的孤子解及其应用

基于推广的立方非线性Klein-Gordon方程对一般形式的变系数非线性Schr(o)dinger方程进行研究,讨论了无啁啾情形的孤子解,发现了包括亮、暗孤子解和类孤子解在内的一些新的精确解. 同时对基本孤子的色散控制方法进行了简单讨论. 作为特例,常系数非线性Schr(o)dinger方程和两类特殊的变系数非线性Schr(o)dinger方程的结果和已知的形式一致.此外,还研究了一个周期增益或损耗的光纤系统,得到了有意义的结果.     

指数型变化有效质量的三维Schrödinger方程的解析解

对随径向坐标指数型变化的有效质量分布,通过坐标变换,得到了与Coulomb型势,Kratzer型势和无限深球方势阱三类势函数相联系的变质量三维Schr(o)dinger方程的解析解,具体给出了这三类系统的能量本征值和本征函数的解析表达式.     

具有Eckart势的Schrödinger方程的任意l波解析解

使用完全量子化规则计算了具有离心项的Eckart势,根据动量积分 和Greene-Aldrich近似化条件,得到了系统的任意l波Schrödinger方程的解析解．讨论了：(1) 基态和激发态下,势能范围参数λ和势阱深度η对具有不同角动量量子数的能量本征值的影响;(2) 径向量子数n和角动量量子数l与能量本征值的关系．     

具有Eckart势的Schrödinger方程的任意l波解析解

使用完全量子化规则计算了具有离心项的Eckart势,根据动量积分 和Greene-Aldrich近似化条件,得到了系统的任意l波Schrödinger方程的解析解．讨论了：(1) 基态和激发态下,势能范围参数λ和势阱深度η对具有不同角动量量子数的能量本征值的影响;(2) 径向量子数n和角动量量子数l与能量本征值的关系．     

具有Manning-Rosen势的Schrödinger方程的任意l波束缚态解析解

使用完全量子化规则计算了具有离心项的Manning-Rosen势，根据动量积分 (其中 )和一种新的近似化条件，得到了系统的任意l波Schrödinger方程的解析解和能谱方程。最后简单讨论了l=0和α=0或1束缚态能谱的有关性质。     

大约1961年1月：Lorenz和蝴蝶效应

对于一般门外汉来说,混沌的概念带给他们的是一种完全任意性的印象,但对于科学家来说,它却是表示因果系统中的随机行为,也就是说,系统对于测量太敏感,以致产生的结果虽然有着根本的秩序,但看起来却很随机。这个表面上很矛盾的观点是由数学家改行成为气象学家的Edward Lorenz所提出的,他在一次意外发现了此现象,随即孕育出现代混沌理论的领域,并永远改变了我们检视天气等非线性系统的方式。     

The Feynman–Kac formula for Schrödinger operators on vector bundles over complete manifolds

Methods from stochastic analysis are combined with functional analytic methods in order to prove a Feynman–Kac formula för Schrödinger type operators with nonnegative locally square integrable potentials on vector bundles over complete Riemannian manifolds. In particular, we obtain a Feynman–Kac–Itô formula on manifolds for Schrödinger operators with magnetic fields.     

Trajectory approach to the Schrödinger–Langevin equation with linear dissipation for ground states

The Schrödinger–Langevin equation with linear dissipation is integrated by propagating an ensemble of Bohmian trajectories for the ground state of quantum systems. Substituting the wave function expressed in terms of the complex action into the Schrödinger–Langevin equation yields the complex quantum Hamilton–Jacobi equation with linear dissipation. We transform this equation into the arbitrary Lagrangian–Eulerian version with the grid velocity matching the flow velocity of the probability fluid. The resulting equation is simultaneously integrated with the trajectory guidance equation. Then, the computational method is applied to the harmonic oscillator, the double well potential, and the ground vibrational state of methyl iodide. The excellent agreement between the computational and the exact results for the ground state energies and wave functions shows that this study provides a synthetic trajectory approach to the ground state of quantum systems.