基于群论及谐振子模型的声传播方法的研究

本文基于谐振子及群论方法,对声在空气中传播建立了一个新的哈密顿数学模型。研究将分子链作为一系列谐振子组合,声传播以能量的形式传播,对于介质里的每个振动的质点而言,它总是在不断地吸收和放出能量。从能量的概念出发,将声传播问题转换成求解谐振子的波函数以及能量本征值的问题。本文将谐振子模型引入到气体声波传播的问题中,在确定气体声波传播过程中的分子振动模式、能级简并时,引入群论,以探索气动声学研究的新方法。     

基于Morse势能函数谐振子模型的声传播研究

对于复杂的体系,组成体系的各粒子间的相互作用决定其能量状态。对于不同势场,描述粒子运动规律的Schrodinger方程是不同的微分方程。本文在原来的Hamilton谐振子数学模型基础上,通过引入Morse势能函数代替原来的简谐振子势模型,研究了一维气体分子链中声传播的问题。从能量的概念出发,将声传播问题转换成求解谐振子的波函数以及能量本征值的问题,建立配分函数与热力学量之间的关系,以Morse势能函数的振子模型构造的声传播模型求取声压及声能量。通过比较,经典方法与量子谐振子模型计算的一维声压吻合。     

具有含时平方反比项的谐振子的路径积分求解

使用含时坐标变换把一个质量含时的带有平方反比项的谐振子系统变换为对应的质量不含时 谐振子系统.利用变换前、后传播子之间的关系并通过Feynman路径积分的方法求出质量含时 的带有平方反比项的谐振子系统的传播子与精确波函数.并对包含有更多的附加势的谐振子 系统进行了讨论. 关键词： 谐振子 坐标变换 路径积分 传播子     

辛和非辛算法求解薛定谔方程误差分析

在哈密尔顿体系下,提出气体声波传播的一种新的谐振子模型,并引入群论确定气体声波传播过程中的分子振动模式、能级简并.新模型将气动声学声传播问题与分子振动关联起来。由于发展高效的薛定谔方程的数值计算方法,有利于联系分子的性质来解释声的传播.本文从此出发,用二阶有限差分格式和生成函数法构造的二阶辛格式分别计算一维定态谐振子势场和含时谐振子势场的薛定谔方程,分析了数值解的误差以及传播能量误差.结果表明辛算法具有明显的优势.     

含时阻尼谐振子的传播子与严格波函数

通过正则化变换技巧,寻找到一种对阻尼系数随时间变化的阻尼谐振子直接量子化方案,进而采用高斯型传播子和费曼路径积分方法求出了含时阻尼谐振子的严格波函数,并对波函数的普遍意义,坐标和动量的零点涨落以及两者的不确定关系作了讨论 关键词： 含时阻尼 传播子 费曼路径积分     

均匀分布费米系统的组态路径积分蒙特卡罗模拟

改进了组态路径积分蒙卡方法,简化采样过程,开发相应程序并对无库伦相互作用谐振子以及均匀电子气进行模拟.谐振子平衡态能量与Fermi-Dirac分布符合较好,均匀电子气平衡态能量和动量基本符合Fermi-Dirac分布规律.对两种费米子系统的研究表明,组态路径积分蒙卡方法对费米子的交换效应具有较好的描述.本文探讨了费米交换符号问题的解决途径,同时对均匀电子气模型进行研究,为后续温热稠密物质的研究奠定基础.     

二维无限深方势阱中粒子运动的路径积分解法

用路径积分的方法计算了二维无限深方势阱中粒子的传播子,并由传播函数推导出二维无限深方势阱中粒子的波函数和能量,进一步体现了路径积分与其他经典量子化方法的等价性,反映了路径积分应用于难以处理的量子力学问题的价值.     

一维无限深势阱中粒子运动的路径积分解法

用路径积分的分析方法求得了一维无限深势阱中粒子的传播函数,并由传播函数导出了粒子的波函数和能量,展示了路径积分与传统方法的等价性,同时还介绍了一种有用的数学函数——雅可比θ_3函数．     

在空间转动变换下的角动量本征态随转角的参变数演变的路径积分

研究在空间转动变换下,直接采用转轴的方向角与转角描述角动量本征态的演变过程,并建立一般角动量转动矩阵元以及多重转动矩阵元随转角θ的参变数λ(0→θ)演变的路径积分;给出处理泛函(路径)积分为普通多重积分的一般方法.     

一类特殊谐振子问题的路径积分研究

考虑存在耦合,且质量不等的双谐振子系统,利用路径积分的方法研究了该系统的费曼振幅。结果表明,这一系统可以等效为退耦合的双谐振子系统,但退耦合的具体方案不具有唯一性。同时,给出了退耦合后谐振子的等效质量与等效频率。这一基本分析思路有望为解决耦合的多谐振子系统的相关问题提供帮助。     

Path integral for the relativistic particle and harmonic oscillators

The action for a massive particle in special relativity can be expressed as the invariant proper length between the end points. In principle, one should be able to construct the quantum theory for such a system by the path integral approach using this action. On the other hand, it is well known that the dynamics of a free, relativistic, spinless massive particle is best described by a scalar field which is equivalent to an infinite number of harmonic oscillators. We clarify the connection between these two—apparently dissimilar—approaches by obtaining the Green function for the system of oscillators from that of the relativistic particle. This is achieved through defining the path integral for a relativistic particle rigorously by two separate approaches. This analysis also shows a connection between square root Lagrangians and the system of harmonic oscillators which is likely to be of value in more general context.     

Path Integral Formulation of Fractionally Perturbed Lagrangian Oscillators on Fractal

Fractional path integration and particles trajectories in fractional dimensional space are motivating issues in quantum mechanics and kinetics. In this paper, a fractional path integral characterized by a fractional propagator is developed based on the framework of the fractional action-like variational approach. A fractional generalization of the free particle problem is found, the corresponding fractional Schrödinger equation is derived and a fractional path integral formulation of harmonic oscillators characterized by a perturbed Lagrangian is constructed after reducing the fractional action to an integral action on fractal. The new fractal-like path integral offers a number of motivating features which are discussed and analyzed. The main outcome is connected to the possibility of constructing on a fractal a path integral for the oscillators characterized by modified ground energy. In particular for low-temperature case, the fractional perturbed oscillator is characterized by a free energy larger than the standard value \( E_{0} = {{\hbar \omega } \mathord{\left/ {\vphantom {{\hbar \omega } 2}} \right. \kern-0pt} 2}.\) Such an increase in the ground energy generalizes the uncertainty principle without involving differentiable paths or even invoking new phenomenological theories based on deformed algebra.     

Time dependence of the average charge and current in a dissipative mesoscopic circuit

Taking into consideration the interactions between electrons and phonons,we have studied the temporal evolution of the average charge and current in a dissipative mesoscopic RLC circuit.Our results show that a mesoscopic RLC circuit can be treated as an interactive system between an electromagnetic harmonic oscillator and many lattice harmonic oscillators;this is called the bathing of the harmonic oscillators.The results also show that the quantum equation of motion of the linear mesoscopic RLC circuit is identical in form to its classical equation of motion,the only difference between them being their respective meanings.In order to thoroughly study the quantum properties of a dissipative mesoscopic circuit,we have to consider not only the electromagnetic energy of the circuit,but also the crystal lattice vibration energy and the interactive energy between electrons and phonons.     

Linearly coupled quantum oscillators with Lévy stable noise

The influence of an external noise on two linearly coupled quantum oscillators is investigated. The noise is described by the probability measure corresponding to a symmetric Lévy stable distribution. In calculations the path integral technique is used. This paper is a continuation of investigations started by Bose, Datta and Feinsilver.     

Path integration on Darboux spaces

In this paper, the Feynman path integral technique is applied to two-dimensional spaces of nonconstant curvature: these spaces are called Darboux spaces D _I-D _IV. We start each consideration in terms of the metric and then analyze the quantum theory in the separable coordinate systems. The path integral in each case is formulated and then solved in the majority of cases; the exceptions being the quartic oscillators where no closed solution is known. The required ingredients are the path integral solutions of the linear potential, the harmonic oscillator, the radial harmonic oscillator, the modified Pöschl-Teller potential, and the spheroidal wave functions. The basic path integral solutions, which appear here in a complicated way, have been developed in recent work and are known. The final solutions are represented in terms of the corresponding Green’s functions and the expansions into the wave functions. We also sketch some limiting cases of the Darboux spaces, where spaces of constant negative and zero curvature emerge.     

精确的量子化条件和不变量

提出并证明了一维量子系统和三维球对称量子系统的一个精确的量子化条件.在此精确量子化条件中, 除了通常的Nπ项外, 还有一积分项, 称为修正项. 发现该修正项正是在超对称量子力学中所谓的有形状不变势的量子系统的一个不变量，它不依赖于波函数的节点数.对这些系统, 可用基态能级和波函数确定此不变量的值, 从而由精确的量子化条件容易算出全部束缚态的能级. 计算得到能级的正确性又反过来验证了在有形状不变势的量子系统中此修正项确实是不变量.计算的有形状不变势的量子系统, 包括一维的有限方势阱、Morse势及其变形、R 关键词： 量子化条件 超对称量子力学 形状不变势 不变量     

Stochastic joint time–frequency response analysis of nonlinear structural systems

A novel approximate analytical approach for determining the response evolutionary power spectrum (EPS) of nonlinear/hysteretic structural systems subject to stochastic excitation is developed. Specifically, relying on the theory of locally stationary processes and utilizing a recently proposed representation of non-stationary stochastic processes via wavelets, a versatile formula for determining the nonlinear system response EPS is derived; this is done in conjunction with a stochastic averaging treatment of the problem and by resorting to the orthogonality properties of harmonic wavelets. Further, the nonlinear system non-stationary response amplitude probability density function (PDF), which is required as input for the developed approach, is determined either by utilizing a numerical path integral scheme, or by employing a time-dependent Rayleigh PDF approximation technique. A significant advantage of the approach relates to the fact that it is readily applicable for treating not only separable but non-separable in time and frequency EPS as well. The hardening Duffing and the versatile Preisach (hysteretic) oscillators are considered in the numerical examples section. Comparisons with pertinent Monte Carlo simulations demonstrate the reliability of the approach.     

Anomalous Dissipation in a Stochastically Forced Infinite-Dimensional System of Coupled Oscillators

We study a system of stochastically forced infinite-dimensional coupled harmonic oscillators. Although this system formally conserves energy and is not explicitly dissipative, we show that it has a nontrivial invariant probability measure. This phenomenon, which has no finite dimensional equivalent, is due to the appearance of some anomalous dissipation mechanism which transports energy to infinity. This prevents the energy from building up locally and allows the system to converge to the invariant measure. The invariant measure is constructed explicitly and some of its properties are analyzed.     

Novel collective effects in integrated photonics

Superradiance, the enhanced collective emission of energy from a coherent ensemble of quantum systems, has been typically studied in atomic ensembles. In this work we study theoretically the enhanced emission of energy from coherent ensembles of harmonic oscillators. We show that it should be possible to observe harmonic oscillator superradiance for the first time in waveguide arrays in integrated photonics. Furthermore, we describe how pairwise correlations within the ensemble can be measured with this architecture. These pairwise correlations are an integral part of the phenomenon of superradiance and have never been observed in experiments to date.