Generation and classification of the translational shape-invariant potentials based on the analytical transfer matrix method

For the conventional translational shape-invariant potentials (TSIPs), it has demonstrated that the phase contribution devoted by the scattered subwaves in the analytical transfer matrix quantization condition is integrable and independent of n. Based on this fact we propose a novel strategy to generate the whole set of conventional TSIPs and classify them into three types. The generating functions are given explicitly and the Morse potential is taken as an example to illustrate this strategy.     

The analytical transfer matrix method for quantum reflection

In this paper, the analytical transfer matrix method (ATMM) is applied to study the properties of quantum reflection in three systems: a sech$^{2}$ barrier, a ramp potential and an inverse harmonic oscillator. Our results agree with those obtained by Landau and Lifshitz [Landau L D and Lifshitz E M 1977 \wx{Quantum Mechanics (Non-relativistic Theory)}{} (New York: Pergamon)], which proves that ATMM is a simple and effective method for quantum reflection.     

Exact solution to the curve crossing problems of two linear diabatic potentials by transfer matrix method

In the present work we have reported a simple exact analytical solution to the curve crossing problem of two linear diabatic potentials by transfer matrix method. Our problem assumes the crossing of two linear diabatic potentials which are coupled to each other by an arbitrary coupling (in contrast to linear potentials in the vicinity of crossing points) and for numerical calculation purposes this arbitrary coupling is taken as Gaussian coupling which is further expressed as a collection of Dirac delta functions. Further we calculated the transition probability from one diabatic potential to another by the use of this method.     

Energy eigenvalues from an analytical transfer matrix method

A detailed procedure based on an analytical transfer matrix method is presented to solve bound-state problems. The derivation is strict and complete. The energy eigenvalues for an arbitrary one-dimensional potential can be obtained by the method. The anharmonic oscillator potential and the rational potential are two important examples. Checked by numerical techniques, the results for the two potentials by the present method are proven to be exact and reliable.     

The rotating Morse potential energy eigenvalues solved by using the analytical transfer matrix method

We study the eigenvalues of the rotating Morse potential by using the quantization condition from the analytical transfer matrix(ATM) method.A hierarchy of supersymmetric partner potentials is obtained with Pekeris approximation,which can be used to calculate the energies of higher rotational states from the energies of lower states.The energies of rotational states of the hydrogen molecule are calculated by the ATM condition,and comparison of the results with those from the hypervirial perturbation method reveals that the accuracy of the approximate expression of Pekeris for the eigenvalues of the rotating Morse potential can be improved substantially in the framework of supersymmetric quantum mechanics.     

On the tunneling time of arbitrary continuous potentials and the Hartman effect

This paper obtains a generalized tunneling time of one-dimensional potentials via time reversal invariance.It also proposes a simple explanation for the Hartman effect using the useful concept of the scattered subwaves.     

Theoretical study of W-shaped optical fiber with a depression in core center by applying analytical transfer matrix method

Analytical transfer matrix method (ATMM) is a method for calculating the propagation constant in weakly guiding optical fiber. By using ATMM, the optical fiber with a depression in the index profile center and a valley in the cladding layer is analyzed. Compared with Wentzel-Kramers-Brillouin (WKB) method, the simulation result of differential mode delay (DMD) by using ATMM fits well with the experimental results obtained by Takahashi. Based on ATMM, by increasing the depth of central depression in a W-shaped index fiber, the improvement of DMD is also discussed.     

Analysis of a right-angle conical reflector resonator by the transfer matrix method

A resonator with a right-angle conical reflector has been proposed to produce high-power CO₂ laser beams. To analyze eigenfields of the right-angle conical reflector resonator, this paper adopts and demonstrates the transfer matrix method. In this paper, the mode-fields and corresponding losses are described as eigenvectors and eigenvalues of a transfer matrix according to the self-reproducing principle of laser field. By solving the transfer matrix for eigenvectors and eigenvalues, we obtain field distributions and losses of the dominant eigenmodes. The calculation results reveal that the right-angle conical reflector resonator could be used for a high-power CO₂ laser to achieve low-order modes. However, the beam quality is reduced due to the residual blind-hole, which is in accord with the experimental result.     

Quantum reflection as the reflection of subwaves

This paper studies quantum reflection with recent research on reflection coefficient. Based on the analytical transfer matrix method, a novel explanation for this phenomenon is proposed that quantum reflection is the reflection of subwaves, which originate inherently from the inhomogeneity of the fields and is always neglected in the semiclassical regime. Comparison with exact formula and the numerical calculations for different potentials has confirmed the reliability and the validity of the proposed theory.     

传输矩阵法在行波管内部反射引起的增益波动计算中的应用

本文分析了由于行波管慢波结构制造误差引入的多个不连续点对小信号增益的影响.行波管内部反射对增益波动的影响,须采用考虑反射波的四阶模型进行分析,用传输矩阵法对节点处的自左至右入射和自右至左入射两种散射类型建立传输矩阵,研究在不同空间电荷参量下,慢波电路的单个反射节点以及慢波电路的皮尔斯速度参量b和增益参量C的多个随机分布不连续性对行波管小信号增益的影响,计算结果与Chernin模型具有很好的一致性.并以G波段行波管为例分析了慢波结构周期长度分布有两个不连续点和周期长度的多个随机分布不连续性带来的小信号增益波动.结果表明,制造误差越大,周期长度分布的两个不连续点相距越远,小信号增益波动越大,多个小的不连续性可以引起较大的增益波动.     

Solving ground eigenvalue and eigenfunction of spheroidal wave equation at low frequency by supersymmetric quantum mechanics method

The spheroidal wave functions are found to have extensive applications in many branches of physics and mathematics. We use the perturbation method in supersymmetric quantum mechanics to obtain the analytic ground eigenvalue and the ground eigenfunction of the angular spheroidal wave equation at low frequency in a series form. Using this approach, the numerical determinations of the ground eigenvalue and the ground eigenfunction for small complex frequencies are also obtained.     

A modified transfer matrix method for the study of the bending vibration band structure in phononic crystal Euler beams

We introduce a modified transfer matrix (MTM) method for the calculation of the bending vibration band structure of one-dimensional phononic crystal (PC) Euler beams. A particular combination of hyperbolic functions and triangular functions is introduced to transform the state parameters of the transfer matrix (TM) method into four initial parameters, which have the explicit meanings of the displacement, rotation angle, bending moment and shear force at one beam end. The method is used to calculate the band structures of two PC Euler beams constructed from aluminum–Lucite and 100 kinds of materials. The effectiveness and high efficiency of the MTM method are demonstrated by the results. Several advantages make it a proper choice for the calculation of the bending vibration band structure of PC Euler beams.     

Community structure discovery method based on the Gaussian kernel similarity matrix

Community structure discovery in complex networks is a popular issue, and overlapping community structure discovery in academic research has become one of the hot spots. Based on the Gaussian kernel similarity matrix and spectral bisection, this paper proposes a new community structure discovery method. First, by adjusting the Gaussian kernel parameter to change the scale of similarity, we can find the corresponding non-overlapping community structure when the value of the modularity is the largest relatively. Second, the changes of the Gaussian kernel parameter would lead to the unstable nodes jumping off, so with a slight change in method of non-overlapping community discovery, we can find the overlapping community nodes. Finally, synthetic data, karate club and political books datasets are used to test the proposed method, comparing with some other community discovery methods, to demonstrate the feasibility and effectiveness of this method.     

基于传输矩阵法讨论光疏介质表面的全反射

探究了光在光密介质与超薄光疏介质界面处的全反射,通过在光密介质中嵌入一光疏介质薄层,利用传输矩阵法,定量地分析了入射角度以及光疏介质的厚度对反射率的影响.数值模拟结果表明,当光疏介质的厚度小于约一个波长时,即使入射角大于全反射的临界角,反射率也不一定会达到1,也就是说,光学隧道效应将会出现.     

Airy传递矩阵法与偏压下多势垒结构的准束缚能级

使用Airy函数和传递矩阵方法精确计算了一维定态薛定谔方程，并推广到多势垒结构，求解出有/无偏压作用的2,3势垒结构的准束缚能级，进一步研究了有/无偏压作用的2,3势垒结构的准束缚能级与有效质量和外加电压的关系，并对结论的正确性进行了验证． 另外，文中 还指出了有些文章中关于Airy传递矩阵法与计算偏压下多势垒结构的准束缚能级的错误陈述 . 关键词： 准束缚能级 有/无偏压作用的多势垒结构 Airy函数 透射系数     

基于槽形函数拟合的刻划光栅衍射特性分析方法

作为对光栅刻划工艺的理论指导，提出了一种基于槽形函数拟合的机械刻划光栅衍射特性分析方法，解决了高刻线密度光栅衍射效率测量值与理论值偏差较大的问题.以刻线密度为1200l/mm的紫外光栅为例，对光栅主截面上的实际槽形纵横坐标扫描取值，进而拟合出槽形函数，经理论计算找到了导致衍射效率偏低的原因，并通过改进工艺，使光栅衍射效率提高了约20%. 关键词： 刻划光栅 槽形函数拟合 衍射特性 分析方法     

求一类非线性偏微分方程解析解的一种简洁方法

通过引入一个变换和选准试探函数，进行求偏导数运算,将非线性偏微分方程化为代数方程，然后用待定系数法确定相应的常数，最后得到其解析 解.不难看出，这种方法特别简洁. 关键词： 非线性偏微分方程 试探函数 待定系数法 解析解     

Comparison of 1D transfer matrix method and finite element method with tests for acoustic performance of multi-chamber perforated resonator

Multi-chamber perforated resonator (MCPR) is a kind of typical silencer element which can both attenuate broadband noise and satisfy specific installation requirements. The one-dimensional transfer matrix method (TMM) and finite element method (FEM) are widely used to predict the transmission loss of the resonators. This paper mainly focuses on the comparison between 1D TMM and FEM in which detailed perforation modeling is applied for the acoustic modeling of MCPRs. Five resonators with different acoustic attenuation frequency ranges are built for simulation and test. In order to verify the results of the above methods, a transmission loss test facility is designed based on two-load method. Through adjusting the distance between microphones, the facility’s effective measurement frequency can be changed. The results show that despite of the complex modeling and calculation, FEM with detailed perforation modeling shows good consistency with test results in both frequency and amplitude within entire frequency range. In comparison, TMM is limited by the cut-off frequency when calculating transmission losses. Besides, accuracy of TMM in low frequency range is also affected by perforation conditions. However, TMM is time-saving in calculation and structure optimization. In MCPRs’ development process, TMM can be used to quickly design and optimize structure parameters while FEM can be used to verify the acoustic performance before prototyping.     

用代数-能量自洽法研究双原子分子解析势能函数

基于计算双原子分子完全振动能谱及离解能的代数方法 (algebraic method,AM)和研究双原子分子解析势能函数的能量自洽法(energy consistent method,ECM),建立了计算双核分子体系精确解析势能函数的代数-能量自洽法(AM-ECM).应用AM-ECM方法研究了7Li+2-23Σg,KH-X1Σ+,NaLi-X1Σ+和NaLi-A1Σ+电子态的解析势能表达形式,并与其他方法的研究结果进行了比较,获得了能正确描述这些电子态在渐近区和离解区的精确解析表达结果.