The string of variable density: Further results

We analyze the problem of calculating the solutions and the spectrum of a string with arbitrary density and fixed ends. We build a perturbative scheme which uses a basis of WKB-type functions and obtain explicit expressions for the eigenvalues and eigenfunctions of the string. Using this approach we show that it is possible to derive the asymptotic (high energy) behavior of the string, obtaining explicit expressions for the first three coefficients (the first two can also be obtained with the WKB method). Finally, using an iterative approach we also obtain analytical expressions for the low energy behavior of the eigenvalues and eigenfunctions of a string with rapidly oscillating density, recovering (in a simpler way) results in the literature.     

Nonlocal potentials and their exact and approximate local and velocity-dependent equivalents

We show that good approximations to the exact equivalent local potential (ELP) and damping factor of a nonlocal Perey-Buck potential can be calculated in the partial wave WKB approximation of Horiuchi. The exact ELP and damping factor are obtained by means of a method previously given by one of us. We also confirm that an approximate ELP proposed by Bauhoff et al. is of comparable accuracy as the Horiuchi approximation. Thesel-dependent ELP's exhibit reduced attraction in the interior and provide a test for higher order WKB approximations. We subsequently obtain an equivalent velocity dependent potential (EVDP) which is even exactly wave function equivalent to the original nonlocal potential. This almost local potential, unlike the trivial equivalent local potential, is smooth and well-behaved and is therefore particularly useful in nuclear reactions where the off-shell behaviour of the potential is important.     

FDTD Simulation of Electromagnetic Wave Transformation in a Dynamic Magnetized Plasma

Finite-Difference Time-Domain (FDTD) algorithms are applied to study the transformation of a pre-existing electromagnetic plane wave by prescribed time variation of a cold magnetoplasma with a magnetic field along the propagation direction. A one-dimensional FDTD code is used to verify the results obtained earlier using analytical approximations based on (a) WKB method for slow switching of the plasma medium and (b) Green's function technique for rapid switching of the plasma medium. A novel successive reduction method has been developed and applied to obtain the amplitudes and the frequencies of the new modes generated by the switching of the medium.     

WKB- and SVEA-approximative methods for slowly varying non-linear dielectric media

For linear media of which the index of refraction changes slowly in the propagation direction, the WKB method is presented as a transformation that enables a simple approach towards higher order approximations. For homogeneous non-linear media the SVEA method considers small amplitude approximations that change slowly in spatial directions. For third order non-linear media with a slowly varying index of refraction, the combination of WKB and SVEA lead to a two-parameter problem; relating the amplitude in a specific way to the measure of the slow medium changes, transformations are given that lead to a NLS equation with slowly varying coefficients and even to NLS with constant coefficients.     

New WKB method in supersymmetry quantum mechanics

In this paper, we combine the perturbation method in supersymmetric quantum mechanics with the WKB method to restudy an angular equation coming from the wave equations for a Schwarzschild black hole with a straight string passing through it. This angular equation serves as a naive model for our investigation of the combination of supersym- metric quantum mechanics and the WKB method, and will provide valuable insight for our further study of the WKB approximation in real problems, like the one in spheroidal equations, etc.     

The generalized WKB method in the three-dimensional case. IV

A method of successive approximations is developed for obtaining solutions of the basic equations of the WKB method in the three-dimensional case. Explicit expressions are found for the first approximations to the energy levels and wave functions of the bound states of a particle in a three-dimensional potential well which were obtained in zeroth approximation in [1, 2].Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 53–60, November, 1976.     

On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-like case

In this paper initial-boundary-value problems for a linear wave (string) equation are considered. These problems can be used as simple models to describe the vertical vibrations of a conveyor belt, for which the velocity is small with respect to the wave speed and is assumed to move with a time-varying speed. Formal asymptotic approximations of the solutions are constructed to show the complicated dynamical behavior of the conveyor belt. It will also be shown that the truncation method cannot be applied to this problem in order to obtain approximations valid on long time scales.     

On variable length induced vibrations of a vertical string

The purpose of this paper is to study the free lateral responses of vertically translating media with variable length, velocity and tension, subject to general initial conditions. The translating media are modeled as taut strings with fixed boundaries. The problem can be used as a simple model to describe the lateral vibrations of an elevator cable, for which the length changes linearly in time, or for which the length changes harmonically about a constant mean length. In this paper an initial-boundary value problem for a linear, axially moving string equation is formulated. In the given model a rigid body is attached to the lower end of the string, and the suspension of this rigid body against the guide rails is assumed to be rigid. For linearly length variations it is assumed that the axial velocity of the string is small compared to nominal wave velocity and the string mass is small compared to car mass, and for the harmonically length variations small oscillation amplitudes are assumed and it is also assumed that the string mass is small compared to the total mass of the string and the car. A multiple-timescales perturbation method is used to construct formal asymptotic approximations of the solutions to show the complicated dynamical behavior of the string. For the linearly varying length analytic approximations of the exact solution are compared with numerical solution. For the harmonically varying length it will be shown that Galerkin?s truncation method cannot be applied in all cases to obtain approximations valid on long timescales.     

On the eikonal approximation in the theory of transition radiation

The transition radiation of relativistic electrons in nonuniform media is considered. Based on the equivalent photon method and the eikonal approximation in wave mechanics, a method for describing this process is proposed. For the case in which the permittivity depends on several coordinates, equations for the spectral-angular density of transition radiation are obtained. The main results obtained in the Born and eikonal approximations of the theory of transition radiation are compared. The equations obtained are used to analyze the transition radiation process for a fiberlike target.     

一种简单精准的渐变折射率分布光波导分析方法

渐变折射率分布的光波导分析对光波导器件的设计和研究至关重要, 近年来已提出了多种分析方法, 然而在简便性或准确性上都存在着不足. 为此, 提出了一种分析渐变折射率分布光波导的方法, 能够结合现有的Wentzel-Kramers-Brillouin近似法和离散化的波动方程, 构建模场分布, 再结合变分运算方程和修正的模式本征方程, 计算出较为精确的有效折射率. 与其他分析方法相比, 该方法较为简单, 而且有一定的精度.     

Higher-order approximations for semiclassical path-coherent states of a spinless relativistic particle

A method has been developed for constructing higher-order approximations for semiclassical path-coherent states of a spinless relativistic particle. Expressions for the higher-order approximations are derived in quadrature form from the classical path of the particle. The method for constructing higher-order approximations is based on Maslov's complex WKB method.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 54–58, July, 1988.     

Three-dimensional Wentzel–Kramers–Brillouin approach for the simulation of scanning tunneling microscopy and spectroscopy

We review the recently developed three-dimensional (3D) atom-superposition approach for simulating scanning tunneling microscopy (STM) and spectroscopy (STS) based on ab initio electronic structure data. In the method, contributions from individual electron tunneling transitions between the tip apex atom and each of the sample surface atoms are summed up assuming the one-dimensional (1D) Wentzel–Kramers–Brillouin (WKB) approximation in all these transitions. This 3D WKB tunneling model is extremely suitable to simulate spin-polarized STM and STS on surfaces exhibiting a complex noncollinear magnetic structure, i.e., without a global spin quantization axis, at very low computational cost. The tip electronic structure from first principles can also be incorporated into the model, that is often assumed to be constant in energy in the vast majority of the related literature, which could lead to a misinterpretation of experimental findings. Using this approach, we highlight some of the electron tunneling features on a prototype frustrated hexagonal antiferromagnetic Cr monolayer on Ag(111) surface. We obtain useful theoretical insights into the simulated quantities that is expected to help the correct evaluation of experimental results. By extending the method to incorporate a simple orbital dependent electron tunneling transmission, we reinvestigate the bias voltage- and tip-dependent contrast inversion effect on theW(110) surface. STM images calculated using this orbital dependent model agree reasonably well with Tersoff-Hamann and Bardeen results. The computational efficiency of the model is remarkable as the k-point samplings of the surface and tip Brillouin zones do not affect the computational time, in contrast to the Bardeen method. In a certain case we obtain a relative computational time gain of 8500 compared to the Bardeen calculation, without the loss of quality. We discuss the advantages and limitations of the 3D WKB method, and show further ways to improve and extend it.     

The CWKB Method of Particle Production in a Periodic Potential

In this work we study the particle production in time dependent periodic potential using the method of complex time WKB (CWKB) approximation. In the inflationary cosmology at the end of the inflationary stage, the potential becomes time dependent as well as periodic. Reheating occurs due to particle production by the oscillating inflaton field. Using CWKB we obtain almost identical results on catastrophic particle production as obtained by others.     

Analysis of planar waveguides with index discontinuity or index slope discontinuity

In this paper, we propose a novel analytic transfer matrix method (ATMM) to the analysis of planar waveguides with index discontinuity or index slope discontinuity, for the cases when the conventional WKB method is no longer valid. We also analyze the physical insight of the approximations in WKB theory, and according to our research, the phase shift at the turning point is not /2, but exactly . Test calculations are done for an index profile with a known solution and the comparison shows that our method gives extremely accurate propagation constant.     

Notes on Application of WKB Method

The notes here presented are of the modifications introduced in the application of WKB method. The problems of two- and three-dimensional harmonic oscillator potential are revisited by WKB and the new formulation of quantization rule respectively. It is found that the energy spectrum of the radial harmonic oscillator, which is reproduced exactly by the standard WKB method with the Langer modification, is also reproduced exactly without the Langer modification via the new quantization rule approach. An alternative way to obtain the non-integral Maslov index for three-dimensional harmonic oscillator is proposed.     

A semiclassical approximation for massive transfer

A semiclassical approximation for the transition matrix for transfer reactions between heavy ions is developed from the distorted-wave Born approximation by means of the three-dimensional WKB approximation. It is designed to be applicable to massive transfer without adopting various assumptions and approximations of previous semiclassical approaches which are reasonable only for few-nucleon transfer. The final expression obtained is of a simple closed form similar to that of the Brink model. It gives a way of seeing the validity of the idea of Brink in massive transfer. Numerical study of illustrative examples indicates promising applicability of the present method to massive transfer.     

Rotating QCD string and the meson spectrum

The spectra of light–light and heavy–light mesons are described by spinless Salpeter equation and Dirac equation respectively, which predict linear dependence of the meson mass squared M² on angular momentum J and number of radial nodes n. Both spectra are computed by the WKB method and shown to agree with exact numerical data within few percent even for the lowest levels. The drawback of Salpeter and Dirac equation is that (inverse) Regge slopes do not coincide with the string ones, 2πσ and πσ respectively, because the string dynamics is not taken into account properly. The lacking string rotation is introduced via effective Hamiltonian derived from QCD which generates linear Regge trajectories for light mesons with the correct string slope.     

String cloud and domain walls with quark matter for a higher dimensional FRW universe in self creation cosmology

In this study, we research a higher dimensional flat Friedmann-Robertson-Walker(FRW) universe in Barber's second theory when strange quark matter(SQM) and normal matter(NM) are attached to the string cloud and domain walls. We obtain zero string tension density for this model. We obtain dust quark matter solutions. This result agrees with Kiran and Reddy, Krori et al, Sahoo and Mishra and Reddy. In our solutions the quark matter transforms to other particles over time. We also obtain two different solutions for domain walls with quark and normal matters by using a deceleration parameter. Also, the features of the obtained solutions are discussed and some physical and kinematical quantities are generalized and discussed. Our results are consistent with Yilmaz, Adcox et al and Back et al in four and five dimensions.