Energy levels of one-dimensional systems satisfying the minimal length uncertainty relation

The standard approach to calculating the energy levels for quantum systems satisfying the minimal length uncertainty relation is to solve an eigenvalue problem involving a fourth- or higher-order differential equation in quasiposition space. It is shown that the problem can be reformulated so that the energy levels of these systems can be obtained by solving only a second-order quasiposition eigenvalue equation. Through this formulation the energy levels are calculated for the following potentials: particle in a box, harmonic oscillator, Pöschl–Teller well, Gaussian well, and double-Gaussian well. For the particle in a box, the second-order quasiposition eigenvalue equation is a second-order differential equation with constant coefficients. For the harmonic oscillator, Pöschl–Teller well, Gaussian well, and double-Gaussian well, a method that involves using Wronskians has been used to solve the second-order quasiposition eigenvalue equation. It is observed for all of these quantum systems that the introduction of a nonzero minimal length uncertainty induces a positive shift in the energy levels. It is shown that the calculation of energy levels in systems satisfying the minimal length uncertainty relation is not limited to a small number of problems like particle in a box and the harmonic oscillator but can be extended to a wider class of problems involving potentials such as the Pöschl–Teller and Gaussian wells.     

Quantum fluctuations,master equation and Fokker-Planck equation

In order to describe quantum fluctuations a general method is developed, which also may be applied to nonstationary systems as well as to states far from thermodynamic equilibrium. After a concise derivation of the master equation quantum mechanically determined dissipation and fluctuation coefficients are introduced, for which several theorems and relations are given. By using these coefficients there is set up a general Fokker-Planck equation for the diffusion of the statistical operator due to quantum fluctuations.     

Determination of the molecular-field coefficient in ferrites from the temperature dependence of the sublattice magnetization

Inverse Brillouin functions can be used to reduce the solution of the system of two transcendental Neel equations describing the temperature dependence of the sublattice magnetization to the solution of two transcendental equations with a single unknown. A method is proposed for evaluating the molecular-field coefficients from the known temperature dependences of the sublattice magnetizations. The method does not necessarily involve the use of an electronic computer. The molecular-field coefficient can be determined with the desired accuracy under two conditions: 1) the molecular-field coefficients being determined differ significantly from those corresponding to the boundary between P and Q curves; 2) the sublattice magnetizations are measured with an accuracy roughly two orders of magnitude better. When the sublattice magnetizations are not measured sufficiently accurately, the system of two linear equations used to determine the molecular-field coefficients by the method of least squares converts into a single equation, so the molecular-field coefficients cannot be determined unambiguously. The linear equation relating the molecular-field coefficients remains essentially the same over a broad range of measurement accuracies, over a broad range of numbers of experimental points, and for a wide variety of distributions of these points. It concluded that the frequent assertion that the Neel theory of the molecular field is inadequate for describing the temperature dependence of the sublattice magnetizations as well as the arguments which lead to this assertion are not convincing.Translated from Izvestiya VUZ. Fizika, No. 5, pp. 7–15, May, 1971.     

Theory of ion reflection from amorphous surface at low glancing angles

A new theory of ion reflection from a smooth amorphous solid surface is developed. A continuous potential is introduced for describing the ion reflection from amorphous surfaces at low glancing angles. It is shown that the difference, between a true surface potential and a continuous one could be taken into account in terms of a correction factor, thus yielding a Fokker-Planck-type equation for a particle distribution in angle and transverse coordinate. Diffusion coefficients included in this equation are calculated with allowance for atom discrete positions and ion-electron collision effect. On the basis of the derived kinetic equation the angular and energy ion distributions in a reflected beam are analyzed. The presented theory, somewhat modified, can be used for describing ion reflection from a crystal surface as well.     

The equation of state of solids

An equation of state for solids, in reduced variables, is obtained within the context of a system-independent formulation of the thermodynamics of high pressures. This formulation is valid for materials obeying a linear relationship between shock and particle velocities. An adequate set of scaling factors for pressure, compression, specific energy, and temperature, is first introduced. A modified Mie-Grüneisen equation, as well as many other thermodynamic relationships and coefficients, are then expressed in terms of reduced variables. Explicit expressions for the temperature along the Hugoniot, and for the equation of state, are obtained. It is also shown that when given in their reduced form, each of the two thermodynamic coefficients appearing in the equation of state can be considered as having the same constant value for many different materials. The possibility and convenience of using a “standard material” is discussed. Numerical results obtained using this reduced variables formalism are in good agreement with those computed or measured, by different authors, for various materials over a wide range of pressures. This is a good indication of the “universality” of the reduced equations obtained, and of the usefulness of the formalism.     

Specific solutions of the generalized Korteweg-de Vries equation with possible physical applications

Solutions for a type of generalized Korteweg-de Vries equation which should have physical impact will be determined here. These types of solutions should have applications in the study of intrinsic localized modes optical waveguide arrays and fluid dynamics. It is shown that trigonometric and hyperbolic solutions can be obtained by matching powers and coefficients of the independent terms in the equation after the assumed solution has been substituted. As well, solutions to the equation in terms of more complicated Jacobe elliptic functions are determined.     

Solutons of a nonisospectral and variable coefficient Korteweg-de Vries equation

A new type of KdV equation with a nonisospectral Lax pair as well as variable coefficients is introduced. Its Lax pair is shown to be invariant under the Crum transformation. This leads to a Bäcklund transformation for the KdV equation and, hence, a method for solutions via an associated nonisospectral variable coefficient MKdV equation. Three generations of solutions are given. The 1-soliton solution shares the novel phenomenology associated with the boomeron, trappon, and zoomeron of Calogero and Degasperis.     

Slip Boundary Conditions for the Compressible Navier–Stokes Equations

The slip boundary conditions for the compressible Navier–Stokes equations are derived systematically from the Boltzmann equation on the basis of the Chapman–Enskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar–Gross–Krook model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.     

InAs/AlSb/GaSb量子阱中的双色光吸收

为了降低噪声对InAs/GaSb量子阱作为双色电探测器性能的影响,设计性能优良的光电探测器,在InAs/GaSb量子阱中加入AlSb夹层,以减少电子和空穴在界面处的复合,从而抑制由于电子和空穴复合引起的噪声。首先应用转移矩阵方法求解薛定谔方程得到量子阱中电子和空穴的能级和波函数,研究AlSb夹层对电子和空穴波函数的影响。应用平衡方程方法求解外加光场条件下的玻尔兹曼方程,研究所有电子和空穴跃迁通道对光吸收系数的贡献,重点研究了AlSb夹层厚度对光吸收系数的影响。结果表明:基于In As/GaSb的量子阱体系可以实现双色光吸收,加入AlSb夹层可以有效抑制电子和空穴在界面处的隧穿,从而降低复合噪声,同时AlSb夹层的加入也对吸收峰有影响。AlSb夹层的厚度达到2 nm即可有效降低电子和空穴复合噪声,双色光吸收峰在中远红外波段,为该量子阱作为性能良好的中远红外光电探测器提供理论支撑。     

Coarse grained approach for volume conserving models

Volume conserving surface (VCS) models without deposition and evaporation, as well as ideal molecular-beam epitaxy models, are prototypes to study the symmetries of conserved dynamics. In this work we study two similar VCS models with conserved noise, which differ from each other by the axial symmetry of their dynamic hopping rules. We use a coarse-grained approach to analyze the models and show how to determine the coefficients of their corresponding continuous stochastic differential equation (SDE) within the same universality class. The employed method makes use of small translations in a test space which contains the stationary probability density function (SPDF). In case of the symmetric model we calculate all the coarse-grained coefficients of the related conserved Kardar–Parisi–Zhang (KPZ) equation. With respect to the symmetric model, the asymmetric model adds new terms which have to be analyzed, first of all the diffusion term, whose coarse-grained coefficient can be determined by the same method. In contrast to other methods, the used formalism allows to calculate all coefficients of the SDE theoretically and within limits numerically. Above all, the used approach connects the coefficients of the SDE with the SPDF and hence gives them a precise physical meaning.     

Integrable systems of partial differential equations determined by structure equations and Lax pair

It is shown how a system of evolution equations can be developed both from the structure equations of a submanifold embedded in three-space as well as from a matrix SO(6) Lax pair. The two systems obtained this way correspond exactly when a constraint equation is selected and imposed on the system of equations. This allows for the possibility of selecting the coefficients in the second fundamental form in a general way.     

Algebraic calculation of stroboscopic maps of ordinary, nonlinear differential equations

The relation between the parameters of a differential equation and corresponding discrete maps is becoming increasingly important in the study of nonlinear dynamical systems. Maps are well adopted for numerical computation and several universal properties of them are known. Therefore some perturbation methods have been proposed to deduce them for physical systems, which can be modeled by an ordinary differential equation (ODE) with a small nonlinearity. An iterative, rigorous algebraic method for the calculation of the coefficients of a Taylor expansion of a stroboscopic map from ODEs with not necessarily small nonlinearities is presented. It is shown analytically that most of the coefficients are small for a small integration time and grow slowly in the course of time if the flow vector field of the ODE is a polynomial in the state variables and if the ODE has a fixed point at the origin. For several nonlinear systems approximations of different orders are investigated.     

阶梯圆盘的设计及其声参数计算

由薄圆盘改进而来的阶梯型薄圆盘声波辐射器在空气中有着很好的应用,但由于阶梯的存在,需要根据在节线处位移连续、位移斜率连续、剪切力连续以及弯矩连续,加之其边界条件才能推导出频率方程。多一条节线,则多四个方程。变量多,方程系数复杂,频率方程难解。本文以单节线阶梯圆盘为例,提出了一种数值计算方法,编制了计算程序。计算结果与实验测试值符合较好。更进一步,可计算得到阶梯盘的声场分布、辐射阻抗等声学特性。本文为阶梯圆盘的设计和其声学性能的计算提供了一种方法。      

Virial series around non-null density: density dependence of virial coefficients

It is shown that the equation of state of fluid systems can be expanded around non-null densities if the well known virial series is generalized by considering its coefficients as density dependent. This in turn leads to a hierarchy of differential equations that describe the coefficients b_j (ρ). Starting from the already known equation of state for hard bodies in d = 0,1,2,3 dimensions this hierarchy is analysed and the behaviour of both the reducible b_j (ρ) and irreducible β～ _j (ρ) cluster integrals is discussed. New virial coefficients b_j (ρ) have been introduced with a simpler density dependence. Their asymptotic (j → ∞) behaviour is discussed.     

Topological solitons of resonant nonlinear Schödinger'sequation with dual-power law nonlinearity by G′/G-expansion technique

This paper considers the resonant nonlinear Schrödinger's equation with dual-power law nonlinearity. The G′/G-expansion method is applied to integrate this equation. The soliton solutions are thus obtained. Both constant coefficients as well as time-dependent coefficients are considered. The results for parabolic law nonlinearity fall out as a special case.     

Soliton solutions for a generalized fifth-order KdV equation with t-dependent coefficients

We consider a generalized fifth-order KdV equation with time-dependent coefficients exhibiting higher-degree nonlinear terms. This nonlinear evolution equation describes the interaction between a water wave and a floating ice cover and gravity-capillary waves. By means of the subsidiary ordinary differential equation method, some new exact soliton solutions are derived. Among these solutions, we can find the well known bright and dark solitons with sech and tanh function shapes, and other soliton-like solutions. These solutions may be useful to explain the nonlinear dynamics of waves in an inhomogeneous KdV system supporting high-order dispersive and nonlinear effects.     

小波自适应方法模拟二维涡旋演化

以涡量方程为控制方程,模拟初始状态涡量分布为高斯分布三个涡旋演化过程.提出一个关联实际流动的小波系数临界值,小波系数分为临界值以上及以下部分,进而涡量可分成尺度系数项、小波系数突出项和小波系数平凡项三部分.只采用尺度系数项和小波系数突出项近似涡量,既可以节约计算量,还可以自动追踪绝大部分的拟涡能.数值结果表明,用不到10%的小波系数,可控制99%以上的拟涡能.     

An Inverse Problem of Determining Coefficients in a One-Dimensional Radiative Transport Equation

We consider an inverse problem of determining unknown coefficients for a one-dimensional analogue of radiative transport equation. We show that some combination of the unknown coefficients can be uniquely determined by giving pulse-like inputs at the boundary and observing the corresponding outputs. Our result can be applied for determination of absorption and scattering properties of an optically turbid medium if the radiative transport equation is appropriate for describing the propagation of light in the medium.     

带强迫项变系数组合KdV方程的无穷序列复合型类孤子新解

为了获得变系数非线性发展方程的无穷序列复合型新解,研究了G′(ξ)G(ξ)展开法.通过引入一种函数变换,把常系数二阶齐次线性常微分方程的求解问题转化为一元二次方程和Riccati方程的求解问题.在此基础上,利用Riccati方程解的非线性叠加公式,获得了常系数二阶齐次线性常微分方程的无穷序列复合型新解.借助这些复合型新解与符号计算系统Mathematica,构造了带强迫项变系数组合KdV方程的无穷序列复合型类孤子新精确解.