Variational approach to the Schrödinger equation with a delta-function potential

We obtain accurate eigenvalues of the one-dimensional Schr\"{o}dinger equation with a Hamiltonian of the form $H_{g}=H+g\delta (x)$, where $\delta (x)$ is the Dirac delta function. We show that the well known Rayleigh-Ritz variational method is a suitable approach provided that the basis set takes into account the effect of the Dirac delta on the wavefunction. Present analysis may be suitable for an introductory course on quantum mechanics to illustrate the application of the Rayleigh-Ritz variational method to a problem where the boundary conditions play a relevant role and have to be introduced carefully into the trial function. Besides, the examples are suitable for motivating the students to resort to any computer-algebra software in order to calculate the required integrals and solve the secular equations.

Description of the Markov chain using the producing function

In article the way of creation of an assumed function of transformation with the help of a generating function and also a method of use of characteristic numbers and vectors for creation of a matrix which elements well describe conditions of process at any moment is described. This approach differs from preceding that, having used a concept of characteristic numbers and characteristic vectors of a matrix of the transitional probabilities, it is possible to simplify considerably calculation of the elements characterizing process. In article methods of stochastic model operation, ways of the description of a generating function, the solution of matrixes of the equations by means of characteristic numbers and vectors are used. Using properties of a generating function, made “dictionary” of z-transformations which helped to define an assumed function of transformation. The generating function of a vector was applied to a research of behavior of a vector of absolute probabilities which elements represent stationary probabilities. For definition of degree of a matrix of transition of probabilities used a concept of characteristic numbers and characteristic vectors of the transitional probabilities. Determined by such way an unlimited set of latent vectors of which made matrixes which describe a condition of a system at any moment. Reception of definition of latent vectors in more difficult examples which is that along with required coefficients of secular equations the system of auxiliary matrixes and an inverse matrix is under construction is also described.

An Eigenvalue Analysis of Damping in Optical Thin Plasmas

In this work, the behaviour of magneto-hydrodynamic waves in optically thin plasmas considering dissipative processes, thermal and magnetic diffusion, a given ionization, and the heating and cooling functions are investigated for several particular cases. A numerical eigenvalues analysis of the dimensionless secular equations according to various cases is performed for the entire set of MHD equations.

SINIF CEBİRİ YAKLAŞIMI ALTINDA 𝑫𝟐𝒅 ve 𝑪𝟑𝒊 NOKTA GRUPLARI

In this study, the point groups 𝐷2𝑑 and 𝐶3𝑖 which belong to tetragonal and trigonal crystal systems, respectively, are handled under the class sum approach. Symmetry groups were formed with symmetry elements that left these point groups unchanged and Cayley tables of related groups were obtained. Using these tables, the conjugates of the elements and the classes of the group were formed. Secular equations are written for each class sum obtained by the sum of the elements that make up the class. By solving these secular equations, the character vectors are obtained. Thus, the character tables were reconstructed with the calculated characters for both point groups under the class sum approach.

Cyclopentadienyl System: Solving the Secular Determinant, π Energy, Delocalization Energy, Wave Functions, Electron Density and Charge Density

In this study, characteristics of Hückel strategy, were abused so as to acquire some significant outcomes, through a theoretical technique with which it is conceivable to get secular equations, π energy, wave functions, electron density and charge density, as an account of cyclopentadienyl system i.e. C5H5+ (cation), C5H5- (anion), and C5H5. (radical) and permitting the expression of delocalization energy of conjugated cyclopentadienyl ring framework. Here, it was presented the secular determinant of the Hückel technique and applied to cyclopentadienyl system framework so as to communicate their orbital energies of cyclopentadienyl system, also to communicate its electron and charge density in terms of stable configuration of a system. It is settled by the Hückel strategy and applied by the assumptions for nearby comparability such as coulomb integrals, exchange integrals and overlap integrals. This simple way hypothetical strategy will allow to graduate and post graduate understudies to understanding the investigation of stable configuration, electron and charge density and also other parameters.

Study of Axi-Symmetric Vibrations in a Micropolar Transversely Isotropic Layer

The present investigation deals with the propagation of circular crested Lamb waves in a homogeneous micropolar transversely isotropic medium. Secular equations for symmetric and skew-symmetric modes of wave propagation in completely separate terms are derived. The amplitudes of displacements and microrotation are computed numerically for magnesium as a material and the dispersion curves, amplitudes of displacements and microrotation for symmetric and skew-symmetric wave modes are presented graphically to evince the effect of anisotropy. Some special cases of interest are also deduced.

Long-term orbital and rotational motions of Ceres and Vesta

Context. The dwarf planet Ceres and the asteroid Vesta have been studied by the Dawn space mission. They are the two heaviest bodies of the main asteroid belt and have different characteristics. Notably, Vesta appears to be dry and inactive with two large basins at its south pole. Ceres is an ice-rich body with signs of cryovolcanic activity. Aims. The aim of this paper is to determine the obliquity variations of Ceres and Vesta and to study their rotational stability. Methods. The orbital and rotational motions have been integrated by symplectic integration. The rotational stability has been studied by integrating secular equations and by computing the diffusion of the precession frequency. Results. The obliquity variations of Ceres over [−20 : 0] Myr are between 2° and 20° and the obliquity variations of Vesta are between 21° and 45°. The two giant impacts suffered by Vesta modified the precession constant and could have put Vesta closer to the resonance with the orbital frequency 2s6 − sV. Given the uncertainty on the polar moment of inertia, the present Vesta could be in this resonance where the obliquity variations can vary between 17° and 48°. Conclusions. Although Ceres and Vesta have precession frequencies close to the secular orbital frequencies of the inner planets, their long-term rotations are relatively stable. The perturbations of Jupiter and Saturn dominate the secular orbital dynamics of Ceres and Vesta and the perturbations of the inner planets are much weaker. The secular resonances with the inner planets also have smaller widths and do not overlap, contrary to the case of the inner planets.

Lamb waves in multilayered media: 3D and 6D formalisms

A mathematical model for analyzing Lamb waves propagating in stratified media with arbitrary elastic anisotropy is worked out. The model incorporates a combined Fundamental Matrix (FM) and Modified Transfer Matrix (MTM) methods. Multilayered unbounded plates with different types of boundary conditions imposed on the outer surfaces are considered. Closed form fundamental matrices and secular equations for dispersion relations are derived.

On degeneracy of dispersive waves at the bulk wave velocities

Degeneracy of the linear dispersion wave equation at the phase velocities coinciding with the bulk wave velocities is observed and analysed. Spectral analysis of Pochhammer – Chree equation is performed. The corrected analytical solutions for components of the displacement fields are constructed, accounting degeneracy of the secular equations and the corresponding solutions.