On the Use of B-Splines as Ritz Variational Basis Functions to Solve the Schrodinger Equation (TISE) for a constrained free Quantum Particle

B-Splines as piecewise adaptation of Bernstein polynomials (aka, B-polys) are widely used as Ritz variational basis functions in solving many problems in the fields of quantum mechanics and atomic physics. In this paper they are used to solve the 1-D stationary Schrodinger equation (TISE) for a free quantum particle subject to a fixed domain length by using the Python software SPLIPY with different sets of computation parameters. In every case it was found that over 60 percent of energy levels had excellent accuracy thereby proving that the use of B-spline collocation is a preferred method.

Electron-Beam Domain Patterning on the Nonpolar Surfaces of LiNbO3 Crystals and in Optical Waveguides Formed on the Nonpolar Surfaces

In this review our recent results on the electron-beam domain writing (EBDW) on the nonpolar surfaces of LiNbO3 crystals of different compositions are presented. The obtained results permitted us to relate the main characteristics of domain formation (the domain sizes and velocity Vf of the frontal motion) to the irradiation conditions (the accelerating voltage U of scanning electron microscopy, EB-current I, the inserted charge Q). The domain depth Td is controlled by U via the electron penetration depth; the domain length Ld increases linearly with Q owing to the domain frontal growth by the viscous friction law. In optical waveguides, the matching of the Td value with the waveguide thickness D provides optimal values of the waveguide conversion to the second harmonic

Numerical Solution for the Extrapolation Problem of Analytic Functions

In this work, a numerical solution for the extrapolation problem of a discrete set of n values of an unknown analytic function is developed. The proposed method is based on a novel numerical scheme for the rapid calculation of higher order derivatives, exhibiting high accuracy, with error magnitude of O(10−100) or less. A variety of integrated radial basis functions are utilized for the solution, as well as variable precision arithmetic for the calculations. Multiple alterations in the function’s direction, with no curvature or periodicity information specified, are efficiently foreseen. Interestingly, the proposed procedure can be extended in multiple dimensions. The attained extrapolation spans are greater than two times the given domain length. The significance of the approximation errors is comprehensively analyzed and reported, for 5832 test cases.

Nearly Single-Cycle Terahertz Pulse Generation in Aperiodically Poled Lithium Niobate

In the present work, an opportunity of nearly single-cycle THz pulse generation in aperiodically poled lithium niobate (APPLN) crystal is studied. A radiating antenna model is used to simulate the THz generation from chirped APPLN crystal pumped by a sequence of femtosecond laser pulses with chirped delays (m = 1, 2, 3 …) between adjacent pulses. It is shown that by appropriately choosing Δtm, it is possible to obtain temporal overlap of all THz pulses generated from positive (or negative) domains. This results in the formation of a nearly single-cycle THz pulse if the chirp rate of domain length δ in the crystal is sufficiently large. In the opposite case, a few cycle THz pulses are generated with the number of the cycles depending on δ. The closed-form expression for the THz pulse form is obtained. The peak THz electric field strength of 0.3 MV/cm is predicted for APPLN crystal pumped by a sequence of laser pulses with peak intensities of the separate pulse in the sequence of about 20 GW/cm2. By focusing the THz beam and increasing the pump power, the field strength can reach values in the order of few MV/cm.

Nearly Single-Cycle Terahertz Pulse Generation in Aperiodically Poled Lithium Niobate

In present work an opportunity of nearly single-cycle THz pulse generation in aperiodically poled lithium niobate (APPLN) crystal is studied. A radiating antenna model is used to simulate the THz generation from chirped APPLN crystal pumped by a sequence of femtosecond laser pulses with chirped delays m (m = 1, 2, 3 …) between adjacent pulses. It is shown that by appropriative choosing m it is possible to obtain temporally overlap of all THz pulses generated from positive (or negative) domains. It results in the formation of a nearly single-cycle THz pulse, if the chirp rate of domain length  in the crystal is sufficiently large. In opposite case, a few cycle THz pulses are generated with the number of the cycles depending on . The closed form expression for THz pulse form is obtained. The peak THz electric field strength of 0.3 MV/cm is predicted for APPLN crystal pumped by the sequence of laser pulses with peak intensity of the separate pulse in the sequence about 20 GW/cm2. By focusing the THz beam and by increasing the pump power the field strength can reach values of an order of few MV/cm.

Stabilization of Traffic Flow With a Leading Autonomous Vehicle

This paper develops boundary control law for autonomous vehicles to stabilize the stop-and-go traffic on freeway. The macroscopic traffic dynamics is described by the Aw-Rascle-Zhang (ARZ) model in a time and state dependent domain. The leading autonomous vehicle aims to regulate the traffic behind it to uniform equilibrium and the domain length of the traffic to a setpoint. The traffic density and speed is governed by second-order, nonlinear hyperbolic partial differential equations (PDEs), coupled with a state-dependent ODE for the leading autonomous vehicle. The actuation is the speed of autonomous vehicle at the moving front boundary of the domain. We linearize the system around a uniform velocity and density reference and certain physical properties are discussed for the model validity. The linearized model describes the dynamics of deviations of density and velocity from the reference. By transforming the linearized system in a moving coordinate, we obtain a domain with a fixed boundary at one end and a state-dependent moving boundary at the other end. The well-posedness of the system is proved and the linear instability of open-loop system is shown. We further map the system to Riemann variables and based on it, propose the boundary feedback control law actuated by the leading autonomous vehicle. The exponential stability of state variables in L2 norm and convergence to the setpoint domain length is achieved for the closed-loop system.