Comparison of Finite Differences and WKB Method for Approximating Tunneling Times of the One Dimensional Schrödinger Equation Student: Yael Elmatad Overview The goal of this research was to examine the quantum mechanical phenomenon of tunneling with respect to the Schrödinger Equation. A quantum mechanical wave is said to “tunnel” when it travels (propagates) through a classically forbidden reg...

Group theoretical methods are used to study some properties of the Riccati equation, which is the only differential equation admitting a nonlinear superposition principle. The Wei–Norman method is applied for obtaining the associated differential equation in the group SL(2,R). The superposition principle for first order differential equation systems and Lie-Scheffers theorem are also analysed f...

in this paper we have solved analytically the schrödinger equation for a microscopic particle in a one-dimensional box with two infinite walls, which the potential function inside it, has a linear form. based on the solutions of this special quantum mechanical system, we have shown that as the quantum number approaches infinity the expectation values of microscopic particle position and square ...

In this paper, the numerical solution methods of one- particale, one – dimensional time- independentSchrodinger equation are presented that allows one to obtain accurate bound state eigen values andeigen functions for an arbitrary potential energy function V(x). These methods included the FEM(Finite Element Method), Cooly, Numerov and others. Here we considered the Numerov method inmore details...

The Principle of Least Action is used with a simple Lagrangian density, involving second-order derivatives of the wave function, to obtain the Schrödinger equation. A Hamiltonian density obtained from this simple Lagrangian density shows that Hamilton’s equations also give the Schrödinger equation. This simple Lagrangian density is equivalent to a standard Lagrangian density with first-order de...

Article history: Received 3 November 2009 Accepted 21 November 2009 Available online xxxx

The problem of solving the time–independent Schrödinger equation for the motion of an electron of mass μ and charge −e (e > 0) in the field of two fixed Coulomb centers has been the subject of extensive studies in theoretical physics and quantum computation. In the present paper, after making a series of coordinate transformations, we apply the qualitative theory of nonlinear differential equat...

The S-wave time-independent Schrödinger equation with an isotropic velocity-dependent potential is considered .We have used perturbation theory to calculate the scattering phase shifts when the energy is changed by a small amount ∆E from an arbitrary unperturbed value E0. The validity of our results were tested by comparing the perturbed phase shifts to those obtained exactly by solving Schrödi...

We study bifurcations in a spatially extended nonlinear system representing population dynamics with the help of analytic calculations based on the time-independent Schrödinger equation for a quantum particle subjected to a uniform gravitational field. Despite the linear character of the Schrödinger equation, the result we obtain helps in the understanding of the onset of abrupt transitions lea...

The aim of this paper is to develop new optimized Schwarz algorithms for the one dimensional Schrödinger equation with linear or nonlinear potential. After presenting the classical algorithm which is an iterative process, we propose a new algorithm for the Schrödinger equation with time-independent linear potential. Thanks to two main ingredients (constructing explicitly the interface problem a...