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...

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...

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

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...

t to )b x2. This corresponds to a time-dependent mass (TM) Schrödinger equation. We give the specific transformations to a different time-dependent quadratic Schrödinger equation (TQ) and to a different time-dependent oscillator (TO) equation. For each Schrödinger system, we give the Lie algebra of space-time symmetries, the number states, the squeezed-state 〈x〉 and 〈p〉 (with their classical mo...