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Schrödinger Equation for a System with Exponential-Type Restoring Force Function

J. Akande, L. H. Koudahoun, Kolawolé Kêgnidé Damien Adjaï, Y. J. F. Kpomahou, M. D. Monsia

Published Sep 1, 2016• 0 citations
OpenAlex

Abstract

In this paper the Schrodinger equation is derived for a dynamical system with exponential-type restoring force function.

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In physics, harmonic motion is among the most representative types of motion. A simple harmonic oscillator is often the source of any vibration with a restoring force proportional to Hooke’s law. Every minimum potential has a solution in the form of the harmonic oscillator potential. Little oscillations at the minimum are characteristic of almost all natural potentials and of many quanta mechanical systems. Harmonic motion is an essential building block for these more complex uses. The Schrödinger equation is a defining feature of the harmonic oscillator. Here, we demonstrate that the time-frequency plane is a useful tool for analyzing their dynamics. We numerically integrate several examples involving different input forces and demonstrate that the oscillations are clearly displayed and easily interpretable in the time-frequency plane.
We construct an explicit solution of the Cauchy initial value problem for the n-dimensional Schrödinger equation with certain time-dependent Hamiltonian operator of a modified oscillator. The dynamical SU(1,1) symmetry of the harmonic oscillator wave functions, Bargmann’s functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankel-type integral transform, and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as by-products.