Advances in Mathematical Physics

Volume 2016 (2016), Article ID 8181927, 20 pages

http://dx.doi.org/10.1155/2016/8181927

## An Efficient Numerical Method for the Solution of the Schrödinger Equation

^{1}School of Information Engineering, Chang’an University, Xi’an 710064, China^{2}Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia^{3}Laboratory of Computational Sciences, Department of Informatics and Telecommunications, Faculty of Economy, Management and Informatics, University of Peloponnese, 221 00 Tripolis, Greece

Received 27 May 2016; Accepted 5 July 2016

Academic Editor: Maria L. Gandarias

Copyright © 2016 Licheng Zhang and Theodore E. Simos. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The development of a new five-stage symmetric two-step fourteenth-algebraic order method with vanished phase-lag and its first, second, and third derivatives is presented in this paper for the first time in the literature. More specifically we will study the development of the new method, the determination of the local truncation error (LTE) of the new method, the local truncation error analysis which will be based on test equation which is the radial time independent Schrödinger equation, the stability and the interval of periodicity analysis of the new developed method which will be based on a scalar test equation with frequency different than the frequency of the scalar test equation used for the phase-lag analysis, and the efficiency of the new obtained method based on its application to the coupled Schrödinger equations.

#### 1. Introduction

The approximate solution of the close-coupling differential equations of the Schrödinger type is studied in this paper. The above-mentioned problem has the following form: where and and the boundary conditions are as follows:where and are the spherical Bessel and Neumann functions. We will examine the case in which all channels are open (see [1]).

Defining a matrix and diagonal matrices , by (see [1])we obtain a new form of the asymptotic condition (3):

In several scientific areas (e.g., quantum chemistry, theoretical physics, material science, atomic physics, and molecular physics) there exists a real problem which is the rotational excitation of a diatomic molecule by neutral particle impact. The mathematical model of this problem can be expressed with close-coupling differential equations arising from the Schrödinger equation. Denoting, as in [1], the entrance channel by the quantum numbers , the exit channels by , and the total angular momentum by , we find that where is the kinetic energy of the incident particle in the center-of-mass system, is the moment of inertia of the rotator, and is the reduced mass of the system.

The above-described problem will be solved numerically via finite difference method of the form of special multistep method.

The multistep finite difference method has the general formwhere are distinct points within the integration area and given by is* the step size or step length of the integration*.

*Remark 1. *A method (8) is called symmetric multistep method or symmetric -step method if and .

If we apply the symmetric -step method , to the scalar test equationwe obtain the difference equationand the associated characteristic equationwhere , is the step length, and are polynomials of .

We give some definitions.

*Definition 2 (see [2]). *For a symmetric -step method with characteristic equation given by (11) one will say that it has an interval of periodicity if, for all , the roots , of (11) satisfywhere is a real function of .

*Definition 3 (see [2]). *One calls P-stable method a multistep method if its interval of periodicity is equal to .

*Definition 4. *One calls singularly P-stable method a multistep method if its interval of periodicity is equal to (where is a set of distinct points).

*Definition 5 (see [3, 4]). *For a symmetric -step method with the characteristic equation given by (11), one defines as the phase-lag the leading term in the expansion of Then, if the quantity as , the order of the phase-lag is .

*Definition 6 (see [5]). *One calls symmetric -step method* phase-fitted* if its phase-lag is equal to zero.

Theorem 7 (see [3]). *The symmetric -step method with characteristic equation given by (11) has phase-lag order and phase-lag constant given by*

#### 2. The New Five-Stage Fourteenth-Algebraic Order P-Stable Two-Step Method with Vanished Phase-Lag and Its First, Second, and Third Derivatives

We consider the following family of five-stage symmetric two-step methods: where , , , , , , , and , , are free parameters.

Application of the above-mentioned method (15) to the scalar test equation (9) leads to the difference equation (10) and the characteristic equation (11) with and

If we request the new method (15) to have vanished phase-lag and its first, second, and third derivatives, then the following system of equations is obtained:where , are given in Appendix A.

Now solving the system of (17) we determine the other coefficients of the new obtained three-stage two-step method:where the formulae , , and are given in Appendix B.

Additionally to the above formulae for the coefficients , , we give also the following Taylor series expansions of these coefficients, for the case of heavy cancelations for some values of in the formulae given by (18): The behavior of the coefficients is given in Figure 1.