Physics Research International

Volume 2015, Article ID 609495, 10 pages

http://dx.doi.org/10.1155/2015/609495

## Evaluated Excited-State Time-Independent Correlation Function and Eigenfunction of the Harmonics Oscillator Cosine Asymmetric Potential via Numerical Shooting Method

^{1}Physics Division, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun 67000, Thailand^{2}Department of Physics, Faculty of Science, Kasetsart University, Bangkok 10900, Thailand^{3}Chemistry Division, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun 67000, Thailand

Received 6 August 2014; Revised 5 January 2015; Accepted 12 January 2015

Academic Editor: Angel Rubio

Copyright © 2015 Artit Hutem and Piyarut Moonsri. 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

We aimed to evaluate the ground-state and excite-state energy eigenvalue (En), wave function, and the time-independent correlation function of the atomic density fluctuation of a particle under the harmonics oscillator Cosine asymmetric potential (Saad et al. 2013). Instead of using the 6-point kernel of 4 Green’s function (Cherroret and Skipetrov, 2008), averaged over disorder, we use the numerical shooting method (NSM) to solve the Schrödinger equation of quantum mechanics system with Cosine asymmetric potential. Since our approach does not use complicated formulas, it requires much less computational effort when compared to the Green functions techniques (Cherroret and Skipetrov, 2008). We show that the idea of the program of evaluating time-independent correlation function of atomic density is underdamped motion for the Cosine asymmetric potential from the numerical shooting method of this problem. Comparison of the time-independent correlation function obtained from numerical shooting method by Boonchui and Hutem (2012) and correlation function experiment by Kasprzak et al. (2008). We show the intensity of atomic density fluctuation in harmonics oscillator Cosine asymmetric potential by numerical shooting method.

#### 1. Introduction

Most problems encountered in quantum mechanics cannot be solved exactly. Exact solutions of the Schrödinger equation exist only for a few idealized systems. To solve general problems, one must resort to approximation methods. A variety of such methods have been developed, and each has its own area of applicability. There exist several means to study them, for example, Wentzel-Kramers-Brillouin [1], perturbation [2], the quasilinearization method [3], the variational method [4], function analysis [5, 6], the eigenvalue moment method [7], the analytical transfer matrix method [8–10], and numerical shooting method [11, 12].

Grobe et al. 1994 [13] proposed a criterion to determine the numerical degree of global correlation function of the multiparities quantum system. They applied this method to several situations, including electron-atom scattering and strong-field photoionization. Lye et al. 2005 [14] discussed the effect of a weak random potential, indicated by stripes in the expanded density profile of the Bose-Einstein Condensate and damped dipole oscillations. Henseler and Shapiro 2008 [15] defined the disorder-induced intensity-intensity correlation function, , for the Bose-Einstein Condensate for Fermi gas. Cherroret and Skipetrov 2008 [16] showed decay of the average atomic density as a function of time. The density reaches a maximum at the arrival time , where is the diffusion coefficient in random potentials. Now, a few works have concerned the expansion of Bose-Einstein Condensate in three-dimensional potentials and evaluate correlation function. Cherroret and Skipetrov 2009 [17] had shown the typical diffusion coefficient of the Bose-Einstein Condensate in a three-dimensional random potential. Beilin et al. 2010 [18] considered diffusion of cold-atomic Fermi gas in the presence of a random optical speckle potential. Pezze et al. 2011 [19] numerically studied the dynamics regimes of classical transport of cold atoms gases in a two-dimensional anisotropic disorder potential. In this paper, we consider approximation methods that deal with stationary states corresponding to time-independent Hamiltonian. To study problem of stationary states, we focus on one approximation method: numerical shooting method useful to evaluate wave function and time-independent correlation function of a particle around attraction by the harmonics oscillator with Cosine asymmetric potential. The scheme of the paper is as follows. In Section 2, we write the basic time-independent Schrödinger equation in terms of finite difference and the harmonics oscillator Cosine asymmetric potential in terms of the new variable is given by where is the Cosine asymmetric potential. In Section 3, we show the idea of writing a program for evaluating energy eigenvalue wave function and correlation function of atomic density for the Cosine asymmetric potential via the numerical shooting method (Asaithambi, Ledoux and van Daele, Boonchui and Hutem [12, 20, 21]). Section 4 contains our conclusions.

#### 2. Time-Independent Schrödinger Equation in Finite Difference Formula for Harmonics Oscillator Cosine Asymmetric Potential

We consider a particle of mass moving on the -axis in a time-independent potential . The time-independent Schrödinger equation corresponding to this one-dimensional motion is where is the total energy eigenvalues of the particle. The solution of this equation yields the allowed energy eigenvalues and the corresponding wave function . To solve this partial differential equation, we need to specify the potential as well as the boundary condition; the boundary condition can be obtained from the physical requirement of the system.

Suppose a particle is bound state to around of attraction by the harmonics oscillator Cosine asymmetric potential (see Figure 1):