Physics Research International
Volume 2015 (2015), Article ID 609495, 10 pages
http://dx.doi.org/10.1155/2015/609495

Research Article

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

Artit Hutem1,2 and Piyarut Moonsri³

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

Figure 1: The harmonics oscillator potential is perturbed by Cosine asymmetric potential, with , , and .

with that is called the Cosine asymmetric potential [22], where , , and are positive constants. Substituting the harmonics oscillator Cosine asymmetric potential from (3) into (2) leads to the following equation:

For our approach (4) can be solved in the numerical shooting method. It is convenient to simplify the arithmetic involved in the shooting solution. We define some new dimensionless variables. Then the position variable is replaced with the dimensionless variable :

With this definition, the second-derivative term can be written as

Substituting in for in terms of and setting and setting into (4), we can rewrite the time-independent Schrödinger equation completely in terms of as follows:

Also, the time-independent potential in terms of the new variable is given by

For the finite difference method, (7) is rewritten in the form of many small segment in the domain length. The second-derivative of the first term in (7) can be approximated in finite difference form as follows:

We can obtain the form of the time-independent Schrödinger equation in terms of finite difference by substituting (9) into (7), and we get

where . The special potential given by harmonics oscillator Cosine asymmetric potential has been used in calculating (10) in the mathematica program (see Section 3).

3. Numerical Shooting Method and Results

We construe the new variable to be used in calculating the ground-state energy eigenvalue, wave function, and the time-independent correlation function of the harmonics oscillator Cosine asymmetric potential.

(1) is the start position in the analysis range.(2) is the ultimate position in the analysis range.(3) is any position in the analysis range.(4) is a number of very small bars in the analysis range.(5) is the length of very small bars so that

The logic of the numerical shooting method evaluation of energy eigenvalue, eigenfunction, and time-independent correlation function for the harmonics oscillator Cosine asymmetric potential is as follows.

(i)Input values and in mathematica program for the harmonics oscillator Cosine asymmetric potential.(ii)Input the period amount.(iii)Input (10) into mathematica program.

Find the initial value for calculation. Input the initial condition by setting for the position imprisons and set from the slope of positions 1 and 2, so that

By inputting and as two initial values for calculation, we can find from (10). In the same way, we can find by substituting and in the equation. As we keep doing this, we can find (see Figure 2 in [12]).

(i)The next task is to calculate wave function in (10) () so that it approaches zero as closely as desired. Normally, we assign a small value as the standard to make sure that wave function in (10) gets close enough to zero. For example, if , we stop the calculation and accept the final energy as the numerical solution.(ii)Plot the wave function by the graph related to .(iii)Plot the wave function that is normalized by the graph related to .(iv)Plot the probability of the average atomic density for the harmonics oscillator Cosine asymmetric potential.(v)Input values and in the mathematica program for the harmonics oscillator potential.(vi)Input equation into the mathematica program for the harmonics oscillator potential.(vii)For example, if , we stop the evaluation and accept the final energy as the numerical solution.(viii)Plot the wave function that is normalized for the harmonics oscillator potential by the graph related to .(ix)Plot the probability of the average atomic density for the harmonics oscillator potential.(x)Plot the time-independent atomic density fluctuation [16] by the graph related to .(xi)Plot the time-independent correlation function [16].

4. Conclusion

In conclusion, we then represented the method by obtaining numerical solution of the one-dimensional harmonic oscillator, perturbed from a set of the Cosine asymmetric potentials. In our calculation, we can obtain the time-independent correlation function corresponding with the Green functions techniques [16]. Although the numerical shooting method does not use complicated formulas, it requires much less computational effort when compared to the Green functions techniques. Generally, regarded as one of the most efficient methods, the numerical shooting method [12] gives very accurate results because it integrates the Schrdinger equation directly, though in the numerical sense.

In this case, the wave function of the harmonics oscillator Cosine asymmetric potential is different from that in case of a typical harmonics oscillator potential () (see Figures 2–4). In this case, the time-independent correlation function () of the harmonics oscillator Cosine asymmetric potential via numerical shooting method and the intensity correlation experiment by reference [23] have the same appearance (see Figures 5–7). From Figures 2(a)–2(c), if the values of the amplitude barrier potential incline, the ground-state energy eigenvalues (En) lessen, but the amplitude of the wave function has supplement and in Figures 5(a)–5(c) the values of the time-independent correlation function () (part of positive) incline and the time-independent correlation function is underdamped motion. From Figures 2(d)–2(f) if the values of increase, the ground-state energy eigenvalues (En) have supplement, but in Figures 5(d)–5(f) the values of the time-independent correlation function lessen.

(a) En = −0.562660496449, , , , and

(b) En = −1.335972183092, , , , and

(c) En = −2.266994942969, , , , and

(d) En = 0.522331835003, , , , and

(e) En = 0.664583592675, , , , and

(f) En = 0.751492211595, , , , and

Figure 2: Figures (a)–(c) show plot of the time-independent wave function for the ground-state energy in case of varying (the amplitude of barrier potential) in harmonics oscillator Cosine asymmetric potential. Figures (d)–(f) show the wave function for the ground-state energy with varying harmonics oscillator Cosine asymmetric potential.

(a) En = 0.357420646119, , , , and

(b) En = 0.287420653214, , , , and

(c) En = 0.197420666459, , , , and

(d) En = 4.299406788515, , , , and

(e) En = 4.229409700958, , , , and

(f) En = 4.139415930374, , , , and

Figure 3: Schematic representation for behavior of the wave function for the ground-state energy and the second excited-state energy in case of the harmonics oscillator Cosine asymmetric potential with varying .

(a) En = 2.331415049731, , , , and

(b) En = 2.261415204498, , , , and

(c) En = 2.171415559947, , , , and

(d) En = 6.260444417595, , , , and

(e) En = 6.190472530201, , , , and

(f) En = 6.100527633912, , , , and

Figure 4: Plot of the wave function for the first excited-state energy and the third excited-state energy in case of the harmonics oscillator Sine asymmetric potential with varying .

(a) = 0.007, = −0.0025, , , , and

(b) = 0.0085, = −0.003, , , , and

(c) = 0.010, = −0.004, , , , and

(d) = 0.0016, = −0.0007, , , , and

(e) = 0.0014, = −0.0006, , , , and

(f) = 0.0013, = −0.0006, , , , and

Figure 5: Figures (a)–(c) show plot of the time-independent correlation function for the ground-state energy in case of varying (the amplitude of barrier potential) in harmonics oscillator Cosine asymmetric potential. Figures (d)–(f) show the correlation function for the ground-state energy with varying harmonics oscillator Cosine asymmetric potential.

(a) = 0.0036, = −0.0014, , , , and

(b) = 0.006, = −0.0025, , , , and

(c) = 0.0085, = −0.0055, , , , and

(d) = 0.0032, = −0.002, , , , and

(e) = 0.005, = −0.0035, , , , and

(f) = 0.0065, = −0.0045, , , , and

Figure 6: Schematic representation for behavior of the time-independent correlation function for the ground-state energy and the second excited-state energy in case of the harmonics oscillator Cosine asymmetric potential with varying .

(a) = 0.0034, = −0.0022, , , , and

(b) = 0.0055, = −0.004, , , , and

(c) = 0.008, = −0.005, , , , and

(d) = 0.0032, = −0.0022, , , , and

(e) = 0.0045, = −0.003, , , , and

(f) = 0.0055, = −0.003, , , , and

Figure 7: Plot of the time-independent correlation function for the first excited-state energy and the third excited-state energy in case of the harmonics oscillator Cosine asymmetric potential with varying .

From Figures 3(a)–3(f), if the values of the parameter increase, the ground-state and the second excited-state energy eigenvalues (En) lessen, but in Figures 6(a)–6(f) the values of the time-independent correlation function have supplement. From Figures 4(a)–4(f), if the values of the parameter increase, the first excited-state and the third excited-state energy eigenvalues (En) lessen, but in Figures 7(a)–7(f) the values of the time-independent correlation function have supplement. From Figures 8(a)–8(c) if the values of the , , and parameters increase, the time-independent atomic density fluctuation for the ground-state has supplement.

(a) , , , , , and

(b) , , , , , and

(c) , , , , , and

(d) , , , , , and

(e)

(f)

Figure 8: Schematic representation for behavior of the time-independent atomic density fluctuation for the ground-state energy in case of the harmonics oscillator Cosine asymmetric potential with varying and .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Artit Hutem and Piyarut Moonsri would like to thank the Institute Research and Development Phetchabun Rajabhat University and Physics Division, Faculty of Science and Technology, Phetchabun Rajabhat University, for partial support.

References

G. A. Dobrovolsky and R. S. Tutik, “Regularization of the WKB integrals,” Journal of Physics A: Mathematical and General, vol. 33, no. 37, pp. 6593–6599, 2000.

M. I. Jaghob, “Perturbation theory for isotropic velocity-dependent potentials: bound-states case,” The European Physical Journal A—Hadrons and Nuclei, vol. 27, no. 1, pp. 99–103, 2006.

E. Z. Liverts and V. B. Mandelzweig, “Accurate analytic presentation of solution for the spiked harmonic oscillator problem,” Annals of Physics, vol. 322, no. 9, pp. 2211–2232, 2007.

N. Saad, R. L. Hall, and H. Ciftci, “Study of a class of non-polynomial oscillator potentials,” Journal of Physics A: Mathematical and General, vol. 39, no. 24, pp. 7745–7756, 2006.

M. C. Zhang, “Exact bound state solutions of the Klein-Gordon particle in Hulthén potential,” Chinese Physics B, vol. 17, no. 9, pp. 3214–3216., 2008.

M. C. Zhang and Z. B. Wang, “Exact solutions of the Klein-Gordon equation with Makarov potential and a recurrence relation,” Chinese Physics, vol. 16, no. 7, article 1863, 2007.

C. R. Handy, H. Hayes, D. V. Stephens, J. Joshua, and S. Summerour, “Application of the eigenvalue moment method to important one-dimensional quantum systems,” Journal of Physics. A. Mathematical and General, vol. 26, no. 11, pp. 2635–2649, 1993.

Z. Cao, “Pressure-induced phase in tetragonal two-dimensional polymeric ,” Physical Review A, vol. 63, no. 5, Article ID 054103, 4 pages, 2001.

A. Hutem and C. Sricheewin, “Ground-state energy eigenvalue calculation of the quantum mechanical well via analytical transfer matrix method,” European Journal of Physics, vol. 29, no. 3, pp. 577–588, 2008.

H. Ying, Z. Fan-Ming, Y. Yan-Fang, and L. Chun-Fang, “Energy eigenvalues from an analytical transfer matrix method,” Chinese Physics B, vol. 19, no. 4, Article ID 040306, 2010.

S. Boonchui and A. Hutem, “Excited-state energy eigenvalue and wave-function evaluation of the Gaussian asymmetric double-well potential problem via numerical shooting method 2,” Journal of Mathematical Chemistry, vol. 50, no. 8, pp. 2103–2119, 2012.

S. Boonchui and A. Hutem, “Excited-state energy eigenvalue and wave-function evaluation of the Gaussian symmetric double-well potential problem via numerical shooting method 1,” Journal of Mathematical Chemistry, vol. 50, no. 6, pp. 1582–1597, 2012.

R. Grobe, K. Rza̧zewski, and J. H. Eberly, “Measure of electron-electron correlation in atomic physics,” Journal of Physics B: Atomic, Molecular and Optical Physics, vol. 27, no. 16, pp. L503–L508, 1994.

J. E. Lye, L. Fallani, M. Modugno, D. S. Wiersma, C. Fort, and M. Inguscio, “Bose-Einstein condensate in a random potential,” Physical Review Letters, vol. 95, no. 7, Article ID 070401, 4 pages, 2005.

P. Henseler and B. Shapiro, “Density correlations in cold atomic gases: atomic speckles in the presence of disorder,” Physical Review A, vol. 77, Article ID 033624, 2008.

N. Cherroret and S. Skipetrov, “Long-range correlations of density in a Bose-Einstein condensate expanding in a random potential,” Physical Review Letters, vol. 101, no. 19, Article ID 190406, 2008.

N. Cherroret and S. E. Skipetrov, “Effect of interactions on the diffusive expansion of a Bose-Einstein condensate in a three-dimensional random potential,” Physical Review A, vol. 79, Article ID 063604, 2009.

L. Beilin, E. Gurevich, and B. Shapiro, “Diffusion of cold-atomic gases in the presence of an optical speckle potential,” Physical Review A, vol. 81, Article ID 033612, 2010.

L. Pezze, M. Robert-de-Saint-Vincent, T. Bourdel et al., “Regimes of classical transport of cold gases in a two-dimensional anisotropic disorder,” New Journal of Physics, vol. 13, Article ID 095015, 2011.

A. Asaithambi, “Numerical solution of the one-dimensional time-independent Schrödinger's equation by recursive evaluation of derivatives,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4400–4405, 2010.

V. Ledoux and M. van Daele, “Automatic computation of quantum-mechanical bound states and wavefunctions,” Computer Physics Communications, vol. 184, no. 4, pp. 1287–1296, 2013.

H. Ciftci, R. L. Hall, and N. Saad, “Exact and approximate solutions of Schrödinger's equation for a class of trigonometric potentials,” Central European Journal of Physics, vol. 11, no. 1, pp. 37–48, 2013.

J. Kasprzak, M. Richard, A. Baas et al., “Second-order time correlations within a polariton bose-einstein condensate in a CdTe microcavity,” Physical Review Letters, vol. 100, no. 6, Article ID 067402, 4 pages, 2008.