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Ciftci Hakan, Kisoglu H F. Application of asymptotic iteration method to a deformed well problem. Chinese Physics B, 2016, 25(3): 030201  
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Application of asymptotic iteration method to a deformed well problem
Ciftci Hakan1, †, , Kisoglu H F2, ‡,
Gazi Üniversitesi, Fen Fakültesi, Fizik Bölümü 06500 Teknikokullar Ankara, Türkiye
>Maritime Faculty, Department of Basic Sciences, Mersin University, Mersin, Turkey

 

† Corresponding author. E-mail: hciftci@gazi.edu.tr

‡ Corresponding author. E-mail: hasanfatihk@mersin.edu.tr

Abstract
Abstract

The asymptotic iteration method (AIM) is used to obtain the quasi-exact solutions of the Schrödinger equation with a deformed well potential. For arbitrary potential parameters, a numerical aspect of AIM is also applied to obtain highly accurate energy eigenvalues. Additionally, the perturbation expansion, based on the AIM approach, is utilized to obtain simple analytic expressions for the energy eigenvalues.

PACS: 02.30.Hq;02.60.Cb;03.65.Ge;03.65.–w;
Keyword:asymptotic iteration method;quasi-exact solutions;perturbation method;approximate solutions;
1. Introduction

The solution of the Schrödinger equation, which is the main scope of quantum mechanics, depends upon the configuration of the system held, i.e., the behavior of the potential. One of the advantages of the asymptotic iteration method (AIM), introduced in Refs. [1] and [2], is that it can be applied to solve the eigenvalue problem exactly or approximately for numerous potentials.[3–14]

In the framework of perturbation theory, AIM has the advantage of obtaining simple analytic expressions for eigenvalues and eigenfunctions. Although, the standard perturbation theory allows us to construct both analytic expressions for the eigenvalues and the eigenfunctions, the eigenfunctions of the unperturbed Hamiltonian (the exactly solvable part) are essential for the construction. It is usually a challenging task to obtain expressions beyond the first order correction of the perturbative expansion. The advantage of AIM is that the coefficients in the perturbation expansions can be calculated directly without a prior knowledge of the exact eigenfunctions of the unperturbed Hamiltonian.[5,8,15–17]

In the present work, the AIM is used to obtain both the quasi-exact and approximate (numeric) solutions of the Schrödinger equation for a deformed well problem. A brief introduction to the AIM is summarized in Section 2. The approximate and quasi-exact eigenvalue solutions of the Schrödinger equation with the potential

are obtained in Section 3, where A and B are real coefficients and γ ranges over the interval (0,∞). In Section 4, we apply the perturbation method in the framework of AIM to construct perturbative expansions for potential (1).

2. Asymptotic iteration method

In this section, we briefly outline the asymptotic iteration method, for more details we refer the reader to the original publication.[1] The AIM was introduced to solve the second-order homogeneous linear differential equations

where the functions λ0 and s0 have continuous derivatives in the domain of definition. According to the asymptotic aspect of the method, if

for sufficiently large n > 0, then the general solution of Eq. (2) reads

where

Eigenvalues E of the Schrödinger equation are the roots (zeros) of the following termination condition:

The eigenvalue problem (2) is said to be an exactly solvable (in a closed form) problem if the termination condition (5) is independent of the variable x. If δn ≡ δn(E;x) = 0, then a suitable initial value of x is necessary to start the AIM iteration. Such starting value may be found as the x coordinate of the minimum point of the potential V(x).[3,5]

3. The deformed well potential

In this section, we focus our attention on the deformed well potential

where A, B, and γ are the potential parameters and γ ∈ [0,∞). The one-dimensional Schrödinger equation then reads

Because of the vanishing of the wave function ψ at the end points, it is appropriate to assume the following form of Ψ:

Upon substituting Eq. (7) into Eq. (6), we directly obtain

This equation can be written using a change of the variable x = arccos y as

where ω = E −(γ + 1)2, ξ = A + 2σB, and 2σ = 2γ + 3. The last equation is in a form suitable to initiate AIM sequences with

3.1. Quasi-exact solutions

The first few iterations of δn(x) = sn(x)λn−1(x) − λn(x)sn−1(x) suggest the necessary condition ξ = −2nB with n = 0,1,2,…. Consequently, the two parameters A and B are related by the equation

Thus for n = 0, we have

while for n = 1, we have

which yields for the eigenvalues the following expressions:

For higher values of n, the expressions for the eigenenergies become messy. However, the exact results can be immediately obtained for numerical values of the parameters (for example, B =1 and σ = 4.5) as shown in the following table.

Table 1.

Exact eigenvalues with B = 1 and σ = 4.5 for n = 0,1,2,…,5 by using ξ = −2nB. Here k is sub-level for each n level and k = 0, 1, 2,…, n.

.
3.2. Highly accurate eigenvalues for arbitrary parameters

For arbitrary potential parameters A, B, and γ, the termination condition (5) can be used to evaluate the eigenenergies with very high accuracy. In this case, however, the termination condition is a function of E and y. Thus an initial value of y = y0 is necessary to start the AIM iteration. For the stability of AIM, it is sufficient to take y0 = 1/2. In Table 2, we report the eigenenergies for n = 0,1,2,…,5 and the parameters (A;B; γ) = (0.167;0.0019;0.0019), (2;0.5;1), (4;1;1.97). As is seen from Table 2, the energy eigenvalues act as ∼(n + γ + 1)2.

Table 2.

Approximately calculated eigenvalues Edirect for (A; B; γ) of (0.167; 0.0019; 0.0019), (2; 0.5; 1), and (4; 1; 1.97). The subscript represents the iteration number.

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4. Perturbation theory within the frame of AIM

The theory of the perturbation method within AIM was introduced earlier in Ref. [17]; we give here a brief summary. Assume that the potential of a quantum system is written in the form

where V0(x) is a solvable (or unperturbed Hamiltonian) potential, and Vp(x) and μ are the perturbed part and the perturbation expansion parameter, respectively. The Schrödinger equation then reads

The eigenvalues En can be written as the expansion of j-th order correction

Using the substitution ψ(x) = ψ0(x)f(x) in Eq. (17), we obtain the following equation for f(x):

The termination condition, Eq. (5), is written as

Taylor’s expansion of δn(x,μ,E) about μ = 0 yields

where

The necessary condition

can be obtained by means of Eq. (20) for each k. According to the perturbation method within the framework of AIM, solving the equation δn(x,0,E) = 0 with respect to (unknown) E gives (eigenvalues of the unperturbed Hamiltonian), equation gives (the first-order correction to En), gives (the second-order correction to En), and so on. The eigenfunctions can be easily found in a similar manner. This is an attractive feature of the AIM that allows us to obtain the eigenfunctions fn(x) without any prior knowledge of the exact solutions of the unperturbed Hamiltonian, given as follows:

where αn(t,μ) ≡ sn(t,μ)/λn(t,μ). Let

where

Thus, the perturbation expansion of fn(x) can be written as

where the k-th order correction to fn(x) is

4.1. Perturbative expansion

In Section 3, we have obtained the eigenvalue equation in the appropriate form for AIM application as

where ω = E −(γ + 1)2, ξ = A + 2σB, and 2σ = 2γ + 3. Parameters A and B of the potential V(x), given by Eq. (1), are related to each other via the relation (11). Suppose that A = 2aB, in this case, ξ = 2(a + σ)B for some values of a, then we have the following differential equation for f(y):

where B can be regarded as a perturbation expansion parameter. By expanding ω as follows:

the general form of the zeroth order correction is obtained

as mentioned previously. It is not difficult to solve this equation for where the first few iterations yield 0, 2σ, 2 + 4σ, 6 + 6σ, 12 + 8σ, 20 + 10σ, and in general, one can write

The first-order correction to ωn is generalized in a similar manner, and it is found that for each n level via the equation The second-order correction to ωn is generalized as

where

If equations (29) and (30) are substituted in Eq. (27), one can obtain

where we use a = A/2B. Using ωn = En −(γ + 1)2, we obtain approximate (perturbed energy) as follows:

Some numeric results of this formula are given in Table 3 for n = 0,1,2,…,5, and in Table 4 for n = 0,1,2,…,10. The comparison of with direct evaluation Edirect using AIM for arbitrary A, B, and γ is also given in Tables 3 and 4. The subscripts represent the iteration numbers in both tables.

Table 3.

Comparison of eigenvalues of perturbed Hamiltonian with numerical ones Edirect for A = 2, B = 2, γ = 4, and n = 0,1,2,…,5.

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Table 4.

Comparison of the eigenvalues of perturbed Hamiltonian with numerical ones Edirect for A = 2, B = 1, γ = 0.0019, and n = 0,1,2,…,10.

.

It is easily seen from Table 3 that the AIM values Edirect are consistent with the perturbation results Also, they become more consistent as the energy level n increases, although the values of Edirect converge more slowly. Similar comments apply to the results reported in Table 4 as well. Furthermore, the eigenvalues act as ∼(n + γ + 1)2 while ωn ≈ n(n + 2) in Table 4. In Table 5, values of Edirect and are inconsonant for A = B = 5 and γ = 0.5.

Table 5.

Comparison of the eigenvalues of perturbed Hamiltonian with numerically calculated ones Edirect for A = B = 5, γ = 0.5, and n = 0,1,2,…,5. The subscripts represent the iteration numbers.

.
5. Conclusion

We have used AIM to solve the Schrödinger equation with a deformed well potential represented by a trigonometric function. The application of AIM to this class of problem allows us to obtain the quasi-exact solutions under certain conditions on the potential parameters. The method was also used to obtain accurate eigenvalues for arbitrary potential parameters. Generally speaking, the energy eigenvalues seem to behave as ∼ (n + γ + 1)2 for larger quantum number n, see Eq. (32). Another usage of AIM is its perturbation version. In the standard perturbation theory, the eigenfunctions of the unperturbed Hamiltonian are essential to constructing the perturbation expansion for either energy eigenvalues or eigenfunctions. Also, determining the third and fourth order correction terms is a serious challenge. The advantage of the AIM perturbative approach is that the coefficients in the perturbation expansion can be calculated directly without the necessity of the analytic solutions of the unperturbed Hamiltonian. The numerical results show high consistency between the eigenvalues as obtained by the perturbation approach and those obtained by direct application of AIM. Although there is a slight shifting in the result obtained by the perturbation expansion for small values of n, such inconsistency tends to decrease as n increases.

Reference
1Ciftci HHall R LSaad N 2003 J. Phys. A: Math. Gen. 36 11807
2Ciftci HHall R LSaad N 2005 J. Phys. A: Math. Gen. 38 1147
3Ciftci HHall R LSaad N 2013 Cent. Eur. J. Phys. 11 37
4Ozer OLevai G2012Rom. Journ. Phys.57582
5Champion BHall R LSaad N 2008 Int. J. Mod. Phys. A 23 1405
6Falaye B JHamzavi MIkhdair S M 2013 Chin. Phys. Lett. 30 020305
7Aygun MBayrak OBoztosun I 2007 J. Phys. B: At. Mol. Opt. Phys. 40 537
8Ciftci HHall R LSaad N 2005 Phys. Rev. A 72 022101
9Turbiner A V 2012 J. Phys. A: Math. Gen. 45 025203
10Ozer O 2009 Progress of Theoretical Physics 121 437
11Yahiaoui S ABentaiba M 2011 Acta Physica Polonica B 42 1755
12Hall R LSaad NSen K DCiftci H 2009 Phys. Rev. A 80 032507
13Ciftci HHall R LSaad NDogu E 2010 J. Phys. A: Math. Theor. 43 415206
14Bayrak OBoztosun I 2006 J. Phys. A: Math. Gen. 39 6955
15Ciftci HHall R LSaad N2005J. Phys. A: Math. Gen.392338
16Barakat T 2006 J. Phys. A: Math. Gen. 39 823
17Ciftci HHall R LSaad N 2005 Phys. Lett. A 340 388