# Application of the Asymptotic Taylor Expansion Method to Bistable Potentials

Authors: Okan Ozer; Halide Koklu; Serap Resitoglu
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

A recent method called asymptotic Taylor expansion (ATEM) is applied
to determine the analytical expression for eigenfunctions and numerical
results for eigenvalues of the Schrödinger equation for the bistable
potentials. Optimal truncation of the Taylor series gives a best possible
analytical expression for eigenfunctions and numerical results for eigenvalues.
It is shown that the results are obtained by a simple algorithm
constructed for a computer system using symbolic or numerical calculation.
It is observed that ATEM produces excellent results consistent
with the existing literature.


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## Body

## 1. Introduction

There is no doubt that an interesting problem in fundamental quantum mechanics for lecturers and advanced undergraduate and graduate students in physics and applied mathematics is to obtain the exact solutions of the Schrödinger equation for any type of potential. It is well known that the Schrödinger equation, proposed by Erwin Schrödinger in 1926, is a second-order differential equation that describes how the quantum state of a physical system changes with time [1]. It is as central to quantum mechanics as Newton's laws are to classical mechanics. It is also well known that the quantum mechanics is established on some certain postulates, and in any introductory quantum physics textbook these postulates can be found with the application of the time-independent Schrödinger equation for a single particle in one dimension as a simple and basic example.Unfortunately, there are not so many potentials that can be solved exactly such as the Coulomb, harmonic oscillator, and Pöschl-Teller potentials. Since one of the source of progress of the science depends on the study of the same problem from different point of view, various methods have been suggested such as numerical calculation [2, 3], the variational [2, 4], the perturbation [5], the WKB [6, 7], the shifted 1/N expansion [8, 9], the Nikiforov-Uvarov (NU) [10, 11], the supersymmetry (SUSY) [12, 13], the generalized pseudospectral (GPS) [14], the asymptotic iteration method (AIM) [15], and other methods [16–21] to find the approximate solutions of the potentials that are not exactly solvable.In this study we will apply a new formalism based on the Taylor series expansion method, namely, asymptotic Taylor expansion method (ATEM) [22], to bistable potentials. These type of potentials have been used in the quantum theory of molecules as a crude model to describe the motion of a particle in the presence of two centers of force [23–29]. It is mentioned in [22] that the taylor series Method [30, 31] is an old one but it has not been fully exploited in the analysis of both physical and mathematical problems in solution. It is also claimed that ATEM can also be easily applied to solve second-order differential equations by introducing a simple Mathematica [32] computer program. Therefore, we focus on the solution of the eigenvalue problems of some type of bistable potentials by using ATEM in this paper.The organization of the paper is as follows. In Section2, we present a brief outline of the method ATEM. In Section 3, the eigenvalues of the bistable potentials are determined by using ATEM. Section 4 is devoted to a conclusion.


## 2. Formalism of Asymptotic Taylor Expansion Method

In this section, we present the solution of the Schrödinger-type equations by modifying Taylor series expansion with the aid of a finite sequence instead of an infinite sequence and its termination possessing the property of quantum mechanical wave function. It is well known in quantum mechanics that the bound-state energy of an atom is quantized and eigenvalues are discrete. Additionally, for each eigenvalues there exist one or more eigenfunctions. If one considers the solution of the Schrödinger equation, discrete eigenvalues of the problem are mainly under investigation. The first main result of this conclusion gives necessary and sufficient conditions for the termination of the Taylor series expansion of the wave function.Following the notation in [22], one can consider the Taylor series expansion of a function f(x) about the point a:
(1)f(x)=f(a)+(x-a)f′(a)+12(x-a)2f′′(a)+16(x-a)3f(3)(a)+⋯=∑n=0∞(x-a)nn!f(n)(a),‍
where f(n)(a) is the nth derivative of the function at a. Taylor series specifies the value of a function at one point, x, in terms of the value of the function and its derivatives at a reference point a. Expansion of the function f(x) about the origin (a=0) is known as Maclaurin's series and it is given by
(2)f(x)=f(0)+xf′(0)+12x2f′′(0)+16x3f(3)(0)+⋯=∑n=0∞xnn!f(n)(0).
Here one can develop a method to solve a second-order linear differential equation of the form
(3)f′′(x)=p0(x)f′(x)+q0(x)f(x).
It is seen that the higher-order derivatives of the f(x) can be obtained in terms of the f(x) and f′(x) by differentiating (3). Then, higher-order derivatives of f(x) are given by
(4)f(n+2)(x)=pn(x)f′(x)+qn(x)f(x),
where
(5)pn(x)=p0(x)pn-1(x)+pn-1′(x)+qn-1(x),qn(x)=q0(x)pn-1(x)+qn-1′(x).
At this point, one can observe that the eigenvalues and eigenfunctions of the Schrödinger-type equations can efficiently be determined by using ATEM. To this end, the recurrence relations (5) allow one to determine algebraic exact or approximate analytical solution of (3) under some certain conditions. Let us now substitute (5) into (1) to obtain the function that is related to the wave function of the corresponding Hamiltonian:
(6)f(x)=f(0)(1+∑n=2m‍qn-2(0)xnn!)+f′(0)(1+∑n=2m‍pn-2(0)xnn!).
After all, one can now obtain useful formalism of the Taylor expansion method. This form of the Taylor series can also be used to obtain series solution of the second-order differential equations. In the solution of the eigenvalue problems, truncation of the asymptotic expansion to a finite number of terms is useful. If the series optimally truncated at the smallest term then the asymptotic expansion of series is known as superasymptotic [33], and it leads to the determination of eigenvalues with minimum error.Since the improper sets of boundary conditions may produce nonphysical results, arrangement of the boundary conditions for different problems becomes very important. When only odd or even power ofx is collected as coefficients of f(0) or f′(0) and vice versa, the series is truncated at n=m; then an immediate practical consequence of these conditions is obtained for qm-2(0)=0 or pm-2(0)=0. In this way, the series truncates at n=m and one of the parameters in the qm-2(0) or pm-2(0) belongs to the spectrum of the Schrödinger equation. Therefore eigenfunction of the equation becomes a polynomial of degree m. Otherwise the spectrum of the system can be obtained as follows. In a quantum mechanical system eigenfunction of the system is discrete. Therefore in order to terminate the eigenfunction f(x) we can concisely write that(7)qm(0)f(0)+pm(0)f′(0)=0,qm-1(0)f(0)+pm-1(0)f′(0)=0.
Eliminating f(0) and f′(0) we obtain
(8)qm(0)pm-1(0)-pm(0)qm-1(0)=0,
again one of the parameters in the equation related to the eigenvalues of the problem.It has been stated that the ATEM reproduces exact solutions to many exactly solvable differential equations and these equations can be related to the Schrödinger equation. It is observed that the process presented in ATEM is iterative and the number of iteration is given bym. The method can be applied to the Schrödinger equation with any type of potential as follows. Using a computer program, one first sets up the iteration number m, say m=30, to obtain the result. Then, setting m=40, another result is obtained. This procedure is repeated for different m values leading to different results. Finally, one can compare the results for each case till desired digits. If the values of the eigenvalue reach their asymptotic values, then one can choose the corresponding m value and truncate the iteration for next calculations. For instance, if one can obtain the values of the eigenvalues for m=60, first few of them—first eight eigenvalues, for example—will reach automatically their asymptotic values. The following comment on the function is considerable. For such a solution it may be suitable to take sum of the first eight terms in (6).


## 3. Applications

We shall illustrate here that (3) with conditions (7) and (8) gives a complete solution for some important Schrödinger-type problems. Through a concrete example we explore the solution of Schrödinger equation (ℏ2=2m=1):
(9)(-d2dx2+V(x))ψ(x)=Eψ(x)
for the harmonic oscillator potential in one dimension V(x)=x2 given as
(10)(-d2dx2+x2)ψ(x)=Eψ(x).
In the limit of large x, the asymptotic solutions of (10) can be taken as any power of x times a decreasing Gaussian function to satisfy the quantum mechanics postulates. With this in mind, one can write the “unnormalized” wave functions as
(11)ψ(x)=exp(-x22)f(x),
where the functions f(x) are to be found by means of the iteration procedure given previously. Substituting (11) into (10), one obtains
(12)L(x)=-f′′(x)+2xf′(x)+(1-E)f(x)=0.
Comparing (3) and (12) one can deduce that
(13)p0(x)=2x,q0(x)=1-E.
Following the procedure given in (8) yields the exact eigenvalues of the harmonic oscillator potential:
(14)En=2n+1,n=0,1,2,3,…,
and the normalized eigenfunctions of the Schrödinger equation in (10) are shown in Figure 1.The first six states normalized eigenfunctions of the Harmonic potential given in (10).
(a)
(b)
(c)
(d)
(e)
(f)As a second application, we consider the quartic anharmonic oscillator potential that has been a great deal of interest in the analytical and numerical investigations of the one-dimensional anharmonic oscillators because of their importance in molecular vibrations [34] as well as in solid state physics [35, 36] and quantum field theories [37]. Schrödinger equation (9) for the one-dimensional quartic anharmonic potential V(x)=x2+gx4 is written as
(15)(-d2dx2+x2+gx4)ψ(x)=Eψ(x),
where g>0. The solution of (15) has been always studied to test accuracy and efficiency of the different methods proposed in [15, 38–41]. Now, we introduce the asymptotic solutions of (15) as
(16)ψ(x)=e-α(x2/2)-β(x4/4)f(x),
and (15) can now be written as
(17)L(x)=-f′′(x)+2(αx+βx3)f′(x)+(α-E+(1-α2+3β)x2+(g-2αβ)x4-βx6)×f(x)=0.
Comparing (3) and (17) one can deduce that
(18)p0(x)=2(αx+βx3),q0(x)=α-E+(1-α2+3β)x2+(g-2αβ)x4-βx6.
By the aid of computer program, one can calculate the eigenvalues En and the corresponding eigenfunctions f(x) for a range of g values, changing from 0.01 to 100, using number of iterations m=20,20,40,60,80,100,120. The term “asymptotic” means the function approaching to a given value as the iteration number m tends to infinity.We present our results carried out for a range ofg values in Table 1 with 7 significant digits, and they are compared with those of supersymmetric perturbation approach by [40] and the ones computed numerically by [42]. In our calculations, we set m=120, α=4, and β=0. It is observed that there is remarkable agreement in the whole range of values for all quantum states for different g values with results of [42] except g=100.Table 1
Eigenvalues of the potential in  (15) for a range of g values. For ATEM results, number of iterations for each g value is set to 120. For SUSY results, numbers in parenthesis  (N) denote the perturbation order used by authors in [40].
g
E
ATEM
E
SUSY [40]
E
exact [42]
0.01
1.007373
1.00737 (N=4)
1.007373
3.036525
3.03653 (N=8)
3.036525
5.093939
5.09609 (N=15)
5.093939
7.178573
7.19832 (N=15)
7.178573
0.05
1.034729
1.03473 (N=4)
1.034729
3.167225
3.16723 (N=8)
3.167225
5.417261
5.42404 (N=15)
5.417261
7.770271
7.83995 (N=15)
7.770271
0.1
1.065286
1.06528 (N=4)
1.065286
3.306872
3.30687 (N=8)
3.306872
5.747959
5.75694 (N=15)
5.747959
8.352678
8.45913 (N=15)
8.352678
0.5
1.2418541
1.24118 (N=4)
1.2418541
4.051932
4.05171 (N=8)
4.051932
7.396900
7.40489 (N=15)
7.396900
11.11515
11.3415 (N=15)
11.11515
1
1.392352
1.39017 (N=4)
1.392352
4.648813
4.64784 (N=8)
4.648813
8.655049
8.65908 (N=15)
8.655049
13.15680
13.4524 (N=15)
13.15680
10
2.449174
2.42910 (N=4)
2.449174
8.599003
8.58582 (N=8)
8.599003
16.63592
16.6188 (N=15)
16.63592
25.80627
26.4698 (N=15)
25.80627
100
4.999410
4.93770 (N=4)
4.999418
17.83000
17.7864 (N=8)
17.83019
34.87117
34.8238 (N=15)
34.87398
54.36576
55.4001 (N=15)
54.38529We also present and compare our results forg=0.1 with those of Bacus et al. [39] in Table 2 with 20 significant digits. The function f(x), for g=0.1, n=5 state for different values of m is found as follow:
(19)m=20,f(x)=x-1.3425x3-1.28331x5-0.249794x7+0.126907x9+0.0974402x11+0.0325753x13+0.00674249x15+0.000816039x17+O(x19),m=40,f(x)=x-0.35941x3-0.998775x5-0.578949x7-0.161047x9-0.0125032x11+0.0088751x13+0.00478244x15+0.00143729x17+O(x19),m=80,f(x)=x-0.328321x3-0.980318x5-0.581291x7-0.167846x9-0.0162743x11+0.00762613x13+0.00449722x15+0.00139268x17+O(x19),m=120,f(x)=x-0.328321x3-0.980318x5-0.581291x7-0.167846x9-0.0162743x11+0.00762613x13+0.00449722x15+0.00139268x17+O(x19).
For the first six states, the plot of the normalized wave functions for g=0.1 are given in Figure 2. As a last example, we consider the symmetric bistable potential defined in [43] as
(20)V(x)=x6-2x4-2x2+1.
Inserting the potential function into Schrödinger equation in (9) and using the ansatz wave function of the form defined in (16), one obtains
(21)L(x)=-f′′(x)+2(αx+βx3)f′(x)+(1+α-2E-(2+α2-3β)x2-2(1+αβ)x4+(1-β2)x6)f(x)=0.
Following the procedure given in (8) for m=120, α=4, and β=1 yields the eigenvalues of the symmetric bistable potential in (20). We note here the selection of adjustable parameters α and β. Since we choose the ansatz wave function given in (16), the form of the superpotential for the partner potentials by supersymmetric quantum mechanics (SUSYQM) [44] is found to be(22)W(x)=-ψ′(x)ψ(x),
and the partner potentials V±(x) are defined as
(23)V±(x)=W2(x)±W′(x).
Since the superpotential W(x) obtained from (16) is
(24)W(x)=αx+βx3,
then the partner potential V-(x) is obtained as
(25)V-(x)=(α2-3β)x2+2αβx4+β2x6-α.
If α2>3β, the potential (25) has one minimum. Investigating this one-minimum case one can select the adjustable parameters α=2 and β=1 that lead to an increase in the iteration number m. Therefore, we select and set the parameters as α=4 and β=1 in our calculations.Table 2
The first ten eigenvalues of the potential in (15) for g=0.1. Number of iteration is set to m=120.
n
E
ATEM
E
[
39
]
0
1.065
285
509
543
717
701
1.065
285
509
543
717
688
1
3.306
872
013
152
913
680
3.306
872
013
152
913
507
2
5.747
959
268
833
563
228
5.147
959
268
833
563
304
3
8.352
677
825
785
754
350
8.352
677
825
785
754
712
4
11.098
595
622
633
043
333
11.098
595
622
633
043
011
5
13.969
926
197
742
799
089
13.969
926
197
742
799
300
6
16.954
794
686
144
150
972
16.954
794
686
144
151
337
7
20.043
863
604
188
462
801
20.043
863
604
188
461
233
8
23.229
552
179
939
290
112
23.229
552
179
939
289
070
9
26.505
554
752
536
617
968
26.505
554
752
536
617
417The first six states normalized eigenfunctions of the quartic potential given in (15) for g=0.1.
(a)
(b)
(c)
(d)
(e)
(f)Our results are presented and compared with the values found by the variational supersymmetric method (EVSQM) [43], state-dependent diagonalization method (ESDD) [45], and by direct numerical integration (EExact) [43], in Table 3. The ATEM results are in a very good agreement, by low percent errors, for all values of energies with the ones obtained by numerical calculation. For lower percent errors, the iteration number m must be increased.Table 3
Comparison of energy eigenvalues of the potential in (20). For ATEM results, number of iterations is set to m=120.
E
ATEM
E
VSQM [43]
E
SDD [45]
E
Exact [43]
0
0
0
0
0.4229446
0.4238512
0.4229446
0.4229511
2.314913
2.319117
2.314913
2.314925
4.503779
4.571588
4.503779
4.503822
7.175475
7.101165
7.175475
7.175509
10.27788
9.861245
10.27789
10.27797
13.75855
12.82074
13.75855
13.75861
17.58421
15.95720
17.58420
17.58434
21.72951
19.25351
21.72942
21.72951
26.17305
22.69614
26.17370
26.17391


## 4. Conclusion

An approximate method based on the asymptotic Taylor series expansion of a function and its fundamental features are presented. It is observed that the method is applicable for obtaining both eigenvalues and eigenfunctions of the Schrödinger-type equations. After applying the method to the one-dimensional harmonic oscillator potential, it is shown that the approach gives accurate results for eigenvalue problems of some certain type of bistable potentials. It is thought that the approach opens the way to the treatment of the Schrödinger equation including large class of potentials of practical interest. As a future study, the method can be developed and applied to the non-Hermitian systems and QES potentials. The direct application of the method to the Fokker-Planck equation for quasi exactly solvable bistable potentials when the drift coefficient has a nonpolynomial nature would also be interesting. Studies along this line are in progress.



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