Research Article  Open Access
Okan Ozer, Halide Koklu, "An Alternative Approach to Energy Eigenvalue Problems of Anharmonic Potentials", Advances in Mathematical Physics, vol. 2014, Article ID 537563, 6 pages, 2014. https://doi.org/10.1155/2014/537563
An Alternative Approach to Energy Eigenvalue Problems of Anharmonic Potentials
Abstract
Energy eigenvalues of quartic and sextic type anharmonic potentials are obtained by using an alternative method called asymptotic Taylor expansion method (ATEM) which is an approximate approach based on the asymptotic Taylor series expansion of a function. It is shown that the energy eigenvalues found by ATEM are in excellent agreement with the existing results.
1. Introduction
It is well known that the exact solution of Schrödinger equation is only possible for very few numbers of potentials and it is required to apply the numerical methods or some approximate schemes in most quantum mechanical systems. In the literature, there are many studies for this purpose such as numerical calculation [1, 2], the perturbation [3], the variational [4], the WKB [5, 6], the shifted expansion [7, 8], the NikiforovUvarov (NU) [9], and the supersymmetry (SUSY) [10] to find the approximate solutions of the potentials that are not exactly solvable. In the last decades, the study of anharmonic oscillator potentials (such as quartic and sextic anharmonic oscillators) has been taking attention because they are open for the theoretical perception of some recently discovered phenomena in different branches of physics [11–13]. On the other hand, these potentials are not exactly solvable which make them very popular for examining the validity of any approach [14–17]. In all these attempts, it is merely required to get a relatively effective and simple approach that gives the energy eigenvalues—and eigenfunctions—to a high degree of accuracy. In this study, an efficient method called asymptotic Taylor expansion method (ATEM) is applied to compute the energy eigenvalues of both the quartic and the sextic anharmonic oscillator potentials in one dimension.
The ATEM based on the Taylor series expansion is proposed by Koç and Sayın [18]. It is claimed that one can easily apply ATEM to solve secondorder differential equations by introducing a simple code in Mathematica [19] computer program. Therefore, it is focused on the solution of the eigenvalue problems of anharmonic oscillator potentials by using ATEM in this paper.
The organization of the paper is as follows. In Section 2, a brief outline of the method ATEM is presented. In Section 3, we obtain numerically the eigenenergies and make comparison with other existing results. Finally, Section 4 is devoted to a conclusion.
2. Asymptotic Taylor Expansion Method
Following the notation in [18], one can consider the Taylor series expansion of a function about the point : where is the th derivative of the function at . If , then it is called Maclaurin’s series and given by It is claimed in [18] that one can construct a method to solve the secondorder linear differential equations in the form of By differentiating (3) with respect to , one can get the higher order derivatives of in terms of and . Thus, one obtains where At this point, it is seen that the eigenvalues and eigenfunctions of the Schrödingertype equations can efficiently be obtained by using ATEM. To this end, the recurrence relations (5) allow one to get analytical or numerical solution of (3) under some certain conditions. After substituting (5) into (1) and obeying the termination condition of the eigenfunction in the boundstate quantum mechanical systems (the reader is addressed to [18]), one can finally write and then eliminating and one gets which will involve a parameter related to eigenvalues of the potential in the study.
It is seen that the method ATEM is iterative and iteration number is given by . It is proposed that the method can be applied to the Schrödinger equation, with any type of potentials, as follows. Using a Mathematica computer program, one can repeat the calculation of the eigenvalues for different values of iteration number such as , till desired digits. When the eigenvalues reach their asymptotic value, then one can choose the corresponding and truncate the iteration for next calculations. For instance, if one obtains the eigenvalues for desired digits when , then the first few eigenvalues—first eightenergy state, for example—will automatically reach their asymptotic values [20].
3. Applications
We consider the onedimensional quartic double well potential as a first application: This potential has great importance in many branches of physics such as molecular vibrations [21], solid state physics [22, 23], quantum field theories [24], and quantum chromodynamics [25]. The Schrödinger equation is written as where . To apply ATEM, now we introduce an ansatz wavefunction as the asymptotic solutions of (9): and (9) can now be written as Comparing (3) and (11) one can deduce that Before proceeding ATEM, we note here some points about the ansatz wave function and the selection of numerical values of adjustable parameters and ; to satisfy the quantum mechanical postulates in the limit of large for the bound systems, the asymptotic solutions of (9) can be taken as the multiplication of any power of with a decreasing Gaussiantype function. Therefore, we suggest the ansatz wave function in the form of (10). Additionally, there is strong relation among the wave function and the potential parameters of , , and , respectively, in the solution of (9). Since the exact analytical solution of the potential in (9) cannot be obtained, then one can look for the effective partner potential solutions; since we introduce (10), then the superpotential function for the partner potentials by supersymmetric quantum mechanics (SUSYQM) [26] is given by and the partner potentials are defined as Since the superpotential obtained from (10) is then the partner potential is obtained as
It is observed that the potential (16) has one minimum if . Searching for this oneminimum case, we select the adjustable parameters as and in this study. Since the iteration number is desired to be as much low as possible, then one has to search for the optimum iteration number in the calculation of eigenvalues of potential in (9). By setting , , and , we search for the optimum iteration number by comparing the results of ATEM with the known exact value of third excited state, , in Table 1. We also obtain the percent errors and then set to truncate the iteration in the following calculations. It is seen that the term asymptotic means the ATEM value approaching to a given value as the iteration number tends to infinity.

We present our results carried out for a range of values in Table 2 and Table 3 with significant digits and they are compared with those of variational supersymmetric approach by [27] and the ones computed numerically by [28]. In our calculations, we set , , and . We observe that the results by ATEM are in very good agreement with numerical ones of [28]. In particular, for , the errors are less than in all cases.


As a second application, we consider the energy eigenvalues of the quasiexactly solvable sextic anharmonic oscillator potentials: Since only energy eigenvalues can be obtained analytically while the other levels remain unknown, these potentials are classified as quasiexactly solvable potentials [29, 30]. Following the same procedure given above and the ansatz wave function in (10) by the same numerical values of , , and , one obtains We present and compare our results in Table 4 for . It is seen from Table 4 that the eigenvalues obtained by the ATEM are acceptably in good agreement with the exact numerical results [31]. We also present the percent errors. It is observed that the errors are less than except the highest energy state—in this study—for values.

As a last example, we consider the quasiexactly solvable potential [6] given as Using the ansatz wave function of the form defined in (10) and following the same procedure given above for the same numerical values of , , and , one obtains We present and compare our results for energy states in Table 5. It is seen that ATEM reproduces excellent results consistent with the existing literature.

4. Conclusion
We have applied relatively simple and efficient method for calculating the energy eigenvalues of some type of onedimensional anharmonic oscillator potentials. The method is based upon the asymptotic Taylor series expansion of a function. It is shown that the optimal truncation of the Taylor series reproduces the numerical results for eigenvalues with higher accuracy. The algorithm constructed for a computer system using symbolic or numerical calculation is relatively simple in ATEM.
It is clear that the determination of the adjustable parameters and directly affects the iteration number . Therefore, the efficiency of the method is then dependent on these parameters. Since we set by choosing and , then the iteration number may seem to be not low enough. One can choose different and values obeying the constraint given in text; then the iteration number is expected to get lower values. On the other hand, one can also investigate the best approximate values of the parameters and by searching the minimum expectation value of energy that satisfies (9), for example, for the ground state. Since we focus on the application of ATEM for the determination of the energy eigenvalues by choosing and values given above, it is thought in our calculations that the percent error ranges for the potentials in this study are acceptable for the selected values of and parameters, by satisfying the minimumcase condition of supersymmetric partnerlike potentials.
It is believed that the simplicity and mathematical facility of the method suggested here can be useful to the treatment of the Schrödinger equation including large class of potentials. For example, the method can be used for other unidimensional potentials and for isotropic potentials in three dimensions. Additionally, one can focus on the potential function given in (8) to search for energy splitting in the wells for value being large enough [33]. Studies along this line are in progress.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The financial supports of Research Fund of Gaziantep University (BAP) and the Scientific and Technological Research Council of Turkey (TÜBİTAK) are acknowledged. The authors are also very much indebted to the referee(s) for useful comments and suggestions.
References
 Y. P. Varshni, “Eigenenergies and oscillator strengths for the Hulthén potential,” Physical Review A, vol. 41, no. 9, pp. 4682–4689, 1990. View at: Publisher Site  Google Scholar
 M. A. Nunez, “Accurate computation of eigenfunctions for Schrödinger operators associated with Coulombtype potentials,” Physical Review A, vol. 47, no. 5, pp. 3620–3631, 1993. View at: Publisher Site  Google Scholar
 P. Matthys and H. de Meyer, “Dynamicalgroup approach to the Hulthén potential,” Physical Review A, vol. 38, no. 3, pp. 1168–1171, 1988. View at: Publisher Site  Google Scholar
 C. Stubbins, “Bound states of the Hulthén and Yukawa potentials,” Physical Review A, vol. 48, no. 1, pp. 220–227, 1993. View at: Publisher Site  Google Scholar
 Y. P. Varshni, “Relative convergences of the WKB and SWKB approximations,” Journal of Physics A Mathematical and General, vol. 25, no. 21, pp. 5761–5777, 1992. View at: Publisher Site  Google Scholar  MathSciNet
 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. View at: Publisher Site  Google Scholar  MathSciNet
 A. Z. Tang and F. T. Chan, “Shifted 1/N expansion for the Hulthén potential,” Physical Review A, vol. 35, no. 2, pp. 911–914, 1987. View at: Publisher Site  Google Scholar
 R. K. Roychoudhury and Y. P. Varshni, “Shifted 1/N expansion and exact solutions for the potential ${V(r)=Z/r+gr+\lambda r}^{2}$,” Journal of Physics A: Mathematical and General, vol. 21, no. 13, pp. 3025–3034, 1988. View at: Publisher Site  Google Scholar  MathSciNet
 A. F. Nikiforov and V. B. Uvarov, Special Functions of Mathematical Physics, Birkhäuser, Basel, Switzerland, 1988.
 S. W. Qian, B. W. Huang, and Z. Y. Gu, “Supersymmetry and shape invariance of the effective screened potential,” New Journal of Physics, vol. 4, p. 13, 2002. View at: Publisher Site  Google Scholar
 P. Maknikowski and A. Radosz, “Comment on “Bounces and the calculation of quantum tunnelling effects for the asymmetric doublewell potential”: [B. Zhou, J.Q. Liang, F.Ch. Pud, Phys. Lett. A 271 (2000) 26],” Physics Letters A, vol. 292, no. 45, pp. 300–302, 2002. View at: Publisher Site  Google Scholar
 B. Zhou, J. Q. Liang, and F. C. Pud, “Quantum tunneling for the asymmetric doublewell potential at finite energy,” Physics Letters A, vol. 281, no. 23, pp. 105–112, 2001. View at: Publisher Site  Google Scholar
 E. Paspalakis, “Physical interpretation of laserinduced suppression of quantum tunneling,” Physics Letters A, vol. 261, no. 56, pp. 247–251, 1999. View at: Publisher Site  Google Scholar
 J. Chen, L. C. Kwek, and C. H. Oh, “Quartic anharmonic oscillator and nonhermiticity,” Physical Review A: Atomic, Molecular, and Optical Physics, vol. 67, no. 1, Article ID 012101, 9 pages, 2003. View at: Google Scholar
 F. Zhou, Z. Cao, and Q. Shen, “Energy splitting in symmetric doublewell potentials,” Physics Letters A, vol. 67, Article ID 062112, 2003. View at: Publisher Site  Google Scholar
 Y. He, Z. Cao, and Q. Shen, “Boundstate spectra for supersymmetric quantum mechanics,” Physics Letters A, vol. 326, no. 56, pp. 315–321, 2004. View at: Publisher Site  Google Scholar
 A. Hutem and C. Sricheewin, “Groundstate energy eigenvalue calculation of the quantum mechanical well $V\left(x\right)=\left(\frac{1}{2}\right){kx}^{2}+{\lambda x}^{4}$ via analytical transfer matrix method,” European Journal of Physics, vol. 29, no. 3, p. 577, 2008. View at: Publisher Site  Google Scholar
 R. Koç and S. Sayın, “Remarks on the solution of the positiondependent mass Schrödinger equation,” Journal of Physics A: Mathematical and Theoretical, vol. 43, no. 45, Article ID 455203, 2010. View at: Publisher Site  Google Scholar
 Wolfram Research, “Mathematica, Version 8.0,” Wolfram Research, Champaign, Ill, USA, 2010. View at: Google Scholar
 O. Ozer, H. Koklu, and S. Resitoglu, “Application of the asymptotic Taylor expansion method to bistable potentials,” Advances in Mathematical Physics, vol. 2013, Article ID 239254, 8 pages, 2013. View at: Publisher Site  Google Scholar
 C. S. Hsue and J. L. Chern, “Twostep approach to onedimensional anharmonic oscillators,” Physical Review D, vol. 29, article 643, 1984. View at: Publisher Site  Google Scholar  MathSciNet
 G. P. Flessas, R. R. Whitehead, and A. Rigas, “On the $\alpha $x^{2} + $\beta $x^{4} interaction,” Journal of Physics A, vol. 16, no. 1, p. 85, 1983. View at: Publisher Site  Google Scholar
 A. R. Bonham and S. L. Su, “Use of hellmann—feynman and hypervirial theorems to obtain anharmonic vibration—rotation expectation values and their application to gas diffraction,” The Journal of Chemical Physics, vol. 45, p. 2827, 1996. View at: Google Scholar
 M. Reed and B. Simon, Methods of Modern Mathematical Physics, IV Analysis of Operators, Academic Press, New York, NY, USA, 1978. View at: MathSciNet
 C. Quigg and J. L. Rosner, “Quantum mechanics with applications to quarkonium,” Physics Reports C, vol. 56, no. 4, pp. 167–235, 1979. View at: Publisher Site  Google Scholar  MathSciNet
 A. Khare and Y. P. Varshni, “Is shape invariance also necessary for lowest order supersymmetric WKB to be exact?” Physics Letters A, vol. 142, no. 1, pp. 1–4, 1989. View at: Publisher Site  Google Scholar
 G. R. P. Borges, A. de Souza Dutra, E. Drigo, and J. R. Ruggiero, “Variational method for excited states from supersymmetric techniques,” Canadian Journal of Physics, vol. 81, no. 11, pp. 1283–1291, 2003. View at: Publisher Site  Google Scholar
 G. Harvey and J. Tobochnik, An Introduction to Computer Simulation Methods: Applications to Physical Systems, AddisonWesley, Reading, Mass, USA, 2nd edition, 1996.
 A. V. Turbiner, “Quasiexactlysolvable problems and $sl(2)$ algebra,” Communications in Mathematical Physics, vol. 118, no. 3, pp. 467–474, 1988. View at: Publisher Site  Google Scholar  MathSciNet
 M. A. Shifman, “New findings in quantum mechanics (partial algebraization of the spectral problem),” International Journal of Modern Physics A, vol. 4, no. 12, pp. 2897–2952, 1989. View at: Publisher Site  Google Scholar  MathSciNet
 P. Roy, R. Roychoudhury, and Y. P. Varshni, “On the application of supersymmetric WKB (WentzelKramersBrillouin) method to quasiexactly solvable problems,” Canadian Journal of Physics, vol. 69, no. 10, pp. 1261–1263, 1991. View at: Publisher Site  Google Scholar  MathSciNet
 R. N. Chaudhuri and M. Mondal, “Eigenvalues of anharmonic oscillators and the perturbed Coulomb problem in Ndimensional space,” Physical Review A, vol. 52, no. 3, pp. 1850–1856, 1995. View at: Publisher Site  Google Scholar
 V. Jelic and F. Marsiglio, “The doublewell potential in quantum mechanics: a simple, numerically exact formulation,” European Journal of Physics, vol. 33, no. 6, pp. 1651–1666, 2012. View at: Publisher Site  Google Scholar
Copyright
Copyright © 2014 Okan Ozer and Halide Koklu. 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.