In this study, the N-dimensional radial Schrodinger equation with an anharmonic sextic potential is solved by the extended Nikirov-Uranov method. We prove that the radial function can be factorised as the product between an exponential function and a polynomial function solution of the biconfluent Heun equation. The approach investigated in this article aims to be an alternative to other known methods of solving, as it has the advantage of dealing with simple, first-order differential and algebraic equations and avoiding numerous and laborious coordinate transformations and series expansions.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
S.H. Dong, Wavefunction ansatz method. In: Wave equations in higher dimensions (Springer, Dordrecht, 2011).
D. Brandon, N. Saad, S.H. Dong, On some polynomial potentials in d-dimension. J. Math. Phys. 54, 082106 (2013)
D. Agboola, Y.Z. Zhang, Unified derivation of exact solutions for a class of quasi-exactly solvable models. J. Math. Phys. 53, 042101 (2012)
A. Azad, M.T. Laradji, Mustafa, Polynomial solutions of differential equations. Adv. Differ. Equ. 11, 58 (2011)
D. Agboola, Y.Z. Zhang, Exact solutions of Schrodinger equation with spherically symmetric octic potential. Mod. Phys. Lett. A 27, 1250112 (2012)
M. Bansal, S. Srivastava, Energy eigenvalues of double-well oscillator with mixed quartic and Sextic anharmonicities. Phys. Rev. A 44, 8012 (1991)
R.L. Hall, N. Saad, Exact and approximated solutions of Schrodinger equation with hyperbolic double-well potentials. Eur. Phys. J. Plus 131, 277 (2016)
G. Campoy, A. Palma, On the numerical solutions of the Schrodinger equation with a polynomial potential. Int. J. Quantum Chem. 30(S20), 33 (1986)
S.H. Dong, The ansatz method for analyzing Schrodinger equation with three anharmonic potentials in D-dimension. Found. Phys. Lett. 15, 385 (2002)
S.H. Dong, On the solutions of the Schrodinger equation with anharmonic potentials: wavefunction ansatz. Phys. Scr. 65, 289 (2002)
T.E. Simos, J. Vigo-Aguiar, A modified Runge-Kutta method with phase-lag of order infinity for the numerical solution of the Schrodinger equation and related problems. Comput. Chem. 25, 275 (2001)
G. Avdelas, T.E. Simos, J. Vigo-Aguiar, An embedded exponentially-fitted Runge-Kutta method for the numerical solution of the Schrodinger equation and related periodic initial-value problems. Comput. Phys. Commun. 131, 52 (2000)
T.E. Simos, J. Vigo-Aguiar, A symmetric high order method with minimal phase-lag for the numerical solution of the Schrodinger equation. Int. J. Mod. Phys. C 12, 1035 (2001)
T.E. Simos, J. Vigo-Aguiar, A new modified Runge-Kutta-Nystrom method with phase-lag of order infinity for the numerical solution of the Schrodinger equation and related problems. Int. J. Mod. Phys. C 11, 1195 (2000)
J. Vigo-Aguiar, H. Ramos, A variable-step Numerov method for the numerical solution of the Schrodinger equation. J. Math. Chem. 37, 255 (2005)
A. Shokri, J. Vigo-Aguiar, M.M. Khalsaraei, R. Garcia-Rubio, A new four-step P-stable Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrodinger equation. J. Comp. Appl. Math. 354, 569 (2019)
J. Vigo-Aguiar, High order Bessel fitting methods for the numerical integration of the Schrodinger equation. Comput. Chem. 25, 97 (2001)
A. Shokri, J. Vigo-Aguiar, M.M. Khalsaraei, R. Garcia-Rubio, A new implicit six-step P-stable method for the numerical solution of Schrodinger equation. Int. J. Comput. Math. 97, 802 (2020)
H. Karayer, D. Demirhan, F. Buyukkilic, Extension of Nikirov-Uranov method for the solution of Heun equation. J. Math. Phys. 56, 063504 (2015)
H. Karayer, D. Demirhan, F. Buyukkilic, Some special solutions of biconfluent and triconfluent Heun equations in elementary functions by extended Nikirov-Uranov method. Rep. Math. Phys. 76, 271 (2015)
R. Budaka, Harmonic oscillator potential with a sextic anharmonicity in the prolate γ-rigid collective geometrical model. Phys. Lett. B 739, 56 (2014)
F.T. Wall, G.J. Glocker, The double-minimum problem applied to the ammonia molecules. Chem. Phys. 5, 314 (1937)
R.L. Somorjai, D.F. Hornig, Double-minimum potentials in hydrogen-bonded solids. J. Chem. Phys. 36, 1980 (1962)
E. Uggerud, The factors determining reactivity in nucleophilic substitution. Adv. Phys. Org. Chem. 51, 1 (2017)
A.V. Nikirov, V.B. Uranov, Special Functions of Mathematical Physics (Birkhauser, Boston, 1988)
T.A. Ishkhanyan, A.M. Ishkhanyan, Solution of the biconfluent Heun equation in terms of the Hermite functions. Ann. Phys. 383, 79 (2017)
W. Robin, On the Rodrigues formula solution of hypergeometric-type differential equation. In International Mathematical Forum 8, 1455 (2013)
A.M. Pupasov-Maksimov, Analytical simulation of double-well, triple-well and multi-well dynamics via rotationally extended harmonic oscillators. J Phys: Conf Ser 670, 012042 (2016)
A. Roseau, On the solutions of the biconfluent Heun equation. Bull. Belg. Math. Soc. 9, 321 (2002)
R.S. Mayer, The 192 solutions of the Heun equation. Math. Comp. 76, 811 (2007)
The author declares no affiliation or involvement with any organisation or entity with any financial interest (e.g. honoraria, educational grants, participation in speakers’ bureaus; membership, employment, consultancy, stock ownership or other equity interest; and expert testimony or patent-licensing arrangements) or non-financial interest (e.g. personal or professional relationships, affiliations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Nanni, L. A new approach to solve the Schrodinger equation with an anharmonic sextic potential. J Math Chem 59, 2284–2293 (2021). https://doi.org/10.1007/s10910-021-01289-5
- Polynomial potential
- Nikirov-Uranov formalism
- Hypergeometric equation
- Schrodinger equation
Mathematics Subject Classification
- Primary: 26C05
- Secondary: 33C15