Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions

J. Morales^1*, J. García-Martínez², J. García-Ravelo³, J. J. Peña¹

¹Universidad Autónoma Metropolitana-Azc., DCB-Area de FAMA, Sn Pablo 180, Mexico City, México.
²Tecnológico de Estudios Superiores de Ixtapaluca, División de Ingeniería Informática e Ingeniería Biomédica, Carretera Ixtapaluca Coatepec Km. 7, Mexico City, México.
³Escuela Superior de Física y Matemáticas, IPN-Zacatenco, UP Adolfo López Mateos, México.

DOI: 10.4236/jamp.2015.311173                

Abstract

In this work, the canonical transformation method is applied to a general second order differential equation (DE) in order to trasform it into a Schr?dinger-like DE. Our proposal is based on an auxiliary function g(x) which determines the transformation needed to find exactly-solvable potentials associated to a known DE. To show the usefulness of the proposed approach, we consider explicitly their application to the hypergeometric DE with the aim to find quantum potentials with hypergeometric wavefunctions. As a result, different potentials are obtained depending on the choice of the auxiliary function; the generalized Scarf, Posh-Teller, Eckart and Rosen-Morse trigonometric and hyperbolic potentials, are derived by selecting g(x) as constant and proportional to the P(x) hypergeometric coefficient. Similarly, the choices g(x)~P(x)/x² and g(x)~x²/P(x) give rise to a class of exactly-solvable generalized multiparameter exponential-type potentials, which contain as particular cases the Hulthén, Manning-Rosen and Woods-Saxon models, among others. Our proposition is general and can be used with other important DE within the frame of applied matematics and physics.

Keywords

Canonical Transformation, Schrödinger-Like Equation, Hypergeometric DE, Exactly-Solvable Potentials

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Sukumar, C.V. (1985) Supersymmetric Quantum Mechanics of One-Dimensional Systems. Journal of Physics A: Mathematical and General, 18, 2917-2936. http://dx.doi.org/10.1088/0305-4470/18/15/020
[2]	De, R., Dutt, R. and Sukhatme, U. (1992) Mapping of Shape Invariant Potentials under Point Canonical Transformations. Journal of Physics A: Mathematical and General, 25, L843-L850. http://iopscience.iop.org/0305-4470/25/13/013
[3]	Setare, M.R. and Karimi, E. (2008) Mapping of Shape Invariant Potentials by the Point Canonical Transformation. International Journal of Theoretical Physics, 47, 891-897.
[4]	Cooper, F., Khare, A. and Sukhatme, U. (1995) Supersymmetry and Quantum Mechanics. Physics Reports, 251 267-385. http://dx.doi.org/10.1016/0370-1573(94)00080-M
[5]	Ho, C.-L. (2009) Simple Unified Derivation and Solution of Coulomb, Eckart and Rosen-Morse Potentials in Prepotential Approach. Annals of Physics, 324, 1095-1104. http://dx.doi.org/10.1016/j.aop.2008.10.004
[6]	Jia, C.-S., Diao, Y.-F., Li, M., Yang, Q.-B., Sun, L.-T. and Huang, R.-Y. (2004) Mapping of the Five-Parameter Exponential-Type Potential Model into Trigonometric-Type Potentials. Journal of Physics A: Mathematical and General, 37 11275-11284. http://dx.doi.org/10.1088/0305-4470/37/46/012
[7]	Jia, C.-S., Yi, L.-Z., Zhao, X.-Q., Liu, J.-Y. and Sun, L.-T. (2005) Systematic Study of Exactly Solvable Trigonometric Potentials with Symmetry. Modern Physics Letters A, 20, 1753-1762. http://dx.doi.org/10.1142/S0217732305017081
[8]	Jia, C.-S., Liu, J.-Y., Sun, Y., He, S. and Sun, L.-T. (2006) A Unified Treatment of Exactly Solvable Trigonometric Potential Models. Physica Scripta, 73, 164-168. http://iopscience.iop.org/1402-4896/73/2/006
[9]	Cooper, F., Khare A. and Sukhatme, U. (2001) Supersymmetry in Quantum Mechanics. World Scientific Publishing Co Pte Ltd.
[10]	Ciftci, H., Hall, R.L. and Saad, N. (2003) Asymptotic Iteration Method for Eigenvalue Problems. Journal of Physics A: Mathematical and General, 36, 11807-11816. http://dx.doi.org/10.1088/0305-4470/36/47/008
[11]	Nikiforov, A.F. and Uvarov, V.B. (1988) Special Functions of Mathematical Physics. Birkhauser, Basel. http://dx.doi.org/10.1007/978-1-4757-1595-8
[12]	Peña, J.J., Morales, J., García-Martínez, J. and García-Ravelo, J. (2008) Exactly Solvable Quantum Potentials with Special Functions Solutions. International Journal of Quantum Chemistry, 108, 1750-1757. http://dx.doi.org/10.1002/qua.21611
[13]	Abramowitz, M. and Stegun, I.A. (1972) Handbook of Mathematical Functions. Wiley and Sons, New York.
[14]	Granville, W.A., Smith, P.F. and Longley, W.R. (1941) Elements of the Differential and Integral Calculus. Ginn & Co., Boston.
[15]	Hulthén, L. (1942) On the Characteristic Solutions of the Schrñdinger Deuteron Equation. Arkiv fñr Matematik Astronomi och Fysik A, 28, art 5: 1-12.
[16]	Chen, G. (2004) Shape Invariance and the Supersymmetric WKB Approximation for the Generalized Hulthén Potential. Physica Scripta, 69, 257-259. http://dx.doi.org/10.1238/physica.regular.069a00257
[17]	Morales, J., Peña, J.J. and Morales-Guzman, J.D. (2000) The Generalized Hulthén Potential. Theoretical Chemistry Accounts, 104, 179-182. http://dx.doi.org/10.1007/s002140000130
[18]	Ahmed, S.A.S. and Buragohain, L. (2010) Exactly Solved Potentials Generated from the Manning-Rosen Potential Using Extended Transformation Method. Electronic Journal of Theoretical Physics, 7, 145-154.
[19]	Fatah, A.H. (2012) Calculation of the Eigenvalues for Wood-Saxon’s Potential by Using Numerov Method. Advances in Theoretical and Applied Mechanics, 5, 23-31.
[20]	Berkdemir, C., Berkdemir, A. and Sever, R. (2005) Polynomial Solutions of the Schrñdinger Equation for the Generalized Woods-Saxon Potential. Physical Review C, 72, 027001-1-027001-4.
[21]	Gñnül, B. and Kñksal, K. (2007) Solutions for a Generalized Woods-Saxon Potential. Physica Scripta, 76, 565-570. http://dx.doi.org/10.1088/0031-8949/76/5/026
[22]	Falaye, B.J., Oyewumi, K.J., Ibrahim, T.T., Punyasena, M.A. and Onate, C.A. (2013) Bound State Solutions of the Manning-Rosen Potential. Canadian Journal of Physics, 91, 98-104. http://dx.doi.org/10.1139/cjp-2012-0330
[23]	Nasser, I., Abdelmonem, M.S. and Abdel-Hady, A. (2013) The Manning-Rosen Potentials Using J-Matrix Approach. Molecular Physics, 111, 1-8. http://dx.doi.org/10.1080/00268976.2012.698026
[24]	Lévai, G. (1989) A Search for Shape-Invariant Solvable Potentials. Journal of Physics A: Mathematical and General, 22, 689-702. http://dx.doi.org/10.1088/0305-4470/22/6/020
[25]	Peña, J.J., García-Martínez, J., García-Ravelo, J. and Morales, J. (2014) l-State Solutions of Multiparameter Exponential-Type Potentials. Journal of Physics: Conference Series, 490, 012199. http://dx.doi.org/10.1088/1742-6596/490/1/012199
[26]	Arfken, G. and Weber, H. (2005) Mathematical Methods for Physicists. 6th Edition, Elsevier AP, Boston.
[27]	Polyani, A.D. and Zaistev, V.F. (2003) Handbook of Exact Solutions for Ordinary Differential Equations. 2nd Edition, Chapman& Hall/CRC, Boca Raton, New York.