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Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions

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DOI: 10.4236/jamp.2015.311173    5,729 Downloads   6,278 Views  


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.

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The authors declare no conflicts of interest.

Cite this paper

Morales, J. , García-Martínez, J. , García-Ravelo, J. and Peña, J. (2015) Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions. Journal of Applied Mathematics and Physics, 3, 1454-1471. doi: 10.4236/jamp.2015.311173.


[1] Sukumar, C.V. (1985) Supersymmetric Quantum Mechanics of One-Dimensional Systems. Journal of Physics A: Mathematical and General, 18, 2917-2936.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11] Nikiforov, A.F. and Uvarov, V.B. (1988) Special Functions of Mathematical Physics. Birkhauser, Basel.
[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.
[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.
[17] Morales, J., Peña, J.J. and Morales-Guzman, J.D. (2000) The Generalized Hulthén Potential. Theoretical Chemistry Accounts, 104, 179-182.
[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.
[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.
[23] Nasser, I., Abdelmonem, M.S. and Abdel-Hady, A. (2013) The Manning-Rosen Potentials Using J-Matrix Approach. Molecular Physics, 111, 1-8.
[24] Lévai, G. (1989) A Search for Shape-Invariant Solvable Potentials. Journal of Physics A: Mathematical and General, 22, 689-702.
[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.
[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.

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