Author(s)

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.

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.

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

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.