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Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions

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DOI: 10.4236/jamp.2015.311173    5,841 Downloads   6,467 Views   Citations

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.

Conflicts of Interest

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.

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