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.
Highlights
And quasi-exactly solvable potential models are important in practically any field of theoretical quantumHow to cite this paper: Morales, J., García-Martínez, J., García-Ravelo, J. and Peña, J.J. (2015) Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions
We present a method to obtain the general canonical form of second order differential equations on the field of theoretical physics
The procedure is similar to the method proposed by Levai [24] and we consider a general Differential Equations (DE) to be converted into a Schrödinger-like equation
Summary
How to cite this paper: Morales, J., García-Martínez, J., García-Ravelo, J. and Peña, J.J. (2015) Exactly Solvable Schrödinger Equation with Hypergeometric Wavefunctions. It becomes clear that the exact solution for the Schrödinger equation is reduced to the study of hypergeometric and/or confluent hypergeometric Differential Equations (DE) At this regard, many efforts have been conducted to find the intermapping between different solvable potentials [1] [2] with the aim to give a unified treatment of partner potentials [3]. In the case of potentials with the hypergeometric wavefunctions, the hexagonal diagram proposed by Cooper et al [4] is very useful to show how all the shape invariant potentials are inter-related It has been proposed a pre-potential approach to study of Eckart-type potentials [5] and a five-parameter exponential-type potential to unify the treatment of exactly solvable trigonometric potential models [6]-[8].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
