Exactly Solvable Schrodinger Equation with Hypergeometric Wavefunctions  

作　　者：J.Morales J.García-Martínez J.García-Ravelo J.J.Pena 

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

出　　处：《Journal of Applied Mathematics and Physics》2015年第11期1454-1471,共18页应用数学与应用物理（英文）

基　　金：supported by the projects UAM-A-CBI-2232004 and 009.JGR thanks to the Instituto Politécnico Nacional for the financial support given through the COFAA-IPN project SIP-200150019.

摘　　要：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.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.

关 键 词：Canonical Transformation Schrodinger-Like Equation Hypergeometric DE Exactly-Solvable Potentials 

分 类 号：O1[理学—数学]