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Exactly Solvable Schrodinger Equation with Hypergeometric Wavefunctions

Journal of Applied Mathematics and Physics 2015年第11期3卷 1454-1471页

作者： J.Morales J.García-Martínez J.García-Ravelo J.J.Pena Universidad Autónoma Metropolitana-Azc. DCB-Area de FAMASn Pablo 180Mexico CityMéxico Tecnológico de Estudios Superiores de Ixtapaluca División de Ingeniería Informática e Ingeniería BiomédicaCarretera Ixtapaluca Coatepec Km.7Mexico CityMéxico Escuela Superior de Física y Matemáticas IPN-ZacatencoUP Adolfo López MateosMéxico

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

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Gradient transformation of Momentum and Single-Valued Vector Potential in Nonrelativistic Dynamics

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Journal of Physical Science and Application 2014年第5期4卷 328-332页

作者： Illia Dubrovskyi Institute for Metal Physics National Academy of Science Ukraine Kyiv 03680 Ukraine

A vector potential of a magnetic field in Lagrangian is defined as the necessary partial solution of a inhomogeneous differential equation. The ＂gradient transformation＂ is an addition of arbitrary general solution of the corresponding homogeneous equation that does not change the Lagrange equations. When dynamics is described by momenta and coordinates, this transformation is not the vector potential modification, which does not change expressions for other physical quantities, but a canonical transformation of momentum, which changes expressions for all fimctions of momentum, not changing the Poisson brackets, and, hence, the integrals of motion. The generating function of this transformation must reverse sign under the time-charge reversal. In quantum mechanics the unitary transformation corresponds to this canonical transformation. It also does not change the commutation relations. The phase of this unitary operator also must reverse sign under the time-charge reversal. Examples of necessary vector potentials for some magnetic fields are presented.

关键词： Magnetic field vector potential momentum gradient transformation canonical transformation unitary transformation.

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