[Article] The Action Of Canonical Transformations On Functions Defined On The Configuration Space

[Article] The Action Of Canonical Transformations On Functions Defined On The Configuration Space

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INVESTIGACI ´ ON REVISTA MEXICANA DE F ´ ISICA 56 (2) 113–117 ABRIL 2010 The action of canonical transformations on functions defined on the configuration space G.F. Torres del Castillo Departamento de F´ ısica Matem´atica, Instituto de Ciencias, Universidad Aut´onoma de Puebla, 72570 Puebla, Pue., M´exico. D.A. Rosete ´ Alvarez and I. Fuentecilla C´arcamo Facultad de Ciencias F´ ısico Matem´aticas, Universidad Aut´onoma de Puebla, Apartado Postal 1152 Puebla, 72001, Pue., M´exico. Recibido el 15 de octubre de 2008; aceptado el 3 de marzo de 2010 The effect of an arbitrary canonical transformation on functions defined on the configuration space is defined in such a way that a solution to the time-independent Hamilton–Jacobi equation is mapped into another solution if the Hamiltonian is invariant under the canonical transformation. Keywords: Canonical transformations; Hamilton-Jacobi equation. Se define el efecto de una transformaci´on can´onica arbitraria sobre funciones definidas en el espacio de configuraci´on en tal forma que una soluci´on de la ecuaci´on de Hamilton–Jacobi independiente del tiempo es enviada en otra soluci´on si la hamiltoniana es invariante bajo la transformaci´on can´onica. Descriptores: Transformaciones can´onicas; ecuaci´on de Hamilton-Jacobi. PACS: 45.20.Jj; 02.20.Qs 1. Introduction In the Hamiltonian formalism of classical mechanics, the so- lution to the equations of motion can be obtained with the aid of the Hamilton–Jacobi (HJ) equation; a complete solution to the HJ equation is the generating function of a canonical transformation that relates the original phase space coordi- nates to a new set of canonical coordinates that are constant in time (see, e.g. , Refs. 1 to 3). The HJ equation is the classical limit of the Schr¨odinger equation and, therefore, the solutions to the HJ equation provide the lowest-order part of the semi- classical approximation for the solutions to the Schr¨odinger equation (see, e.g. , Ref. 4). The HJ equation and the Schr¨odinger equation involve only the time and one half of the phase space coordinates (usually a set of coordinates of the configuration space). In the framework of quantum mechanics, the problem of find- ing the effect of an arbitrary change of coordinates has been recognized, and has only been solved in the case of point transformations or of linear canonical transformations (see, e.g. , Refs. 5, 6, and the references cited therein). The aim of this paper is to give a natural definition for the action of a canonical transformation on Hamilton’s character- istic function (which satisfies the time-independent HJ equa- tion), or on any function defined on the configuration space.
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[Article] The Action Of Canonical Transformations On Functions Defined On The Configuration Space

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