Sistemas integrables y superintegrables en bajas dimensiones: el potencial Morse

Abstract

Este trabajo estudia los sistemas integrables y superintegrables en bajas dimensiones proponiendo un método para su obtención. Se particulariza dicho análisis a la MASA nilpotente de su(1,1), la cual nos lleva al bien conocido potencial Morse. Formulado este, se plantean las factorizaciones tanto clásica como cuántica del Hamiltoniano asociado a dicho potencial.

The aim of the study we are about to conduct is to go in depth in the properties of integrable low dimensional systems by the construcction, factorization, and resolution of the Morse 1-dimensional one. In this way, we will begin with a review of the basics that justify and define this analysis. Here, we will discuss about the Algebra, the Lie groups, and other mathematical notions closely linked with the symmetries that characterize this systems.We will remember also, the standart elements typical of the hamiltonian mechanic, from the classic point of view as well as the quantum one, which we will work with in our following factorization. Then, we will delve into this integrable systems, showing his properties, relating them with the superintegrable ones and suggesting a general method to build them. After this introductory parragraph, we will fix this notions to our Morse potential offer. In the next sections, we will use the Hamiltonian built to split it with a classical and quantum factorization, considering the interchange operators, the obtaining and ploting of eigenstates, and the determination of the symmetry relations and their group behavior. Finally, we will conclude our study with a few considetations about the whole proces, the techniques applied, etc.