El método de mallado de Lagrange y aplicaciones en Física Cuántica

Abstract

Solving the Schrödinger equation is essential for understanding quantum physical systems. With the exception of a few systems, most are described by Hamiltonians whose mathematical resolution does not provide analytical solutions. Therefore, the application of numerical methods is convenient for their study. This paper presents a method that allows us to solve the time-independent Schrödinger equation in matrix form. The formalism used is based on states located in configuration space (CLS). Its construction begins with a family of orthogonal polynomials. With these, we will calculate relevant physical quantities such as energy or wave functions. First, the results will be compared with potentials with analytical solution such as the Morse potential. Second, it will be applied to potentials of interest in Molecular Physics, such as the Kratzer-Fues, Deng-Fan, and Varshni potentials. Finally, the method will be extended to be applied to 3D systems and the ro-vibrational espectrum of 𝑂2 molecule will be studied. The following pages will describe the theoretical framework of the method and the obtained results with a code implemented in Matlab (which can be consulted in appendix A). Both the code used to implement the method and the visualization of the results were created entirely by the author.