Sobre o método semi-analítico de Chhajlany e Malnev para soluções aproximadas não-perturbativas da equação de Schrödinger com potencial polinomial par

Abstract

We propose, as part of the introductory study of Quantum Mechanics at undergraduate level, to employ the Chhajlany and Malnev method (MCM) for obtaining approximate semi-analytical solutions of the unidimensional Schrödinger equation (energy eigenvalues and eigenfunctions), as an alternative to perturbative methods (which bring issues of convergence of infinite series), to methods like WKB (of semi-classical character, and not completely quantum), and to methods such as the finite difference method (purely numerical), for the study of polynomial potentials with even powers. Polynomial potentials appear, for instance, as effective potentials in the study of oscillations about local minima of a given potential. In the present work, we develop in detail the MCM for the harmonic and quartic anharmonic quantum oscillators, using a Fortran code that implements that method, for those cases. Pre-requisites for understanding the MCM are the familiarity with Schrödinger equation, as well as with basic tools of integral-differential calculus and linear algebra.