Solução da equação de Schrodinger pelo método variacional com redes neurais artificiais

Abstract

This work aims to solve the time-independent Schrodinger equation through the variational method, which consists of using the expected value of the Hamiltonian operator as a functional of the system's wave functions, minimizing this functional to find the eigenfunctions. At this point we introduce multilayer perceptron artificial neural networks, which can be used to approximate a real function of several variables, this result is guaranteed by the universal approximation theorem. We use the neural networks in the functional of the expected value of H, and minimize this functional in relation to the neural network parameters, thus obtaining an approximation for each of the eigenfunctions of the system. We employ this method in one-dimensional quantum systems, the harmonic oscillator and the square well, we find the first three eigenfunctions of these systems and also their first three eigenergies. We used the Python programming language and its libraries to computationally implement the method and perform the minimization. The results found were very positive and show that the method is efficient in solving one-dimensional systems.