Solving schrödinger equation for a one dimensional system in a harmonic potential well using artificial neural network

Abstract

It is well known that the solution of the Schrödinger equation beyond the hydrogen atom is a non-trivial problem. Our main objective in this report is to use artificial neural network to solve Schrödinger equation. In the study we considered one dimensional Schrödinger wave equation in harmonic potential well as a model system. Also, the ANN is faster than the Numerov method. In our project, we have considered two and three non-interacting fermion systems. We have got the wavefunctions and probability densities for the two and three fermion systems using the Numerov algorithm. Our target is fitting of kinetic energy density functional using artificial neural network. For this, we made a dataset of about 2000 probability densities using Numerov method. The dataset is obtained by randomly changing the value of the force constant in harmonic potential well. Also, the input of the artificial neural network would be the probability densities and output would be the total energy of the system. Such models can be used to calculate the energies of one dimensional Schrödinger equation in a similar way but unknown potential energy well without really solving the Schrödinger equation analytically.