Machine learning approach to solve schrodinger equation for a one dimensional system in a gaussian potential well and in a harmonic potential well

Abstract

It is well known that the solution of the Schrodinger equation beyond the hydrogen atom is a non-trivial problem. Our main objective in the thesis is to use machine learning techniques to solve Schrodinger equation. In the study we considered one dimensional Schrodinger wave equation in Gaussian potential well and also in harmonic potential well as a case study to use machine learning techniques. Since the 1- dimensional Schrodinger equation in Gaussian potential well has no analytical solution, we used the Numerov method to solve it. Similarly, the Numerov method is used to solve the 1- dimensional Schrodinger wave equation in harmonic potential well. Using the numerov method, we calculated wave functions, probability densities and energies for systems with single electrons, 2 electrons, 3 electrons and 4 electrons. Now our aim is to map the probability densities to energies using artificial neural networks. For this we made a dataset of about 5000 probability densities using the Numerov method and train these probability densities to the known energies obtained. The dataset is obtained by randomly changing the parameters of Gaussian potential well and the force constant values in harmonic potential well. The inputs for the artificial neural networks will be probability densities and the output will be total energies of the system. Such models can be used to calculate energies of 1- dimensional Schrodinger equation in a similar but unknown potential energy well without really solving the Schrodinger equation analytically.