Numerical solution of Schrodinger equation: automatic generation of wave functions and densities that can be used as input to machine learning methods

Abstract

I know that the Schr¨odinger equation is solvable for simple systems like particle in a box, harmonic oscillator and hydrogen atom. But, solution of the Schr¨odinger equation for complex systems is a nontrivial problem. Our main objective in the thesis is to solve the Schr¨odinger equation by using numerical methods for One and Two dimensional systems. In our project, I have implemented two different type numerical methods one is Numerov and the other one is Finite-difference method. Firstly, I employed both these methods to solve the Schr¨odinger equation for Harmonic potential well. I have compared the results with analytical solution and it is found that the our results for Harmonic potential well is quite accurate. So in the next we employed the above numerical methods for the Gaussian potential well to get the eigenvalues and eigenvectors. This project has taken as a prototype for typical machine learning problems. I have automated this scheme to generate wave functions and densities for 10000 random potentials.