EFFECTS OF EXTERNAL POTENTIALS ON THE SOLUTIONS OF LINEAR SCHRODINGER EQUATION

Abstract

The linear Schrodinger equation is a fundamental equation in quantum mechanics, which describes the dynamics of a particle’s wave function in space and time. This project focuses on numerically studying how the addition of an external potential influences the motion of the wave function governed by the linear Schrodinger equation on the real line. We implement Python codes and compare several numerical methods - Fourier, Chebyshev, Crank-Nicolson, Leapfrog, Lax-Wendroff Methods as well as Duhamel’s pinciple and Time-discretization method-to study their performance in terms of its accuracy, stability and time efficiency. By analyzing how solutions change under attractive (ε > 0) and repulsive (ε < 0) forces, we gain insight into the interaction between wave behavior and potential energy, as well as the effectiveness of different numerical approaches for such systems.