Numerical Solution of the 1D-Schr\"odinger Equation with Pseudo-Delta Barrier Using Numerov Method

TL;DR

This paper develops a numerical approach using the Numerov method to solve the 1D Schrödinger equation with pseudo-delta potentials, demonstrating accurate results for complex quantum systems with delta barriers.

Contribution

It introduces a numerical solution for the Schrödinger equation with pseudo-delta potentials using the Numerov method, including systems with multiple delta functions and position-dependent mass.

Findings

Numerical results agree well with analytical solutions.

Successfully modeled systems with multiple delta potentials.

Extended to systems with harmonic potential and position-dependent mass.

Abstract

In this work, aiming to solve numerically the Schr\"odinger equation with a Dirac delta function potential, we use the Numerov method to solve the time independent 1D-Schr\"odinger equation with potentials of the form V(x) + deltap(x), where deltap(x) is a pseudo-delta function, a very high and thin barrier. The numerical results show good agreement with analytical results found in the literature. Furthermore, we show the numerical solutions of a system formed by three delta function potentials inside of an infinite quantum well and the harmonic potential with position dependent mass and a delta barrier in the center.