Numerical determination of the eigenenergies of the Schrödinger equation in one dimension

Abstract

In most approaches to teaching quantum mechanics, discrete energy levels are introduced through the example of the infinite square well. Here, the boundary conditions at the walls, namely a vanishing amplitude of the wave function, lead to quantized energy levels. This contribution extends the discussion to more realistic, smooth potentials in one dimension, including non-symmetric potentials. A numerical solution algorithm for the corresponding time-independent Schrödinger equation illustrates quantization in an intuitive manner by use of simple forward-shooting and bisection methods. It also demonstrates how computational physics can go beyond finding approximate solutions to a specific problem and truly aid the understanding of a basic concept in physics.