The Schrodinger Equation: Origins and Numerical Solutions

Abstract

The classical laws ofmotion and Newtonian thought met considerable opposition in the late 1800s and early 1900s as a result of several observations that were inexplicable by then current means. Several breakthroughs were made by notable historic figures such as Louis de Broglie, Werner Heisenberg, Max Planck, Albert Einstein, and ofparticular interest, Erwin Schrodinger. The new ideas and innovations ofthese and other individuals helped forge the beginnings of Quantum Mechanics and paved the way for significant understanding. Erwin Schrodinger confronted the problem of electrostatic attraction between the nucleus and electrons within atoms. His approach greatly utilized the developing theory ofwave-particle duality, introduced by Planck and Einstein, and the quantization of energy. The result of Schrodinger’s work was a wave model that fundamentally stated that all possible locations for a particle can be represented by a wave. The Schrodinger Equation establishes a mathematical relation between a particle’s energy and its wavefunction, and is considerably generalized, making it adaptable to various applications. This investigation aims to understand the observations and discoveries leading up to and resulting from the Schrodinger equation, as well as look into some simple solutions of the equation in one dimension. Specifically, five well-known instances of classical mechanical failures are presented followed by a basic approach to solving the single particle systems of a particle in a box and a simple harmonic oscillator. Numerical methods are utilized to solve the Schrodinger equation and discover allowed particle states within various potentials.