Schrödinger Equation Solutions That Lead to the Solution for the Hydrogen Atom

Abstract

Particle in the box (PIB) solutions to the Schrodinger equations are not new to physical chemistry. Many have been developed to demonstrate complex quantum mechanical techniques in a simplified form. This article outlines two PIB solutions to the Schrodinger equation that increase gradually in complexity and lead to the solution for the H atom. In addition to the PIB solutions, laboratory exercises are cited that can be modeled by the PIB solutions. The two PIB solutions are a particle in a cylindrical box (two-dimensional problem) and a particle in a spherical box (three-dimensional problem). The student is exposed to new functions (i.e., Legendre, Bessel, and spherical Bessel) not commonly seen in undergraduate physical chemistry classes. Pre-H atom quantum theory students should benefit from these exercises, which provide a more thorough introduction to the advanced mathematics encountered in the H atom development but not found in most Cartesian coordinate PIB treatments.