Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Atom©
Mark David Ellison
Wittenberg University
P. O. Box 720
Springfield, OH 45501
United States
mail to: mellison@wittenberg.edu

Thomas S Kuntzelman
Department of Chemistry
Spring Arbor University
Spring Arbor, MI 49283
United States
mail to: tkuntzle@arbor.edu

John Tippin
Deparment of Mathematics
Spring Arbor University
Spring Arbor, MI 49283
United States

This Mathcad document allows students to explore the characteristics of the components of the angular electronic wave function of the hydrogen atom,Ylm(θ, φ). Within this exercise, Ylm(θ, φ) is separated into three parts: a normalization constant, a polynomial function and an exponential function. After choosing quantum numbers l and m, users can evaluate graphically display the component functions of Ylm(θ, φ) separately. After combining these components, the entire wave function is constructed and a graph of its associated probability density is displayed. Users have the opportunity to discover when, why and how linear combinations of the exponential portion of Ylm(θ, φ) are used in constructing orbital plots that are normally displayed in textbooks. In addition, students gain a mathematical understanding of why the quantum number m can only vary from -l to l. This document illustrates a method by which one may generate and present graphs of the probability distributions of the angular electronic wave function for the hydrogen atom "from scratch". No attempt is made to derive this wave function.
Editor's Commentary
Audiences: Upper-Division Undergraduate
Pedagogies: Computer-Based Learning
Domains: Physical Chemistry
Topics: Mathematics / Symbolic Mathematics, Quantum Chemistry
File NameDescriptionSoftware TypeSoftware Version
AngularWavefunctions11.mcd JCE Mathcad Computational Document Mathcad 11
JCE JCE Subscribers only: name and password or institutional IP number access required.
Kuntzleman, T. S.; Ellison, M. D.; Tippin, J. J. Chem. Educ. 2007, 84, 1886
©Copyright 2007 Journal of Chemical Education