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Construction of the Electronic Angular Wave Functions and Probability Distributions of the Hydrogen Atom

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Abstract

This Mathcad document allows students to explore the characteristics of the components of the angular electronic wave function of the hydrogen atom, Y_l^m(Θ, φ). Within this exercise, Y_l^m(Θ, φ) is separated into three parts: the normalization constant, a polynomial function and an exponential function. After choosing quantum numbers l and m, users can evaluate and graphically display the component functions of Y_l^m(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 Y_l^m(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.

Figure 1. The first five Legendre Polynomials, Leg(order,x) as a function of x.

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Mathcad 11	AngularWavefunctions11.mcd

Version	Download
Mathcad 11	AngularWavefunctions11.pdf

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Keywords

Audience: Upper-Division Undergraduate;

Domain: Physical Chemistry;

Pedagogy: Computer-Based Learning;

Topics: Chemometrics; Mathematics / Symbolic Mathematics; Quantum Chemistry;

JCE Citation