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Exploring the Harmonic Oscillator Wave Function Components

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Abstract

The goal of this document is to have students explore the solutions to the quantum mechanical harmonic oscillator Schrödinger equation. Students do this by examining the exponential component, generating the various Hermite polynomials, and then creating the final HO wave functions as a product of a normalization factor, the exponential component, and a Hermite polynomial. Students also prepare plots of several harmonic oscillator wave functions and probability densities. The plots created with Mathcad provide an effective way for students to draw their own figures and master understanding of the concepts involved. Finally, students can examine the plots at both high and low quantum numbers and correlate their observations with classical harmonic oscillator predictions of maximum and minimum kinetic energy and maximum probability of the extension of the oscillator.

Figure 1. Plots of 1st, 2nd, and 4th harmonic oscillator wave functions drawn with Mathcad. Students can generate many such plots and explore their properties.

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Mathcad 11	HOExploration11.mcd

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Mathcad 11	HOExploration11.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