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JCE SymMath: Symbolic Mathematics in Chemistry
edited by
Theresa Julia Zielinski
Monmouth University West Long Branch, NJ 07764-1898
Demystifying Some Advanced Undergraduate Chemical Computations
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There are very good reasons to recommend the use of modern mathematical tools in physical chemistry. First, these tools can increase efficiency and effectiveness of learning physical chemistry. A Symbolic Mathematics Engine (SME) is one way to have students develop greater appreciation of the mathematical models of physical chemistry without the tedium of mathematical manipulations that don’t seem to be chemically intuitive. This column introduces four symbolic mathematics worksheets for learners of physical chemistry and quantitative/ instrumental analysis. These templates are specifically designed to take advantage of the capabilities of the SME software in the service of more efficient and effective instruction.
In Descriptive Statistics, students enter a set of data points and obtain the average, standard deviation, and confidence intervals for the data. Comparative Statistics permits students to calculate the average, standard deviation (N and N21 weighted), and confidence intervals for each data set, and also t-test for comparison of means of two data sets. These templates encourage students to understand how the number of samples, the standard deviation, and the required certainty affect the ability to identify when there is a significant difference between two averages and explain why statistical tests are stated as using a null hypothesis. These Mathcad documents could be used in physical chemistry, analytical chemistry, or instrumental methods of analysis.
Constructing Hydrogen-like Radial Wave Functions
Harmonic Oscillator Wave Functions
In Construction of the Electronic Radial Wave Functions and Probability Distributions of Hydrogen-like Systems, Tom Kuntzleman leads students to examine in detail the components of the solution of the radial function for the hydrogen atom. The polynomial portion of the radial function solution and the associated Laguerre functions are demystified through examination of their mathematical generation, form, and plots. Coupling these to the radial solution normalization factor and exponential factor enables students to examine the complete radial function and gain an understanding not possible by just examining an equation and a table in a text. The directions in the template could easily be completed by the typical junior-level chemistry major. Faculty and graduate students would also find this template enlightening.
In Exploring Harmonic Oscillator Wave Functions Theresa Zielinski brings together a set of exercises that help students comprehend the components of the solutions to the harmonic oscillator Schrödinger equation. The components include the normalization factor, the Gaussian exponential, and the Hermite polynomials. Students build each component and examine its properties. The components are then multiplied to get the HO solutions. A large number of HO solutions can be examined interactively. Their probability density plots can easily be prepared and analyzed. Students can then examine the relationship between the quantum mechanical harmonic oscillator and the classical harmonic oscillator at both small and large quantum numbers. When students discover that the probability density increases at the extrema of the oscillator extensions as the quantum number increases, they can see more clearly the connection between classical and quantum oscillators.
The Gaussian Distribution, Sample Statistics, and Student’s t Statistic Scott van Bramer, in his collection, A Brief Introduction to the Gaussian Distribution, Sample Statistics, and the Student’s t Statistic, provides a suite of templates that serve as an introduction to important statistical concepts for undergraduate students. Using the Gaussian template, students build a basic understanding of the Gaussian function and its connection to probability. The Sample Statistics template introduces students to small-sample statistics to help them build an understanding of statistical techniques. In the Student’s t Statistic, students increase their skills to include determination of the confidence interval for an average determined from a small number of samples. They also gain an understanding of how the number of samples, the standard deviation, and the required certainty affect the confidence interval. They can also gain skill in deciding whether an average determined from a small number of samples is significantly different from a known or expected value. Two other documents round out the suite: Descriptive Statistics and Comparative Statistics.
Particle-in-a-Box Dynamics In the Particle-in-a-Box Dynamics template, Hanson and Zielinski present an introduction to time-dependent quantum mechanics. The focus is two particles trapped in a one-dimensional box (PIB). The first particle is described by the n 5 1 PIB wave function. The second particle is described by a linear combination of the n 5 1 and n 5 2 PIB wave functions. In the template students examine the two wave functions, both their real and imaginary parts, using the animation feature of Mathcad. For particle two, a time-dependent radiation field at resonance with the n 5 1 and n 5 2 states drives the molecule back and forth between states 1 and 2. The oscillation is shown in the probability density as a function of time for particle two. This document provides an important first exposure to time dependent quantum mechanics for undergraduate physical chemistry students. The template would also be useful for beginning graduate students specializing in spectroscopy.
1230 Journal of Chemical Education • Vol. 84 No. 7 July 2007 • www.JCE.DivCHED.org
