 Demystifying Some Advanced Undergraduate Chemical Computations
Often students come to their first physical chemistry courses with rusty mathematics skills. Consequently introducing a symbolic mathematics engine, SME, may not seem like the best method for quickly enhancing mathematics skills. One could also argue that since physical chemistry is not a computer science course, we should not be teaching students with SME software. Similar arguments could be use with respect to physical chemistry's role as a writing intensive course in a chemistry curriculum. I don't think anyone would recommend giving up the writing component of physical chemistry courses.
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. An SME is one way to have students develop greater appreciation of the mathematical models of physical chemistry without the tedium of non-chemically intuitive mathematical manipulations. In this column I introduce four symbolic mathematics worksheets that meet the needs of 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.
Constructing Hydrogen-like Radial Wave Functions
In the document Construction of the Electronic Radial Wave Functions and Probability Distributions of Hydrogen-like Systems by Tom Kuntzleman students are lead through the process of examining 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. When coupled to the radial solution normalization factor and exponential factor students can 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.
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. With the Gaussian template students build 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 if 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. In the Descriptive Statistics document 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 N-1 weighted), and confidence intervals for each data set, and also t-test for comparison of means of two data sets. Through use of these templates students should be encouraged 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.
Harmonic Oscillator Wave Functions
In Exploring Harmonic Oscillator Wave Functions Theresa Zielinski brings together a set of exercises that help students to comprehend the components of the solutions to the harmonic oscillator Schrodinger Equation. The components include the normalization factor, the Gaussian exponential, and the Hermite polynomials. Students build each component and examine the properties of each. The components are then multiplied together 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 increasing at the extrema of the oscillator extensions as the quantum number increases they can see more clearly the connection between classical and quantum oscillators.
Particle in a Box Dynamics
In the template Particle in a Box Dynamics 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 = 1 PIB wave function. The second particle is described by a linear combination of the n = 1 and n = 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 = 1 and n = 2 states drives the molecule back and forth between the 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.
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