 Uncovering the Fourier Transform
In our vigorous teaching of concepts and skills to students, we may cover (hide) more than we uncover, obscuring significant relationships between mathematical models and their associated chemical concepts with excessive mathematical derivations. To set the record straight, I find that mathematical treatments of physical phenomena are beautiful and elegant. Students should know from where the equations and simplifications leading to them arise. They should know the limits of the equations in order to use them properly. However, this can be the Siren's song. For example, the mathematical representation of the Fourier transform and its significance as presented in most texts are too brief to convey understanding to the typical undergraduate student. Furthermore, a few hand calculations would not permit deeper probing of the method and its intimate link to spectroscopy. The Fourier transform is a very good example of how symbolic equation software can help to uncover the science by making the mathematical manipulations easier and the mathematical concepts more accessible.
Gaining Greater Insights with Technology
Not so long ago, when symbolic software was not readily available for student use, many could show students the mathematical foundation of the Fourier Transform but few of their students had the time, energy, or programing skills to explore the concepts independently. The Fourier Transform was for all practical purposes a 'black box' into which they could peak but into which they could not reach for hands on exploration. Symbolic mathematics software puts power into the hands of students.
At the 2nd Physical Chemistry Using Mathcad workshop organized by Sidney Young (University of South Alabama), Jeffry Madura (Duquesne University), and Andrzej Wierzbicki (University of South Alabama), Peter Atkins gave a keynote address. In this address he characterized mathematics in chemistry as the art of rendering the qualitative quantitative. The rendering role of mathematics in physical chemistry is a stumbling block for many students and a challenge for their teachers. How much mathematics should students know and be able to do easily in a first course in physical chemistry and sebsequent courses that have physical chemistry as a prerequisite? What can we do to give students the mathematical skills and insights needed to better understand the quantitative aspects of chemistry? Here the power of using symbolic equation engines becomes apparent.
Mathcad and other symbolic mathematics engines permit concept development by appealing to the visual sensitivity of the chemistry student. With symbolic software, faculty can hide difficult and tedious mathematical processes from beginning students. The software can make the mathematics easier, opening the course for a dramatic infusion of concept building, deeper understanding, critical thinking, and more penetrating insights into physical phenomena through mathematical modeling.
The Fourier Transform: A Central Mathematical Tool for Chemists
Fourier transforms are central to our major spectroscopic techniques and to the determination of molecular structure by diffraction. Providing students with interactive symbolic mathematics documents with which to explore this topic is an excellent use of software in the aid of teaching.
In the February 1999 appearance of the Mathcad in the Chemistry Curriculum column we present the work of three teaching chemists. Each document or a set of documents develops the concepts of the Fourier transform. The documents are similar but different. They will help students to develop essential skills and insights and to better appreciate the spectroscopic instruments that are part of the routine practice of chemists.
Scott van Bramer sets the stage with a single interactive tutorial on the Fourier transform. Through a series of presentations and interactive lessons students can capture the essence of the Fourier transforms and how they lead to recognizable signals recorded in the various spectroscopic experiments. van Bramer’s tutorial document is accompanied by an instructor’s document that can be used for interactive presentations before an entire class.
W. Tandy Grubbs provides a different perspective by focusing on the vibrational frequencies of a diatomic molecule. His three documents examine harmonic and anharmonic potential functions and extend students´ skills by solving the equations of motion using the Runge-Kutta method.
Mark Iannone also provides an introduction to the Fourier transform in his four documents, which overlap with the van Bramer document in some content but not in approach. The style of each author clearly illustrates the rich diversity of application and implementation possible with symbolic mathematics software. The documents presented here provide instructors with an array of tools with which to increase student understanding of the Fourier Transform.
The Mathcad documents require Mathcad Plus 6 or higher. They have been tested with Mathcad 7. All of the documents are accompanied by pdf files showing the content. My hope is that the pdf files will be used as models by supporters of other symbolic software packages to create interactive learning materials for their students.
Acknowledgement
The author acknowledges the National Science Foundation for support of the 1997 NSF-UFE Workshop on "Numerical Methods in the Undergraduate Chemistry Curriculum Using the Mathcad Software" and the organizers Jeff Madura, Andrzei Wierzbicki, and Sidney Young of the University of South Alabama. The author also acknowledges that additional partial support was provided by the National Science Foundation's Division of Undergraduate Education through grant DUE #9455928, the New Traditions project at the University of Wisconsin - Madison.
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