Using Mathematical Software To Introduce Fourier Transforms in Physical Chemistry To Develop Improved Understanding of Their Applications in Analytical Chemistry

Abstract

This manuscript presents an exercise that utilizes mathematical software to explore Fourier transforms in the context of model quantum mechanical systems, thus providing a deeper mathematical understanding of relevant information often introduced and treated as a "black-box" in analytical chemistry courses. The exercise is given to undergraduate students in their third year during physical chemistry, thus providing a theoretical foundation for the subsequent introduction of such material in analytical instrumentation courses. With the reinforcement of familiar concepts such as the Heisenberg Uncertainty Principle, classical correspondence, and linear combinations in the context of both position and momentum space for a particle in a box, a better understanding of the mathematical implications of the Fourier transform is fostered. Subsequent analysis of a time-dependent function constructed via a linear combination and its transformation to the frequency domain provides a practical example relating to the Fourier processes applied in analytical spectroscopy. The final portion of the exercise returns to the position/momentum conjugate pair and explores how the construction of a narrow wavepacket via a sum of cosines illustrates the Uncertainty Principle once the probability density functions of each coordinate are analyzed. This exercise has been shown to not only reinforce fundamental concepts necessary for a true appreciation of quantum mechanics, but also help demystify the Fourier transform process for students taking analytical chemistry.