The story of change starts with Farrington Daniels in the 1920s at the University of Wisconsin–Madison. There Daniels was assigned to teach advanced physical chemistry and a course in calculus for chemists. Daniels, never having had a course in calculus, kept one step ahead of the students and eventually wrote the first book on calculus for physical chemistry (1–3). A more recent textbook contribution to mathematics for physical chemistry is by Francl (4). Examination of the two volumes shows a striking overlap. Francl calls calculus for physical chemistry “guerilla math”. I am sure this is what most students think as well. Nevertheless mathematics is essential for progress in physical chemistry, even for those who will not become physical chemists. Mathematics may also be one reason why many students do not choose to be chemistry majors.

Curriculum change incorporating symbolic mathematics software allows instructors to provide students with more meaningful mathematical experiences (even students with only two semesters of calculus), leading to better understanding of core concepts. The templates available in symbolic mathematics software furnish the tools students need to advance their mathematical skills, appreciate mathematical models in science, and apply understanding of models to the practice of spectroscopy. Well-crafted templates permit exploration and discovery of concepts free from the drudgery of programming or numerous error-prone hand calculations and plots. Using these templates encourages students to focus on concepts and promotes instructor–student discussion.

Visualizing Particle-in-a-Box Wavefunctions

In this column we introduce two new Mathcad documents. The first, “Visualizing Particle-in-a-Box Wavefunctions Using Mathcad”, provides a series of laboratory exercises that explore the properties of wavefunctions for the simple particle in a box, a step potential, and a double-well potential. The template is arranged with clear instructions that permit students to proceed after learning only a few basic Mathcad operations. Students learn how the wave-functions look and the effect of the size of the box on energy level separation. They learn that the boundary condition forces quantization and can verify the relationship between length of the box and particle energy. The template uses the Runge–Kutta method for numerically solving the differential equations. It provides a concrete introduction to the tunneling concept and the relationship between energy and the curvature of the wavefunction. Students are asked to find the eigenvalue and wavefunction by iteratively varying the initial energy and testing the resulting wavefunction to see if it matches the boundary properties of a quantum mechanical wavefunction. An instructor’s document details the method used in the template and provides sample wavefunction plots for the three cases studied. The template would be an ideal laboratory assignment early in the semester when students would be learning Mathcad and beginning to see the solution to simple box models in the lecture portion of the course.

Femtosecond Spectroscopy

The second document takes students on an exploration of femtosecond spectroscopy. This template would be a valuable addition to the latter part of the traditional quantum chemistry semester for undergraduates and first-year graduate students. Here the focus is the harmonic oscillator in both the time-independent and time-dependent forms. Students first practice determining energy levels for the time-independent harmonic oscillator. They then move to explore the distribution of energies in an ultra-short laser pulse and see how these energies contribute to the population of multiple excited states of a harmonic oscillator. The lesson includes consideration of the Franck–Condon factors and the probability of any particular excited state being populated. Both 100- and 350-fs laser pulses are considered, and the time dependence of the population of the excited state probability distribution is examined. The author provides an animation, developed using Mathcad, for the 100-fs time evolution of the populated states. The ‘Femtochemistry’ template enables students to work with models based on the work of Baskin and Zewail (5). All of the concepts presented in this document are easily within the reach of the typical undergraduate student, and working with this template makes it clear how knowledge of quantum chemistry models is important for modern spectroscopy.

Literature Cited

1. Daniels, F. J. Chem. Educ. 1958, 35, 322–333.
2. Daniels, F. J. Chem. Educ. 1929, 6, 254–259.
3. Daniels, F. Mathematical Preparation for Physical Chemistry; McGraw-Hill: New York, 1928.
4. Francl, M. Survival Guide for Physical Chemistry; Physics Curriculum and Instruction, Inc.: Lakeville, 2001.
5. Baskin, J. S.; Zewail, A. H. J. Chem. Educ. 2001, 78, 737–751.