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Mathcad in the Chemistry Curriculum
Theresa Julia Zielinski Monmouth University West Long Branch, NJ 07764-1898
Mathematics in Physical Chemistry We live in a wonderful time for teaching physical chemistry. Using symbolic mathematics software appropriately, we can effectively implement the physical chemistry curriculum, bringing about even more comprehensive chemistry teaching and learning. Change has been occurring in the physical chemistry curriculum for over 75 years but its pace has increased phenomenally in the last decade due to the availability of computers. 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 wavefunctions 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 tunnel580
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ing 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 timeindependent 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. Supplemental Material Files accompanying this column are available in Mathcad versions 6 and/or 8 and 2001i. An Adobe Acrobat PDF file is provided for users of other symbolic mathematics software to aid their development of adaptations of this material. W
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.
Journal of Chemical Education • Vol. 80 No. 5 May 2003 • JChemEd.chem.wisc.edu
Information • Textbooks • Media • Resources
Visualizing Particle-in-a-Box Wavefunctions Using Mathcad: pib6.mcd, pib8.mcd, pib2001i.mcd, pib.pdf, pibInstructor.pdf Edmund L. Tisko, Department of Chemistry, University of Nebraska at Omaha, Omaha, NE 68182;
[email protected] Using the built-in differential equation solvers and graphical capabilities of Mathcad, students can visualize the wavefunctions of the particle-in-a-box potential. By applying the mathematical requirements of the wavefunction, the particle in a box is seen to have quantized energies. By plotting possible solutions, students are able to visualize the consequences of requiring the wavefunction to be continuous. Also, the step potential and barrier potential can be examined, thus allowing students to see how requiring the wavefunction to be finite results in the quantum mechanical phenomenon of tunneling. The instructor notes include Mathcad graphics illustrating some of the peculiar features of the simple particle in a box, the step potential, and the double-well barrier potential.
Figure 1. Mathcad plot (in arbitrary units) of wavefunction for quantum number n = 3 showing tunneling for the step potential V(x) = 0 at 0 < x < 2 and V(x) = 20 at 2 ≤ x < 4. Z 0 is the vector containing the values of x and Z1 is the vector containing the values of the wavefunction at x.
Femtochemistry: Femtochemistry8.mcd, FemtochemistryStudent8.mcd, Femtochemistry2001i.mcd, FemtochemistryStudent2001i.mcd, Femtochemistry.pdf, 100FS.avi Mark David Ellison, Wittenberg University, P.O. Box 720, Springfield, OH 45501;
[email protected] The goal of this document is to have students explore a simple solution to the time-dependent Schrödinger equation. This is done in the context of understanding the work, commonly called femtochemistry, of recent Nobel Prize winner Ahmed Zewail. This document is suitable for use once the students have been introduced to the time-dependent Schrödinger equation and the harmonic oscillator model. An incomplete document is given to the students in which they first review the harmonic oscillator model for molecular vibration, which is a solution to the time-independent Schrödinger equation. (A completed version of the document is available for instructors.) The students then study the process of exciting molecules with an ultra-fast laser pulse. The molecules thus excited are in a superposition that has timedependent behavior. The wavefunction of the superposition is determined, and students use Mathcad to model the timedependent behavior of the system. The accompanying animation shows the time evolution of a superposition of excited vibrational states. It models the result of a 100-fs laser pulse exciting an iodine molecule. The graph shows the probability distribution plotted as a function of displacement from the equilibrium bond length. Initially, the probability of finding the bond stretched is high. As time elapses, the wavefunction changes, and therefore, so does the probability distribution. This illustrates the ability to study a molecule’s reactivity as a function of delay time after the excitation laser pulse. Prior to the exercise students should read “Freezing Atoms in Motion: Principles of Femtochemistry and Demonstration by Laser Spectroscopy” by J. S. Baskin and A. H. Zewail ( J. Chem. Educ. 2001, 78, 737–751). The instructor should ensure that students grasp the important aspects of this article.
Figure 2. Probability distribution at time = 0 as a function of x, the harmonic oscillator displacement in meters, for the superposition of vibrational states in iodine excited by a 100-fs laser pulse.
JChemEd.chem.wisc.edu • Vol. 80 No. 5 May 2003 • Journal of Chemical Education
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