Using Computers to Visualize and Reason with Quantum Concepts Peter Garik, 1 Alan Crosby, 2 Dan Dill, 2 Alex Golger, 2 Morton Z. Hoffman 2 1) School of Education 2) Department of Chemistry Boston University Boston, Massachusetts

The Challenge-I Quantum Concepts are among the most challenging topics in general chemistry. Some instructors are uncomfortable about teaching that material. Our objective is to present the time- dependent interaction of light with matter to science and pre-medical general chemistry students (CH101).

The Challenge-II Students are weak in their understanding of the energetics of waves and the nature of fields. They generally do not know that EM waves have electric and magnetic fields associated with them. They do not understand how light interacts with electrons. They do not associate the energy of emitted or absorbed light with a difference in electronic energy levels.

Our Approach Learning cycle-based activities: data collection, analysis, extension. Use of bridging analogies whenever possible with computer visualization and representation. Guided-inquiry approach and software Interactive graphical renderings of time- dependent atomic orbitals and their interaction with light without mathematics! Visualizations of the beats that correspond to dipole excitations of atoms.

Learning Objectives Spatial and time dependence of normal modes. Spatial and time dependence of the complex wavefunction. The Planck relationship for electrons: |E| = h. The superposition of wavefunctions and the resulting oscillation at the difference between the two Planck frequencies (beating).

Guided Inquiry Software Used in conjunction with lecture demonstrations, lecture/discussion workshops, lab exercises, and homework. Project 1: spectroscopy of atomic hydrogen and hydrogen-like ions. Project 2: introduction to the normal modes of one- (cable) and two-dimensional (square and circular membranes) waves with analogy to the modes of a bound electron. Project 3: time-dependent behavior of electron orbitals and their interaction with light.

Waves

Harmonics of an Oscillating Cable Students studied the harmonics of an oscillating cable by measuring the amplitude, wavelength, period, and frequency. The connection between the spatial and temporal aspects of the wave was emphasized through the use of two related graphs.

Energy Density for an Oscillating Cable The total energy density, as well as the kinetic and potential energy densities at a point, are connected by the interactive graph of displacement and the energy histogram.

Representations of a One- Dimensional Harmonics Eventually, students work with a visual representation in which the intensity is colored coded. The two representations are presented side-by-side in order to build familiarity.

Two-Dimensional Normal Modes For the 2-D modes, two representations are initially provided. Each has its own advantage for displaying the frequency of the mode. The 3-D display of the 2-D mode can be rotated in space.

Beats in Two-Dimensions To understand the phenomena of beats, students are asked to measure the frequency of superposed harmonic modes. Visually this is difficult for arbitrary modes; however we restrict the activity to those for which they can be successful.

Harmonics of an Oscillating Disk The activities performed for the rectilinear geometries are now repeated but with cylindrical symmetry. The aim is to help students develop an understanding of rotational degeneracy.

Harmonics of an Hydrogenic Electron Students are now expected to extend their understanding of harmonics to the normal modes of hydrogenic orbitals. To assist them, a phasor indicating the phase both by color and complex number is provided.

Beating of a Hydrogen Atom In a lecture demonstration, the students heard the beats of superposed sound waves. Now they will strive to “see” the connection of emitted light to superposed orbital frequencies.

Assessing the Efficacy of the Approach Pre- and post-tests and student interviews were used. The sophistication of the questions demonstrated our level of expectation to the students. The test results from more than 500 general chemistry students suggest that they can master the concepts that underlie the modern quantum model of chemistry, spectroscopy, and nanotechnology.

Energy Distribution in a Wave Pre Test B (N=304) Post Test B (N=288) a) The energy of vibration is uniformly distributed over the length of the string. At this instant, all the energy is kinetic energy of motion. 47.2%3.2% b) The energy is uniformly distributed over the length of the string. At this instant, all the energy is potential energy. 17.5%8.1% c) The energy is non-uniformly distributed over the length of the string. At this instant, all the energy is kinetic energy. 18.2%7.7% d) The energy is non-uniformly distributed over the length of the string. At this instant, all the energy is potential energy. 6.3%64.5% e) None of the above.9.6%13.7%

Learning About the Complex Nature of the Wavefunction On the post-test we asked the students the question, “At a specific time, one lobe of a 2p orbital has a phase angle corresponding to 1 + i. Which of the following complex numbers corresponds to the phase of the second lobe?” This is a question which would have been meaningless on the pre-test. 58.3% of the general chemistry students (N = 550) selected -1 – i. Since the complex nature of the wavefunction had not been discussed in lecture in the general chemistry course, this result suggests success for our visualization methods for the complex phase.

Time Dependence of the Wavefunction 8.1 Select the best explanation for the time-dependence of an electron in an atomic orbital. a) The electron has a velocity that corresponds to its kinetic energy, which varies as it approaches or recedes from the nucleus. b) The time-dependence for an electron in an orbital corresponds to the uncertainty in its position and, therefore, its instantaneous velocity. c) An atomic orbital has complex values with periodic oscillation in its value with time. d) The square of the values of an atomic orbital corresponds to a probability density for finding an electron at a point in space; this does not vary in time, so there is no time-dependence for the orbital. e) None of the above.

Time Dependence of the Wavefunction c) An atomic orbital has complex values with periodic oscillation in its value with time. Pre-test A (N=235):18.7% Post-test A (N=201):51.6% Post-test B (N=274):62.4% Physical chemistry Post-test (N=22):22.7%

Conclusions-I Our results support the conclusion that general chemistry students can learn about quantum concepts through the use of guided-inquiry interactive graphics and visualizations.

Conclusions-II The vocabulary of time-dependent electron orbitals provides new insights for the students about the absorption and emission of electromagnetic radiation across the spectrum, van der Waals interactions, and London dispersion forces.

Acknowledgements Peter Carr, Programmer Joshua Csehak and Lars Travers, Ace Coders Programming Judith Kelley and Russell Faux, Project Evaluators Funding, U.S. Department of Education Fund for the Improvement of Post Secondary Education (FIPSE Grant P116B020856)