| 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 time-dependent behavior. The wavefunction of the superposition is determined, and students use Mathcad to model the time-dependent 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..
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| Figure. 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. |
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