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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 1. 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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Citation
