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Particle-in-a-Box Dynamics

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Abstract

The movement of an electron or excitation energy (an exciton) through space from one molecule (or part of a molecule) to another is a common and important process, for example in photosynthesis, in enzyme-mediated redox reactions, as well as in semiconductors. Time-dependent quantum mechanics is needed to understand and describe such phenomena. This activity provides a very simple example of a quantum mechanical description of electron or exciton motion. The document is designed for use in junior–senior level quantum chemistry courses. Successful use of the document requires that students be familiar with expectation values and the physical interpretation given to a wave function, complex numbers and functions, and solution of the time-independent and time-dependent Schrödinger equations for a one-dimensional particle in a box. The document can be used by students in group study either in class or as homework. Instructors can use the document for lecture demonstration and discussion. Clear goals and objectives focus student attention on the key question, electron or exciton motion. Ample questions and exercises further help students master the concepts. The document concludes with a set of reflection questions to help students retain concepts practiced while doing the exercise.

Figure 1. The probability densities for the two particles at T = 0. Students can animate this graph to see how the probability densities change with time. A movie of the animation is provided at this link.

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Version	Download
Mathcad 11	DynamicsMCD11.mcd

Version	Download
Mathcad 11	dynamics.pdf

Animation	ParticleDynamics.avi
	Two non interacting particles are trapped in a 1-D box. Particle-a is described by the n=1 wave function Ψ 1(X,T). Particle-b is described by the wave function Ψ 3(X,T), which is a linear combination (i.e. sum) of the n = 1 and n = 2 wave functions Ψ 1(X,T) and Ψ 2(X,T). The time-dependent radiation field at resonance with 2 states drives the molecule back and forth between the states 1 and 2. This is described by the changing probability density shown in the animation for the probability density of Ψ 3(X,T) shown as Pb(X,T) in the animation.

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Keywords

Domain: Physical Chemistry;

Pedagogy: Computer-Based Learning; Inquiry-Based / Discovery Learning;

Topics: Mathematics / Symbolic Mathematics; Quantum Chemistry; Theoretical Chemistry;

JCE Citation