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Exploring the Uncertainty Principle

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Abstract

This Mathcad worksheet uses two Gaussian functions to guide students as they explore the uncertainty principle. The standard deviations of Gaussian functions provide convenient measures of position operator uncertainty. Students are asked to build a linear combination of orthogonal particle-in-a-box eigenfunctions to represent Gaussian functions. The document permits students to discover that a Gaussian function with smaller standard deviation requires more eigenfunctions in its expansion than a Gaussian function with a larger standard deviation. The magnitude of momentum uncertainty is related to both the number of eigenfunctions (required in the expansion of the Gaussian function) and the quantum numbers of these eigenfunctions. Through the many document exercises, students are guided to discover the relationship between position and momentum uncertainty. The document can also be used as a classroom instruction tool. Answers are embedded in the document. These can be deleted in copies of the document provided to students for in-class work or homework.

Figure 1. Plots of f(x) as a function of x for two Gaussian functions where the solid line is for the function with σ² = 0.25 and the dotted line is for the function with σ² = 2.5.

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Mathcad	TheUncertaintyPrinciple.mcd

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Mathcad	TheUncertaintyPrincip.pdf

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Keywords

Domain: Physical Chemistry;

Pedagogy: Computer-Based Learning;

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

JCE Citation