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Exploring Orthonormal Functions

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Abstract

In this document students explore orthonormal functions using particle-in-a-box wave functions. The particle-in-a-box wave functions, which form a complete orthonormal set of functions, are also used for fitting a linear combination of these functions to a target function. Students compare the quality of the fit between the target function and the linear combination of particle-in-a-box functions and calculate standard deviations. Thus students obtain an introduction to the concept of the expansion of a target function in terms of the components of a complete set of orthonormal functions. The document in this way serves as a preliminary foundation for studying Fourier series expansions. The use of a set of functions as a basis set further prepares students to understand linear combinations of atomic orbitals and basis sets used in ab initio calculations. An introductory overlap matrix exercise is included in this template. This provides a simple introduction to the concept of the overlap integral and the matrix representations of overlap in quantum calculations.

Figure 1. Fitting a linear combination of the first two particle-in-a-box Fy (dotted curve) wave functions to the target function fy_i = y_i (l − y_i)³ (solid curve). When using the document students discover that using 7 particle-in-a-box wave functions brings the two curves into coincidence with a standard deviation of 0.003.

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Mathcad 11	orthonormal11.mcd

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Mathcad 11	orthonormal11.pdf

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Audience: Upper-Division Undergraduate;

Domain: Physical Chemistry;

Pedagogy: Computer-Based Learning;

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

JCE Citation