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# 1.5: Formal definition of a complete, orthonormal basis set


Consider a basis set $$|i_n \rangle$$. It is orthonormal if $$\langle i_n | i_m \rangle = \delta_{mn}$$. It is complete if any wavefunction can be written as $$|\phi \rangle = \sum_n c_n | i_n \rangle$$ and the $$c_n$$ are uniquely defined. If the wavefunction cannot be so written, the basis set is incomplete, if there exists more than one possible set of $$c_n$$, the basis set is overcomplete. Choosing a basis set in a Hilbert space (see 1.7) is analogous to choosing a set of coordinates in a vector space. Note that completeness and orthonormality are well defined concepts for both vector spaces and function spaces.

This page titled 1.5: Formal definition of a complete, orthonormal basis set is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Graeme Ackland via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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