1.5: Formal definition of a complete, orthonormal basis set
- Page ID
- 28666
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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.