Eigenfunction prior

For the construction of spin eigenfunctions see, for example, Ref. [22], There are obviously many parallels to the multiconfiguration self-consistent field (MCSCF) methods of MO theory, such as the restriction to a relatively small active space describing the chemically most interesting features of the electronic structure. The core wavefunction for the inactive electrons, 4>core, may be taken from prior SCF or complete active space self-consistent field (CASSCF) calculations, or may be optimised simultaneously with the and cat. 

From prior experience with the one-dimensional quantum mechanical harmonic oscillator [4], we know that its eigenfunctions and eigenvalues are... 

This example shows that the statistical case applies if the wave function just prior to the measurement is not an eigenfunction of the operator corresponding to the variable being measured. 

Consider the nondegenerate case, that each eigenfunction of the operator A has a distinct eigenvalue. Since the set of eigenfunctions is assumed to be a complete set, the wave function immediately prior to a measurement of A can be represented as a linear combination of eigenfunctions of A, as in Eq. (16.3-34) ... 

This theorem shows that every eigenfunction of a Hermitian operator is orthogonal to every other eigenfunction with a different eigenvalue. Furthermore, it is possible to prove that eigenfunctions of a Hermitian operator can be constructed to be orthogonal even if some have like eigenvalues. This is a corollary to the prior theorem which we consider without a detailed proof. 

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