Eigenfunction orthonormality

Here (1/2%) exp(ikx) is the normalized eigenfunction ofF =-ihd/dx corresponding to momentum eigenvalue hk. These momentum eigenfunctions are orthonormal ... 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

It should be noted that the Hartree-Fock equations F ( )i = 8i ([)] possess solutions for the spin-orbitals which appear in F (the so-called occupied spin-orbitals) as well as for orbitals which are not occupied in F (the so-called virtual spin-orbitals). In fact, the F operator is hermitian, so it possesses a complete set of orthonormal eigenfunctions only those which appear in F appear in the coulomb and exchange potentials of the Foek operator. The physical meaning of the occupied and virtual orbitals will be clarified later in this Chapter (Section VITA)... 

The basis functions constructed in this manner automatically satisfy the necessary boundary conditions for a magnetic cell. They are orthonormal in virtue of being eigenfunctions of the Hermitian operator Ho, therefore the overlapping integrals(6) take on the form... 

The eigenfunctions Uj. constitute what is called an orthonormal system ... 

The total number of the eigenfunctions is equal to (iVi — 1)(VV2 1) = N. These constitute an orthonormal system... 

Along these lines, we can immediately write down the eigenvalues and the orthonormal eigenfunctions... 

Let N be the dimension of a finite-dimensional space H, Xf. be eigenvalues and be orthonormal eigenfunctions of the problem (see Chapter 1, Section 1 and Chapter 2, Section 1)... 

Consider a set of orthonormal eigenfunctions 0, of a hermitian operator. Any arbitrary function / of the same variables as 0, defined over the same range of these variables may be expanded in terms of the members of set 0,-... 

The coefficients a, are evaluated by multiplying (3.27) by the complex conjugate 0 of one of the eigenfunctions, integrating over the range of the variables, and noting that the 0,s are orthonormal... 

If the eigenvalues of N are represented by the parameter X and the corresponding orthonormal eigenfunctions by (pxi( ) or, using Dirac notation, by Xi), then we have... 

From equations (4.34) and the orthonormality of the harmonic oscillator eigenfunctions n), we find that the matrix elements of a and are... 

In this case, the operator FT(1, 2,. .., A) is obviously symmetric with respect to particle interchanges. For the N particles to be identical, the operators H i) must all have the same form, the same set of orthonormal eigenfunctions and the same set of eigenvalues where... 

To prove the variation theorem, we assume that the eigenfunctions 0 form a complete, orthonormal set and expand the trial function 0 in terms of that set... 

The quantity k > is the unperturbed Hamiltonian operator whose orthonormal eigenfunctions and eigenvalues are known exactly, so that... 

We assume in this section and in Section 10.2 that equation (10.6) has been solved and that the eigenfunctions Q) and eigenvalues k(Q) are known for any arbitrary set of values for the parameters Q. Further, we assume that the eigenfunctions form a complete orthonormal set, so that... 

The electronic Hamiltonian and the corresponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on qx. The index i in Eq. (9) can span both discrete and continuous values. The v /f, ad(r q J form a complete orthonormal basis set and satisfy the orthonormality relations... 

According to the postulates of QM, any if/ representing a physical state of the system can be expressed as a linear combination of energy eigenfunctions forming an infinite orthonormal set ... 

For cn = 1, or An = j2]l the eigenfunctions, like the unit vectors of particle mechanics, are normalized. Thus, for Sn = y/2/lsinmrx/l, the functions are said to be orthonormal and the normalization condition reads... 

Because of these properties of Hermitian functions it is accepted as a basic postulate of wave mechanics that operators which represent physical quantities or observables must be Hermitian. The normalized eigenfunctions of a Hermitian operator constitute an orthonormal set, i.e. 

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