The Orthogonality Relations

Earlier, for Fourier series, we had the orthogonality relation among the Fourier functions ... 

Using the orthogonality relation exp(27rzj(m — s)/N) = N6ms, where 6 ... 

The orthogonality relations satisfied by solutions corresponding to the energies Et and Ef, are found from (10-393) and (10-395) to be... 

Since the Dirac notation suppresses the variables involved in the integration, we re-express the orthogonality relation in integral notation... 

As before, we apply the hermitian property of introduce the abbreviation and use the orthogonality relation (9.26) to obtain... 

We solve the recurrence relation (E.4) for Pi p), multiply both sides by Pi p), integrate with respect to p from —1 to +1, and note that one of the integrals vanishes according to the orthogonality relation (E. 18), so that... 

As a consequence, such examples show that the orthogonality relations (between vectors in different subspaces) alone, do not fix the S subspace. To do so, one would need some previous additional information on the basis which spans S and Sk That is to say, one would need to constrain the set of recovered O s to form a basis of the occupied subspace. This would then make additional orthogonality constraints within the subspace to take into account in the search for a K formula,... 

Since the transition moments of the antisymmetric and symmetric CH2 stretching vibrations and the methylene chain axis are mutually perpendicular, the average orientation angle y of the hydrocarbon chain axis around the surface normal is obtained to be 27° by the orthogonal relation... 

It is obvious that the character is the same for all elements in a class, and that it is invariant under similarity transformations of the representation. For the characters, the orthogonality relations (1) and (2) take the form... 

Summing over s and using the orthogonality relation (1), we find... 

Substitution of this into (17), summation over t, followed by k and l, leads, with the help of the orthogonality relations, the representation property of the D, and the definition (10), to the result... 

The Clebsch-Gordan coefficients satisfy the orthogonality relations... 

It is well known that for lossless media, all squared effective indexes are real, and for any transversally limited structure they form a discrete nongrowing sequence. The field distributions / (O and /i (0 of the m-th mode are mutually orthogonal, have the same phase at each point of the cross-section, and the set of functions corresponding to all modes is complete. The orthogonality relations can be taken in the form... 

The orthogonality theorem The inequivalent irreducible unitary matrix representations of a group G satisfy the orthogonality relations... 

In the first section of this chapter, we shall prove that CS is semisimple if the characteristic of the field C does not divide any of the integers s with s 6 S. This enables us to refer to some of the results about semisimple rings which we obtained in Section 8.5. We shall also see that C C Z(CS), so that we may speak about characters of CS and refer to results about characters which we obtained in Section 8.6. The results of the first section include the orthogonality relations for fields the characteristic of which does not divide any of the integers s with s S S. 

In Section 9.5, we shall apply the orthogonality relations in order to investigate the case where 5 < 5. 

The equations in the following theorem are called the orthogonality relations for schemes with finite valency. 

We shall now see that the assumption that C is algebraically closed gives rise to a generalization of the orthogonality relations. 

For all the following considerations it is an important fact that within the CC of interest mutual chromophore wave function overlap and electron exchange effects among different chromophores do not take place (absence of the Dexter mechanism). Therefore, we may assume the orthogonality relation Pvnf Pnb) — A/ ,i A,.h to be valid, where A, / - /A,) denotes the electronic... 

Using integration by parts it is easily verified that for any compact field there is the orthogonality relation... 

Other forms of normalization, as well as forms denoting irreducible tensor operators may be found in [304] and in Appendix D. With the aid of the orthogonality relation one may easily express the quantum mechanical polarization moments fq and Pq through the elements of the density matrix /mm and... 

Applying the orthogonality relation eq. (3.63) to the vector parts allows us to write ... 

This is the orthogonality relation of the two Lanczos polynomials Q (m) and Qm(u) with the weight function, which is the residue dk [48]. We recall that the sequence Q = (Q ( z-) coincides with the set of eigenvectors of the Jacobi matrix (60). 

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