Orthonormal modes

Propagation constant and phase velocity 11-2 Symmetry properties of the field components 11-3 Field representation on nonabsorbing waveguides 11-4 Orthogonality relations and normalization 11-5 Orthonormal modes... 

In some applications it is convenient to use modal fields that have unit normalization. Modes satisfying this condition are called orthonormal modes, and are constructed from modes with arbitrary normalization by setting... 

The fields of forward- and backward-propagating orthonormal modes are related by Eq. (11-7) with a introduced over each field component, and the orthogonality of backward-propagating orthonormal modes is handled by the procedure laid down below Eq. (11-10). 

Here we examine the excitation of other local modes by the /th local mode. We start by determining the contribution daj to the amplitude of theyth forward-propagating local mode due to the induced current of Eq. (22-30) within one section. If we re-express Eq. (21-2a) in terms of the orthonormal modes of Section 11-5, then... 

The two waveguides have free-space permeability Po, the translationally invariant refractive-index profile isn(x, yX and the profile of the perturbed waveguide is n(x, y, z). For convenience we use the orthonormal modes of Sections 11—5 and 25—4, and set... 

In a real chain segment-segment correlations extend beyond nearest neighbour distances. The standard model to treat the local statistics of a chain, which includes the local stiffness, would be the rotational isomeric state (RIS) [211] formalism. For a mode description as required for an evaluation of the chain motion it is more appropriate to consider the so-called all-rotational state (ARS) model [212], which describes the chain statistics in terms of orthogonal Rouse modes. It can be shown that both approaches are formally equivalent and only differ in the choice of the orthonormal basis for the representation of statistical weights. In the ARS approach the characteristic ratio of the RIS-model becomes mode dependent. 

The system is free of oscillations when Zp approaches zero. Since the acoustic mode satisfies the orthonormal property, Eq. (22.22) can be simplified as... 

The elements of D represent the sum over all unit cells of the interaction between a pair of atoms. D has 3n x 3n elements for a specific q and j, though the numerical value of the elements will rapidly decrease as pairs of atoms at greater distances are considered. Its eigenvectors, labeled e ( fcq), where k is the branch index, represent the directions and relative size of the displacements of the atoms for each of the normal modes of the crystal. Eigenvector ejj Icq) is a column matrix with three rows for each of the n atoms in the unit cell. Because the dynamical matrix is Hermitian, the eigenvectors obey the orthonormality condition... 

Exercise. Solve (6.2). [Hint Determine the normal modes obeying the boundary conditions, find an orthonormality relation between them and their adjoints, and assume that they constitute a complete set in the interval (L, R) so that the initial condition can be satisfied.]... 

The changes in structure that must occur create a barrier to electron transfer. In order to understand the origin of the barrier and to treat it quantitatively, it is necessary to recall that the structural changes at each reactant can be resolved into a linear combination of its normal vibrational modes. The normal modes constitute a complete, orthonormal set of molecular motions into which any change in intramolecular structure can be resolved. 

Returning to equation (25), evaluation of the total vibrational overlap integral, (Xj X7), is less formidable than it appears. The vibrational wavefunctions are a complete orthonormal set for which ( 1 0 )= where S is the Kronecker delta. For the vast majority of normal modes, S (and AQe) = 0. For these modes the vibrational overlap integrals become (yjy,/) = 1 if v = v, and = 0 if v v . Except for the requirement that the vibrational quantum number must... 

Of course, using the fact that the operators involved in the ACF do not work on the space of the fast mode, because of the orthonormality, the above ACF reduces after a circular permutation within the trace and making the trace explicit, to... 

Besides, it is possible to use the fact that the kets and the bras of the fast mode do not act on the operators belonging to the space of the slow mode. Then, because of the orthonormality properties leading to simplifications, the ACF in representation III takes the following form ... 

Because experimental error is always present in a measured data matrix, the corresponding row-mode eigenvectors (or eigenspectra) form an orthonormal set of basis vectors that approximately span the row space of the original data set. Figure 4.14 illustrates this concept. The distance between the endpoints of a and a is equal to the variance in a not explained by x and y, that is, the residual variance. 

The geometrical meaning is that the Eckart subspace pf is perpendicular to the three-dimensional manifold of rigid-body rotation at the reference configuration z.si. The Eckart subspace is Euclidean since the conditions in Eq. (32) are linear. Therefore this space can be spanned by vectors , (p = 1,..., 3m — 6) that specify the 3m 6 directions of (vibrational) normal modes at the reference configuration zVI- in the (3m — 3(-dimensional configuration space. The vectors (m,m are orthonormal as Mi > = [Pg.107]

At this point, if one were to consider only the axial properties of the system, the analysis would be equivalent to the extended onedimensional analysis discussed above. The energies of the bands—now effectively one dimensional—are the same as given by eq (44) for the simple collection of harmonic wells above. The full analysis, as can be seen by considering the consequences of Lowdin orthonormalization, mixes character of the x and y coordinates into that of the axial modes, although the smallness of the off-diagonal matrix elements essentially preserves the block-diagonal character of the problem. 

This decomposition of the properly matricized array X ensures an orthonormal basis (V) for a loading plot of the A-mode (US). Similar reasoning leads to orthonormal bases for the B and C loadings and for the nonorthogonal loadings from Tucker3 models. 

Direct matrix diagonalization. The time-honored way to compute a spectrum is worth a brief review. Assume that we begin with a real-valued orthonormal basis set with dimension N, j), j = 1, 2,. .., N, where each member could represent a multi-mode function. We assume that this basis is sufficient to represent both the initial state i /,) and the eigenstates ijja) that make a major contribution to the spectrum. The expansion coefficients of the initial state in this basis set,... 

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