Vector spherical harmonics orthogonality

The vector spherical harmonics YjtM form an orthogonal system. The state of the photon with definite values of j and M is described by a wave function which in general is a linear combination of three vector spherical harmonics... 

The orthogonality of all the vector spherical harmonics, which was established in the preceding section, implies that the coefficients in the expansion (4.23) are of the form... 

Using the orthogonality relations of vector spherical harmonics we see that the expansion coefficients amn and bmn are given by... 

Because the system of vector spherical harmonics is orthogonal and complete in L Q), the series representation (1.40) is valid for any tangential vector field. Taking into account the expressions of the vector spherical harmonics (cf. (B.8) and (B.9)) we deduce that the expansions of Vp and Va are given by... 

The orthogonality relations of the vector spherical harmonics on the unit sphere give... 

An interesting feature of the null-field method is that all matrix equations become considerably simpler and reduce to the corresponding equations of the Lorenz-Mie theory when the particle is spherically. For a spherical particle of radius R, the orthogonality relations of the vector spherical harmonics show that the QP matrices are diagonal... 

The system of vector spherical harmonics is orthogonal and complete in Llnim (the space of square integrable tangential fields defined on the unit sphere f ) and in terms of the scalar product in we have... 

Since Imn and Vmn are linear combinations of nimn and Umn, we deduce that the system of vector spherical harmonics of left- and right-handed type is also orthogonal and complete in... 

Then the moment induced by the electric vector of the incident light is parallel to that vector resulting in complete polarization of the scattered radiation. The A lg i>(CO) mode of the hexacarbonyls provides a pertinent example08. Suppose we have a set of coupled vibrators, equidistant from some origin. Then it must be possible to express the basis functions for the vibrations in terms of spherical harmonics, for the former are orthogonal and the latter comprise a complete set. The polarization of a totally symmetric vibration will be determined by its overlap with the spherically symmetrical term which may be taken as r2 = x2 + y1 + z2. Because of the orthogo-... 

Evidently, correlation functions for different spherical harmonic functions of two different vectors in the same molecule are also orthogonal under equilibrium averaging for an isotropic fluid. Thus, if the excitation process photoselects particular Im components of the (solid) angular distribution of absorption dipoles, then only those same Im components of the (solid) angular distribution of emission dipoles will contribute to observed signal, regardless of the other Im components that may in principle be detected, and vice versa. The result in this case is likewise independent of the index n = N. Equation (4.7) is just the special case of Eq. (4.9) when the two dipoles coincide. 

The boundary conditions (4.39), the orthogonality of the vector harmonics, and the form of the expansion of the incident field dictate the form of the expansions for the scattered field and the field inside the sphere the coefficients in these expansions vanish for all m = = 1. Finiteness at the origin requires that we take y (kjr), where kj is the wave number in the sphere, as the appropriate spherical Bessel functions in the generating functions for the vector harmonics inside the sphere. Thus, the expansion of the field (Ej,H,) is... 

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