Spherical harmonics orthogonality

Expansion Polynomials.—The techniques to be discussed here for solving the Boltzmann equation involve the use of an expansion of the distribution function in a set of orthogonal polynomials in particle velocity space. The polynomials to be used are products of Sonine polynomials and spherical harmonics some of their properties will be discussed in this section, while the reason for their use will be left to Section 1.13. 

Because both and Lz are hermitian, the spherical harmonics Yim(0, q>) form an orthogonal set, so that... 

Cauchy function 276 Cauchy s ratio test 35-36 central forces 107,132-135 spherical harmonics 134-135 spherical polar coordinates 132-133 chain rule 37, 57, 160 character 153,195,197 orthogonality 197, 204 tables 198-200... 

The orientation is not strictly identical for all structural units and is rather spread over a certain statistical distribution. The distribution of orientation can be fully described by a mathematical function, N(6, q>, >//), the so-called ODF. Based on the theory of orthogonal polynomials, Roe and Krigbaum [1,2] have shown that N(6, generalized spherical harmonics that form a complete set of orthogonal functions, so that... 

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... 

Spherical harmonics closely resemble normal Fourier harmonics except that they are functions of both the latitude and the longitude instead of the linear abscissa on a standard axis. Bi-dimensional Fourier analysis on a plane exists but is inadequate since the most desirable property of the requested expansion is the orthogonality of its components upon integration over the surface of the Earth, assumed to be spherical for most practical purposes. 

Given a finite number of measurements at a given latitude (90° — 6) and longitude on the surface of the Earth, we look for a smooth function that could be fitted to the data and represent their variations to within any desired precision. Spherical harmonics are suitable because they make an orthogonal set of functions which can... 

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-... 

Surface Spherical Harmonics. From the two sets of orthogonal functions ITU (cos 0), cos ( up) we can form a third set of functions... 

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. 

Here, the permutations of j, k,l,... include all combinations which produce different terms. The multivariate Hermite polynomials are listed in Table 2.1 for orders < 6. Like the spherical harmonics, the Hermite polynomials form an orthogonal set of functions (Kendal and Stuart 1958, p. 156). 

Since the associated Legendre polynomials (and the spherical harmonics) form an orthogonal set, only terms with / = / and m = m do not vanish in the integral of Eq. (3.41). Furthermore, for the 6 integration,... 

For a given value of n, the functions httk are identical to a sum of spherical harmonics with l = n, n — 2, n — 4,..., (0,1) for n > 1. The relationships are summarized in Table 3.8. For n = 0,1, the Hirshfeld functions are identical to the spherical harmonics with / = 0, 1, but, starting with the n = 2 functions, lower-order spherical harmonics are included for each n value. Unlike the spherical harmonics, the hnl functions are therefore not mutually orthogonal. As the radial functions in Eq. (3.48) contain the factor r", quite diffuse s, p, and d functions are included in the n = 2, 3, and 4 sets. For n <4 there are 35 deformation functions on each atom, compared with 25 valence-shell density functions with / < 4 in the multipole expansion of Eq. (3.35). 

The orthogonality of the spherical harmonics results in only s-states having non-zero values for Anm. We can then drop the Yoo (integrating this term will only result in unity) in... 

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... 

If the states have different angular momentum character then the angular integration over the spherical harmonics guarantees orthogonality. But if the states have the same angular momentum character then the orthogonality... 

With knowledge of the full OPDF in Eq. (6), the averages in Eq. (8) can be evaluated by integration with respect to the angles f and , which is straightforward due to the orthogonality properties of the spherical harmonics.46... 

The product Y2m> Ylm" can be represented by a summation of spherical harmonics with no terms having /> 4. Thus the orthogonality property of Yim makes all terms in Eq. (24) with /> 4 give no contribution to VMM>. Therefore, we can truncate the expansion for Vc at the 1=4 terms. For rare-earth ions with / electrons, the expansion must be truncated at the 1=6 terms. 

Proof. Since each is finite-dimensional, we can use Proposition 3.5 to define the orthogonal projection 13, V L (S ) onto the subspace Since V is not trivial, Proposition 7.4 impUes that V is not orthogonal to all of the spherical harmonics. Hence there must be at least one I such that the orthogonal projection n,>[ V] is not trivial. 

This integral vanishes because of the orthogonality of the spherical harmonics. Hence (4.38) are the correct zeroth-order functions. 

The expansion of the plane wave into partial waves yields F m(K)I m(f) components. If these are multiplied by loo V 71) t ie orthogonality condition for spherical harmonics then leads to the result that only the (/ = 0)-component remains. Hence, there is no dependence on k in equ. (4.74b). Similarly, in equ. (4.75b) only the ( f = l)-component is proved. Note, in addition, the typical overlap property of these integrals if a Coulomb wave with Z = 2 were used for the continuum electron the result of the integration would be proportional to Z — Zeff and would vanish for Zeff = 2. In other words, the dependence on Z — Zeff reflects the fact that the final and initial wavefunctions belong to different sets of... 

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