Spherical harmonics orthogonality property

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. 

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. 

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. 

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... 

Contrasting with the one-centre expansion for the inter-particle distance which always terminates when used with AOs because of the orthogonality properties of the spherical harmonics of the first kind. 

The shape of such a surface may be described in a mathematical way which has several important advantages. The shape is analysed into harmonic components functions much as one might analyse a wave into a Fourier series. The spherical harmonics, as these components are called, have the necessary property of orthogonality however, their form is more complicated than the cos(njc) type of component of a Fourier series for a plane wave. The spherical harmonics are functions of the polar coordinate angle, referred to the director axis. The first four components, abbreviated Pq, P2(cosa), P4(cos Of) and PeCcos a) or simply Pq, P2, P4, Pe, are defined below and drawn in Fig. 4. 

The integral over Q is easily evaluated with the aid of the orthogonality properties of the spherical harmonics, if we note that Pniti) F (Q)//f . With this reduction, (7.351) takes the form... 

These functions form a complete orthogonal set and are equal to the complex conjugate spherical harmonics for k or m equal to zero, except for a normalizing constant. They possess the useful property that use of the spherical basis yields... 

Now using the orthogonality property of the spherical harmonics we see that the electric dipole matrix elements of equation (5.10) will vanish unless... 

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