Vector spherical harmonics definition

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... 

It is not possible to ascribe a definite value of the orbital angular momentum to a photon state since the vector spherical harmonic YjM may be a function of different values of . This provides the evidence that, strictly speaking, it... 

For each nonnegative integer f, the space of spherical harmonics of degree f (see Dehnition 2.6) is the vector space for a representation of 50(3). These representations appear explicitly in our analysis of the hydrogen atom in Chapter 7. Recall the complex scalar product space L (S ) from Definition 3.3. 

Recall the vector space y of spherical harmonics from Definition 2.6 y is the set of restrictions to of homogeneous harmonic polynomials. Recall also the definition of spanning (Defiiution 3.7). The set y of spherical harmoiucs spans... 

By definition the components of the second-rank Cartesian tensor ax transform under rotation just like the product of coordinates xy (e.q., see Jeffreys, 1961) The motivation for what ensues springs from the observation that the spherical harmonics Ym (0, ft) (where 6, ft) are the polar and azimuthal angles of the unit vector (r/1 r )) can be written in terms of the coordinates (x, y, z) of the vector r, for example,... 

Our definition of spherical harmonics Yq k) and Clebsch-Gordan coefficients are in accord with the standard references. The quantity A(K, K ) is the product of radial integrals, and various factors which arise from the vector coupling coefficients of n equivalent electrons. The latter component of A(K, K ) is essentially embodied in the Racah tensor. Let... 

In the usual texts a multipole expansion involving spherical Bessel functions and spherical vector harmonics is also introduced [16,23,23,26]. The fields from electric and magnetic dipoles correspond to the lowest-order terms ( =1) in the expansion. If we define dipole by this expansion then our toroidal antenna is an electric dipole. In any event, the fields away from the source are the same. This is perhaps a matter of consistency in definitions. 

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