Spherical harmonics orthogonality relation

The explicit form of the spherical harmonics of low-order numbers (, m) are given in Table 3.1. It may be verified by direct integration that they also obey the orthogonality relation (Problem 3.4)... 

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

In the following, we pay special attention to the connections among the spherical, Stark and Zeeman basis. Since in momentum space the orbitals are simply related to hyperspherical harmonics, these connections are given by orthogonal matrix elements similar (when not identical) to the elements of angular momentum algebra. 

As is well known, conventional hydrogenoid spherical orbitals are strictly linked to tetradimensional harmonics when the atomic orbitals for the tridimensional hydrogen atom are considered in momentum space. We have therefore studied an alternative representation, providing the Stark and Zeeman basis sets, related to the spherical one by orthogonal transformation, see eqs. (12) and (15). The latter can also be interpreted as suitable timber coefficients relating different tree structures of hyperspherical harmonics for R (Fig. 1). 

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