Spherical harmonics complex conjugate

In general, the Q m are complex quantities.1- The Y(m are spherical harmonics an asterisk designates the conjugate complex. Equation 2.41... 

As usual, there are the spherical harmonics or the complex conjugates (identified by the ). This expression can be inserted into the intermediate scattering function where one gets the following expression containing the regular Bessel functions ji ... 

The irreducible representations may be classified according to whether j is an integer or half of an odd integer. We shall consider here the former, which are the potentially real representations 22 b, p. 287). These representations in contrastandard form 4) can be transformed into real form by a constant unitary matrix, i.e. the same matrix for every element in the group. The elements of the constant matrix will be chosen such that the contrastandard, self-conjugate sets which form the bases for the potentially real representations in complex standard form are connected to the sets of the usual real spherical harmonics which form the bases for the real standard representations. From the constant matrix, the vector coupling coefficients pertaining to the real functions will be deduced. 

Elements of contrastandard, self-conjugate sets have been chosen in Ref. (4). The basis functions jm> are for integery represented by the complex standard spherical harmonics, which differ from the spherical harmonics Yjm= 0 jtn) 0 m) used by Condon and Shortly... 

Einally, we also remind that the definition of the spherical harmonics in the standard phase convention implies complex conjugation, as... 

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... 

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