The spherical harmonic functions Y _?,m( ?, ) are eigenstates of both the square of the angular momentum operator L ², as well as of L _z, the projection of on some specific axis z (see Eq. (3.26))

(B.1)

The Y _{? ,m}( ?, ) are products of periodic functions of ? and of that are often encountered in quantum mechanics and in other areas where we seek solutions to problems with spherical symmetry. The Y _?,m( ?, ) can be written in terms of associated Legendre polynomials P _?,m(cos ?) and exponentials in as

(B.2)

where the associated Legendre functions are given by

(B.3)

with x= cos ?. The P _?,m( x) are defined such that the spherical harmonics obey the following normalization relation over the full solid angle

(B.4)

where the ? _nm are the Kronecker symbols (see Eq. (10.21)). It follows from (B.2) that

(B.5)

Some of the low-order spherical harmonics are

(B.6)