The associated Legendre functions Pl|m| (w) are defined by

Verify that P00(w) = 1, P01(w) = w, P11(w) = (1 - w2)1/2 and find P02(w), P12(w), and P22(w). It can be shown that (Pauling and Wilson, page 129)

Equations (5.146) and (5.145) give the explicit formula for the normalized theta factor in the angular-momentum eigenfunctions. The normalized eigenfunctions of L2 and Lz (the spherical harmonics) are given by Eqs. (5.146) and (5.99) as

The phase of the normalization constant of the spherical harmonics is arbitrary (see Section 1.7). Many texts use a different phase convention than in (5.147). Thus Yml differs from text to text by a minus sign.

dwitlm (5.146) (5.147)