The Condon-Shortley phase is the factor of 
that occurs in some definitions of the spherical
harmonics (e.g., Arfken 1985, p. 682) to compensate for the lack of inclusion
of this factor in the definition of the associated
Legendre polynomials (e.g., Arfken 1985, p. 669).
Using the Condon-Shortley convention in the definition of the spherical harmonic after omitting it in the definition of 
gives
 |
(Arfken 1985, p. 692), whereas using the definition of 
that already includes it gives
 |
(e.g., the Wolfram Language).
The Condon-Shortley phase is not necessary in the definition of the spherical harmonics, but including it simplifies the treatment of angular moment in quantum
mechanics. In particular, they are a consequence of the ladder operators 
and 
(Arfken 1985, p. 693).
