TOPICS
Condon-Shortley Phase
The Condon-Shortley phase is the factor of
that occurs in some definitions of thespherical harmonics (e.g., Arfken 1985, p. 682) to compensate for the lack of inclusion of this factor in the definition of theassociated 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., theWolfram Language).
The Condon-Shortley phase is not necessary in the definition of thespherical 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).
See also
Associated LegendrePolynomial,Spherical HarmonicExplore with Wolfram|Alpha
References
Arfken, G.Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 682 and 692, 1985.Condon, E. U. and Shortley, G.The Theory of Atomic Spectra. Cambridge, England: Cambridge University Press, 1951.Shore, B. W. and Menzel, D. H.Principles of Atomic Spectra. New York: Wiley, p. 158, 1968.Referenced on Wolfram|Alpha
Condon-Shortley PhaseCite this as:
Weisstein, Eric W. "Condon-Shortley Phase."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/Condon-ShortleyPhase.html