TOPICS
Search

Spherical Harmonic Closure Relations

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook

The sum of the absolute squares of the spherical harmonics Y_l^m(theta,phi) over all values of m is

sum_(m=-l)^l|Y_l^m(theta,phi)|^2=(2l+1)/(4pi).
(1)

The double sum over m and l is given by

sum_(l=0)^(infty)sum_(m=-l)^(l)Y_l^m(theta_1,phi_1)Y^__l^m(theta_2,phi_2)=1/(sintheta_1)delta(theta_1-theta_2)delta(phi_1-phi_2)
(2)
=delta(costheta_1-costheta_2)delta(phi_1-phi_2),
(3)

where delta(x) is the delta function.

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Spherical Harmonic Closure Relations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalHarmonicClosureRelations.html

Subject classifications