Spherical harmonics

The spherical harmonics $Y_l^m(\theta ,\varphi )$ are the angular portion of the solution to Laplace's equation in spherical coordinates. They are given by

$Y_l^m(\theta ,\varphi )=(-1{)}^m\sqrt{\frac{2n+1}{4\pi }\frac{(n-m)!}{(n+m)!}}{P}_n^m(\cos \theta )e^{im\varphi },$

where $P_n^m$ is the associated Legendre function.

The first few spherical harmonics are given by:

$Y_0^0(\theta ,\varphi )=\frac{1}{2}\frac{1}{\sqrt{\pi }}$

$Y_1^{-1}(\theta ,\varphi )=\frac{1}{2}\sqrt{\frac{3}{2\pi }}\sin \theta e^{-i\varphi }$

$Y_1^0(\theta ,\varphi )=\frac{1}{2}\sqrt{\frac{3}{\pi }}\cos \theta$

$Y_1^1(\theta ,\varphi )=-\frac{1}{2}\sqrt{\frac{3}{2\pi }}\sin \theta e^{i\varphi }$

See also

• Wigner D-matrix
• Spherical Harmonic -- from Wolfram MathWorld

References

• M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) Appendix III
• I. Nezbeda, J. Kolafa and S. Labík "The spherical harmonic expansion coefficients and multidimensional integrals in theories of liquids", Czechoslovak Journal of Physics 39 pp. 65-79 (1989)