Spherical harmonics

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

$Y_{l}^{m}(\theta ,\phi )=(-1)^{m}{\sqrt {{\frac {2n+1}{4\pi }}{\frac {(n-m)!}{(n+m)!}}}}P_{n}^{m}(\cos \theta )e^{im\phi },$

where $P_{n}^{m}$ is the associated Legendre function.

The first few spherical harmonics are given by:

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

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

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

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

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)