Spherical Harmonics Facts For Kids

Pierre-Simon Laplace, 1749–1827

Real (Laplace) spherical harmonics Y ℓ m {displaystyle Y_{ell m}} for ℓ = 0 , … , 4 {displaystyle ell =0,dots ,4} (top to bottom) and m = 0 , … , ℓ {displaystyle m=0,dots ,ell } (left to right). Zonal, sectoral, and tesseral harmonics are depicted along the left-most column, the main diagonal, and elsewhere, respectively. (The negative order harmonics Y ℓ ( − m ) {displaystyle Y_{ell (-m)}} would be shown rotated about the z axis by 90 ∘ / m {displaystyle 90^{circ }/m} with respect to the positive order ones.). Rotation added for a better visual of the harmonic.

Alternative picture for the real spherical harmonics Y ℓ m {displaystyle Y_{ell m}} .

Plot of the spherical harmonic Y ℓ m ( θ , φ ) {displaystyle Y_{ell }^{m}(theta ,varphi )} with ℓ = 2 {displaystyle ell =2} and m = 1 {displaystyle m=1} and φ = π {displaystyle varphi =pi } in the complex plane from − 2 − 2 i {displaystyle -2-2i} to 2 + 2 i {displaystyle 2+2i} with colors created with Mathematica 13.1 function ComplexPlot3D

The rotation of a real spherical function with m = 0 and ℓ = 3. The coefficients are not equal to the Wigner D-matrices, since real functions are shown, but can be obtained by re-decomposing the complex functions

Schematic representation of Y ℓ m {displaystyle Y_{ell m}} on the unit sphere and its nodal lines. ℜ [ Y ℓ m ] {displaystyle Re [Y_{ell m}]} is equal to 0 along m great circles passing through the poles, and along ℓ−m circles of equal latitude. The function changes sign each time it crosses one of these lines.