Eigenfunctions of the angular Laplacian — the basis for atomic orbitals and multipole expansions
Eigenvalue Equation
Spherical harmonics solve:
L²Y_lm = ℏ²l(l+1)Y_lm
L_z Y_lm = ℏm·Y_lm
The angular Laplacian: ∇²_Ω = (1/sinθ)∂_θ(sinθ∂_θ) + (1/sin²θ)∂²_φ
Real form (|m|>0):
Y_l,|m|^cos = N·P_l^|m|(cosθ)·cos(mφ)
Y_l,|m|^sin = N·P_l^|m|(cosθ)·sin(mφ)
Nodal lines: l−|m| latitude circles, |m| longitude pairs.
Physical Significance
Atomic orbitals: ψ_nlm(r,θ,φ) = R_nl(r)·Y_lm(θ,φ)
— hydrogen wavefunctions factorize into radial × angular parts.
Multipole expansions: any function on a sphere decomposes as
f = Σ_{lm} a_lm Y_lm
(CMB anisotropy, gravitational potential, radiation fields)
Orthogonality:
∫Y_lm* Y_l'm' dΩ = δ_ll' δ_mm'
Completeness: any square-integrable function on S² expands in Y_lm.