Ylm

Ylm refers to the spherical harmonics, a set of complex-valued functions defined on the surface of a sphere. They are denoted Y_l^m(θ, φ), where θ is the polar angle (0 ≤ θ ≤ π) and φ is the azimuthal angle (0 ≤ φ < 2π). The integers l ≥ 0 and m with -l ≤ m ≤ l index the functions. Spherical harmonics arise as the angular part of solutions to Laplace’s equation in spherical coordinates and form an orthonormal basis for square-integrable functions on the sphere.

A common explicit form is Y_l^m(θ, φ) = sqrt((2l+1)/(4π) · (l−m)!/(l+m)!) · P_l^m(cos θ) · e^{i m φ}, where P_l^m are associated Legendre polynomials. The Condon–Shortley phase, a factor of (−1)^m, is often included in this convention, though alternatives exist. Real spherical harmonics can be obtained by combining Y_l^m and Y_l^{−m} into real and imaginary parts, yielding a real-valued basis.

Orthogonality and completeness follow from the integral over the sphere: ∫∫ Y_l^m(θ, φ) [Y_{l'}^{m'}(θ, φ)]* sin θ dθ dφ = δ_ll' δ_mm'. These functions transform under rotations via Wigner D-matrices and satisfy(L^2 Y_l^m = l(l+1) Y_l^m, L_z Y_l^m = m Y_l^m), making them eigenfunctions of angular momentum operators in quantum mechanics.

Applications span quantum mechanics (hydrogen atom, angular wavefunctions), geophysics and astronomy (gravitational and magnetic field modeling), and computer graphics (lighting with spherical harmonics).