spherical harmonics
Spherical harmonics, written Yℓm, are the standard set of wave patterns that live on the surface of a sphere. Just as a vibrating guitar string settles into a handful of clean standing-wave shapes, the angular part of a quantum wavefunction settles into these harmonics. Each one is labelled by the two integers ℓ and m, and together they form a complete alphabet from which any pattern on a sphere can be spelled out.
In quantum mechanics they are the natural home of orbital angular momentum. When a problem has a central point of symmetry — an electron around a nucleus, a molecule rotating freely — the wavefunction splits into a radial part, telling you how far out the particle is, and an angular part, which is always one of the spherical harmonics. The number ℓ sets the magnitude of angular momentum and m its orientation, exactly the quantum numbers these functions carry.
Their shapes are the origin of the familiar orbital pictures. The simplest harmonic, Y00, is a uniform glow over the whole sphere — the spherical s orbital. Higher harmonics carry lobes and nodal lines, giving the dumbbell of a p orbital or the cloverleaf of a d orbital. Far from being abstract, spherical harmonics are the mathematical fingerprints of every orbital shape a chemist ever draws.
In a central-force problem the wavefunction factors into a radial part and a spherical harmonic.
The orbital pictures in textbooks usually plot the squared harmonic, the probability density, or real combinations of Yℓm. The raw Yℓm are complex-valued, so the tidy real-lobed shapes are partly a matter of presentation.