partialwave

Partial waves refer to components of a wave with definite angular momentum in a spherical-wave expansion. In a quantum mechanical problem with a central potential V(r), the Schrödinger equation separates into radial and angular parts. The angular dependence is expanded in spherical harmonics Y_l^m(θ,φ), and for a fixed angular momentum quantum number l the radial function u_l(r) satisfies a one-dimensional radial equation with an effective potential l(l+1)/r^2. The full wave is a sum over l (and m) of these partial waves, reflecting the rotational symmetry of the problem.

In scattering theory, partial waves provide a convenient basis for analyzing interactions. The incoming wave is decomposed into partial waves; each l accumulates a phase shift δ_l due to the potential. At large distances, the radial solution behaves like a free-wave form with this phase shift. The scattering amplitude can be written as f(θ) = (1/2ik) ∑_{l=0}∞ (2l+1) [e^{2iδ_l} - 1] P_l(cos θ), where P_l are Legendre polynomials. The differential cross section is dσ/dΩ = |f(θ)|^2, and the total cross section is σ = (4π/k^2) ∑_{l=0}∞ (2l+1) sin^2 δ_l. These expressions link observable scattering to the underlying angular-momentum structure.

Practically, partial-wave analysis reveals which angular momenta contribute at a given energy. At low energies, s-wave (l = 0) often dominates; as energy increases, more partial waves contribute, requiring truncation at some l_max. The concept is widely used in nuclear, atomic, and particle physics, as well as in acoustics and electromagnetism, to analyze scattering and resonance phenomena.