Hard-Sphere Scattering Cross Section
Classically πa². Quantum: 4πa² at low energy, 2πa² at high energy. Neither equals classical.
ka (wavenumber×radius)
1.0
ℓ_max shown
8
σ_tot / πa² vs ka
At ka =
1.0
: σ/πa² =
—
Low energy limit →
4
(quantum, 4×classical)
High energy limit →
2
(Babinet's principle)
Partial Wave Contributions (2ℓ+1)sin²δ_ℓ
Current ka =
1.0
| ℓ
max
needed ≈
—