Partial-wave scattering

Partial-wave scattering decomposes a scattering state into partial waves with definite orbital angular momentum. For a central potential, each channel has its own scattering phase shift and Partial-wave S-matrix.

The partial-wave scattering channel with orbital angular momentum quantum number one. Its angular dependence is built from degree-one spherical harmonics. For a finite-range potential away from a threshold pole, the finite-range partial-wave threshold law gives phase shift ${\delta }_1=O(k^3)$ and a partial elastic scattering cross-section proportional to $k^4$ as the relative wavenumber $k$ tends to zero.

For a real central potential and the outgoing convention $e^{i\pi /4}f(\theta )e^{ikr}/\sqrt{kr}$, the Partial-wave S-matrix is $S_m=e^{2i{\delta }_m}$. The cosine form of the Jacobi-Anger expansion fixes $w_0=1$ and $w_m=2$ for $m\ge 1$. Comparing the outgoing part of each phase-shifted radial cosine with the incident plane wave gives the displayed scattering amplitude. Integrating $|f{|}^2/k$ gives the displayed scattering cross-section, since the constant angular mode has squared norm $2\pi$ and each positive-order cosine has squared norm $\pi$. Reversing the multiplicities would already fail to reproduce the free plane wave at zero scattering phase shift.