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For the exterior wavethe partial-wave scattering matrix isThus $\tan {\delta }_0=-k/\lambda$, the scattering length is $1/\lambda$, and the pole at $k=i\lambda$ represents a bound state of energy $-{\hslash }^2{\lambda }^2/(2m)$ whose exterior wavefunction is proportional to $\mathrm{sech}(\lambda r)/r$.

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.

At incident wave number $k$, low kinetic energy means $ka\ll 1$: the de Broglie wavelength is much larger than the scattering region. A generic finite-range scatterer can then be described to leading order without resolving its angular structure. In partial-wave scattering, this is leading S-wave scattering. The $\ell =0$ spherical harmonic is constant, whereas the centrifugal term ${\hslash }^2\ell (\ell +1)/(2m{r}^2)$ impedes penetration to the potential for $\ell >0$.

For a central potential with regular threshold behaviour, the finite-range partial-wave threshold law gives ${\delta }_{\ell }=O(k^{2\ell +1})$. Thuswhich is independent of scattering angle. For finite nonzero scattering length $a_s$, $f\to -a_s$ and $d\sigma /d\Omega \to a_s^2$ is isotropic. The potential range $a$ and scattering length $a_s$ are different quantities. The claim describes the generic leading S-wave scattering limit; tuned vanishing S-wave scattering amplitude or exceptional threshold resonances require separate analysis rather than a universal angular conclusion.

Assume a real, central, short-range potential and elastic quantum scattering. At large radius, normalize the stationary wavefunction aswhere $k=\sqrt{2mE}/\hslash$ and $\theta$ is the scattering angle. Equal incoming and outgoing speeds give the differential scattering cross-section $d\sigma /d\Omega =|f(\theta ){|}^2$.

The partial-wave scattering expansion separates the spherical harmonics. The incident plane wave has expansion ${\sum }_{\ell \ge 0}{i}^{\ell }(2\ell +1)j_{\ell }(kr)P_{\ell }(\cos \theta )$, with Spherical Bessel functions $j_{\ell }$ and Legendre polynomials $P_{\ell }$. The regular radial solution outside the potential has asymptotic form $u_{\ell }\propto \sin (kr-\ell \pi /2+{\delta }_{\ell })$. Its real scattering phase shift ${\delta }_{\ell }$ changes the ratio of outgoing to incoming waves to the Partial-wave S-matrix $S_{\ell }=e^{2i{\delta }_{\ell }}$. Matching outgoing coefficients yieldsHere $\ell$ labels the orbital angular momentum, with squared magnitude ${\hslash }^2\ell (\ell +1)$. The assumption of purely elastic scattering is what makes every $S_{\ell }$ have unit modulus.

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.

Quantum scattering relates incoming and outgoing states of an interaction problem. For a short-range potential in three dimensions, a stationary asymptotic solution is often written $\psi \sim e^{ikz}+f(\theta ,\phi )e^{ikr}/r$: the incident plane wave and outgoing spherical wave define the scattering amplitude $f$.

For equal incoming/outgoing speeds and the stated normalization, $d\sigma /d\Omega =|f{|}^2$ connects that amplitude to detector counts. Phases matter because alternative contributions interfere before their probabilities are formed. A normalizable wave packet supplies a physical incident preparation, while ideal stationary states simplify the asymptotic calculation. Long-range Coulomb interactions require modified asymptotic treatment, and internal excitations create additional channels. Partial-wave scattering uses angular momentum to organize a central-potential problem into simpler radial channels.

For elastic scattering by a real central potential, the phase shift ${\delta }_l(k)$ measures the phase difference between the asymptotic radial wave in angular-momentum channel $l$ and the corresponding free wave. Its channel Partial-wave S-matrix is $S_l=e^{2i{\delta }_l}$.

The phase shift parametrizes elastic partial-wave scattering, rather than general scattering phenomena.

For time dependence $e^{-i\omega t}$, the three-dimensional radial solutions $e^{ikr}/r$ and $e^{-ikr}/r$ describe outgoing and incoming spherical waves away from the origin. Their inverse-radius amplitudes make their flux through spheres independent of radius. They form the asymptotic radial ingredients of partial-wave scattering.