Partial-wave S-matrix

For a central potential, each angular-momentum channel has a scalar scattering coefficient $S_l(k)=e^{2i{\delta }_l(k)}$.

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 $S_l=e^{2i{\delta }_l}$.

For a reflectionless scattering wavefunction whose two asymptotic phases differ by $\delta (k)$, a large periodic box gives $k_{n}L+\delta (k_n)=2\pi n$. Compared with the free value $k_n^{(0)}=2\pi n/L$, $k_n-k_n^{(0)}=-\delta (k_n^{(0)})/L+O(L^{-2})$ away from thresholds. Discrete bound states and phase-branch changes alter mode indexing and must be counted when comparing complete spectra.

An S wave is the rotationally invariant angular-momentum channel $l=0$. At sufficiently low energy it normally dominates short-range scattering because higher partial waves are suppressed by their centrifugal barriers.

For a central potential and the convention $S_l=1+2i{f}_l=e^{2i{\delta }_l}$, the scattering amplitude isThe identity $|S_l|=1$ is equivalent to $\mathrm{Im}{f}_l=|f_l{|}^2$.

For elastic scattering by a central potential,At low momentum, the S-wave normally supplies the leading term.

Partial-wave unitarity bounds each angular-momentum component of a scattering amplitude because its elastic S-matrix eigenvalue has modulus one. Perturbative violation of this bound signals that interactions become strong or that additional degrees of freedom must enter.

For a real central potential and real $k$, radial-current conservation gives $S_l(k{)}^{\ast }{S}_l(k)=1$. Invariance of the radial equation under $k\mapsto -k$ gives $S_l(k)S_l(-k)=1$, so a continuous phase convention has ${\delta }_l(-k)=-{\delta }_l(k)$.

The $s$-wave scattering length isand the low-energy total cross-section is $4\pi a_s^2$.

For a barrier of radius $a$ and potential ${\hslash }^2{\gamma }^2/(2m)$, matching the regular inner hyperbolic-sine wave to the outer sine wave gives $a_s=a-\tanh (\gamma a)/\gamma$. The low-energy $s$-wave scattering amplitude is $-a_s$.

A zero-energy scattering resonance occurs when an S-wave bound state reaches the continuum threshold. The zero-energy radial solution then approaches a nonzero constant rather than a generic linear function, and the scattering length diverges and changes sign.

A pole at $k=i\kappa$ with $\kappa >0$ represents a normalizable bound state of energy $-{\hslash }^2{\kappa }^2/(2m)$. Resonance poles instead lie away from the imaginary axis on the analytically continued unphysical sheet.

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 $\mathrm{sech}(\lambda r)/r$.

A scattering resonance is a metastable state represented by a pole of the analytically continued scattering matrix. Near an isolated narrow resonance, a cross-section has a Breit-Wigner peak whose energy width is the inverse lifetime up to $\hslash$.