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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 asymptotic scattering wavefunction isIn each partial wave, the Partial-wave S-matrix is the ratio of its outgoing coefficient to its incoming coefficient. With the conventions in the question,For elastic scattering by a real central potential, unitarity gives $|S_l|=1$, so $S_l=e^{2i{\delta }_l}$ for a real scattering phase shift ${\delta }_l$. HenceUsing the Orthogonality of Legendre polynomials in ${\sigma }_T=\int |f(\theta ){|}^{2}d\Omega$ yields the partial-wave total scattering cross-section

For the S wave, write the reduced radial wavefunction aswhere regularity at $r=0$ selected the hyperbolic sine. Continuity of $u$ and $u^{\prime }$ at $r=a$, equivalently logarithmic-derivative matching, givesorFor $ka\ll 1$, let the scattering length be $a_s=-{\lim }_{k\to 0}\tan {\delta }_0/k$. Expanding the matching relation givesOnly the S wave contributes at leading order, soIn the hard-sphere limit $\gamma a\to \infty$, penetration is suppressed, $a_s\to a$, andThis is four times the geometric area $\pi a^2$, a wave effect associated with diffraction from an impenetrable sphere.

Write $u=kr-l\pi /2$. Sincethe outgoing-to-incoming coefficient ratio is $-S_l$. Hence the Partial-wave S-matrix element and scattering phase shift areThe scattering length from a partial-wave S-matrix is

For the stated S-wave scattering solution, $\tanh (\lambda r)\to 1$ as $r\to \infty$, soUsing the same incoming/outgoing convention,Therefore

The analytic continuation of $S_0$ has a bound-state pole of the scattering amplitude at $k=i\lambda$ in the upper half-plane. Since $E={\hslash }^2{k}^2/(2m)$, this givesTo extract the bound-state wavefunction, multiply the scattering solution by $k-i\lambda$ and take the pole limit:Thus the unnormalized exterior solution isThese data are collected in the Hyperbolic-tangent S-wave exterior solution.

The scattering amplitude is defined byFor a central potential its partial-wave expansion isUsing orthogonality of the Legendre polynomials in ${\sigma }_T=\int |f{|}^{2}d\Omega$ gives

A scattering phase shift measures the change of a partial wave's outgoing phase relative to the free radial solution. For a real central potential and elastic scattering, probability conservation gives a unit-modulus Partial-wave S-matrix $S_l=e^{2i{\delta }_l}$. Indeedso the incoming amplitude is unchanged and the outgoing one acquires $S_l$.

Expanding an incident plane wave in Legendre polynomials and comparing the outgoing spherical wave givesUse $d\sigma /d\Omega =|f{|}^2$ and $\int d\Omega P_l{P}_{l^{\prime }}=4\pi {\delta }_{l{l}^{\prime }}/(2l+1)$ to remove cross terms:This is the elastic total scattering cross-section for the one-particle central-potential problem, with no identical-particle symmetrization factor.

Let $w_0=1$ and $w_m=2$ for $m\ge 1$, as required by the cosine form of the Jacobi-Anger expansion. For a central real potential, the Partial-wave S-matrix is $S_m=e^{2i{\delta }_m}$. Splitting each shifted radial cosine into incoming and outgoing exponentials leaves its incoming coefficient unchanged and multiplies its outgoing coefficient by $S_m$. Subtracting the incident plane wave givesComparing with the prescribed scattering amplitude convention yields the two-dimensional partial-wave scattering formula,Orthogonality gives ${\int }_0^{2\pi }1d\theta =2\pi$ and ${\int }_0^{2\pi }{\cos }^2(m\theta )d\theta =\pi$ for $m\ge 1$, while all cross terms vanish. HenceThe multiplicities printed in this part are reversed. With those printed values and every ${\delta }_m=0$, the purported free wave would be $\frac{1}{2}{e}^{ikr\cos \theta }+\frac{3}{2}{J}_0(kr)$, rather than the required incident plane wave. The extra $J_0$ contains an incoming cylindrical wave, which cannot be absorbed into $f$. The formulas above use the corrected multiplicities and therefore satisfy the specified scattering boundary condition.

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.

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 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.