Partial-Wave Numerics

Partial-wave scattering reduces a central-potential problem to radial equations, one for each angular momentum. The numerical task is to propagate the regular solution without losing its physical branch, match it to the correct exterior functions, and include enough partial waves for the requested observable. The partial-wave expansion explains the angular decomposition.

Required background. Radial boundary conditions and matching methods supply the ODE framework.

The regular radial initial condition

For reduced mass $\mu$, $E=\hbar^2k^2/(2\mu)$, and real central potential $V(r)$,

$$u_\ell''= \left[\frac{\ell(\ell+1)}{r^2} +\frac{2\mu}{\hbar^2}(V-E)\right]u_\ell.$$

For a potential finite at the origin, let $q_0=2\mu[V(0)-E]/\hbar^2$. The regular Frobenius start is

$$u_\ell(r)=r^{\ell+1} \left[1+\frac{q_0r^2}{2(2\ell+3)}+\cdots\right].$$

Differentiating this series supplies $u'$. Starting with only $u\propto r^{\ell+1}$ is often adequate, but its error must be reduced by shrinking the starting radius or adding terms. Singular potentials require their own local analysis.

For large $\ell$, $r^{\ell+1}$ may underflow. Scale the initial pair so that one component is order one while preserving its logarithmic derivative.

Stable propagation and matching

A high-order ODE solver or Numerov-type recurrence can propagate $u,u'$, with occasional common rescaling. In classically forbidden regions a tiny irregular admixture may grow rapidly. Log-derivative or renormalized propagation avoids tracking exponentially disparate amplitudes.

If $Y=u'/u$, the Riccati equation is $Y'=Q-Y^2$ with $Q$ the bracket in the radial equation. $Y$ has poles at wavefunction nodes; a naive generic integrator applied across those poles is not automatically stable. Established log-derivative propagators manage the transformation through finite-step algebra.

Match beyond the short-range tail with the Wronskian phase algorithm. Vary radial step, starting radius, and matching radius independently.

A finite square-well test

For $V=-V_0$ with $V_0>0$ inside radius $a$, the $s$-wave interior momentum is $q=\sqrt{k^2+2\mu V_0/\hbar^2}$. At the interface, $u\propto\sin(qr)$ gives logarithmic derivative $q\cot(qa)$. Matching to the exterior yields

$$\tan(ka+\delta_0)=\frac{k}{q}\tan(qa).$$

Implement the interface exactly as a domain boundary rather than smearing the discontinuity across an uncontrolled stencil. Compare $S=e^{2i\delta}$ to avoid ambiguity from tangent branches. As $k\to0$,

$$a_s=a-\frac{\tan(\kappa_0a)}{\kappa_0}, \qquad \kappa_0=\sqrt{2\mu V_0}/\hbar.$$

Divergence near $\kappa_0a=(n+\frac12)\pi$ exposes threshold sensitivity and tests adaptive energy resolution.

Angular-momentum cutoff and observable error

The heuristic $\ell\sim kR_{\rm int}$ identifies waves whose centrifugal turning points reach the interaction region. It is a starting estimate, not a certified cutoff. Increase $\ell_{\max}$ until total and differential cross sections stabilize over the requested angle range.

Weak high-$\ell$ phases may contribute coherently in forward directions even when their individual total-cross-section terms are small. Cross-section computation therefore checks angular observables separately.

For real elastic potentials, verify $|S_\ell|=1$ and free-potential phases. Strong absorption, inelastic channels, or long-range tails change those checks and must be declared. Coupled channels require matrix propagation with threshold-dependent exterior functions.

Exercises

Derive the coefficient of $r^{\ell+3}$ in the regular origin series.

Solution

Insert $u=r^{\ell+1}(1+cr^2)$ into the radial equation with $Q=\ell(\ell+1)/r^2+q_0+\cdots$. The coefficient difference at order $r^{\ell+1}$ is $c[(\ell+3)(\ell+2)-\ell(\ell+1)]=2c(2\ell+3)$. Equating it to $q_0$ gives $c=q_0/[2(2\ell+3)]$.

References

• Johnson, B. R., “The multichannel log-derivative method for scattering calculations,” Journal of Computational Physics 13, 445–449 (1973), doi:10.1016/0021-9991(73)90049-1 — stable radial propagation.
• Taylor, John R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Publications (2006; original edition 1972) — scattering normalization, amplitudes, and analytic structure.