Scattering Computations

A numerical scattering calculation must extract an observable from a wave that continues beyond the interaction region. This chapter’s phase-shift example solves a radial equation for a short-range central potential, matches to known asymptotic waves, and converts the result into partial-wave observables. Related labs compare stationary scattering with wave-packet motion and with the Born approximation.

Helpful background. Phase shifts defines the asymptotic phase; partial-wave cross sections explains the channel sum. Computational Quantum Mechanics develops the numerical methods and their error analysis.

Start with a radial phase shift

The phase-shift extraction notebook includes a runnable s-wave calculation for a Gaussian potential. It begins with the regular solution near the origin and integrates outward to a radius where the potential is negligible at the desired accuracy.

Outside that region, known free radial functions describe the solution. Matching the numerical wave and derivative fixes the phase shift without requiring a physically meaningful overall normalization of the integrated wave. The logarithmic derivative $u'/u$ is convenient, but it has poles at zeros of $u$; those poles are not automatically scattering resonances. Matching with the pair $(u,u')$ or using a stable propagation method avoids mistaking a normalization singularity for physics.

The phase is defined modulo $\pi$. When scanning energy, choose a continuous branch before interpreting rapid phase motion. A jump produced by the principal inverse tangent is not itself a resonance.

Vary the controls separately

Numerical choice	What to test
Starting radius and origin expansion	Move closer to the origin or include more terms, while avoiding singular numerical evaluation.
Grid spacing or ODE tolerances	Refine until the phase shift and target cross section stabilize.
Matching radius	Move it outward until the neglected potential tail no longer changes the answer appreciably.
Largest angular momentum	Increase it until the requested angular distribution or integrated cross section converges.
Energy sampling	Resolve narrow structures and track the phase branch continuously.

A small ODE residual or solver success flag does not check all of these choices. Likewise, constructing $S_\ell=e^{2i\delta_\ell}$ from a real numerical phase automatically gives $|S_\ell|=1$; it does not prove that the phase is accurate.

Use the zero-potential limit and the exact square-well or hard-sphere phases to check matching signs and normalization. For a weak smooth potential, compare with the Gaussian Born result. Agreement should improve as the interaction is weakened within a fixed, appropriate regime.

Compare stationary waves with packets

The wave-packet scattering lab follows a packet through a rectangular barrier. Its late-time transmitted probability averages the energy-dependent transmission over the packet’s momentum distribution. It need not equal the transmission at the mean momentum when the packet is broad or the barrier response changes rapidly.

The Born-approximation lab compares an analytic weak-potential prediction with radial scattering. It separates numerical convergence from the physical error introduced by neglecting repeated interactions.

When modifying either calculation, keep units, potential parameters, boundary conditions, and refinement settings with the output. Reproducible computation explains how to make that comparison repeatable.

Exercises

An automatic check. Two programs return different real phase shifts but both produce $|e^{2i\delta_\ell}|=1$. What has this check established, and what comparison would be stronger?

Solution

It establishes the elastic form of each constructed channel matrix, not the phase’s accuracy. Compare the phases modulo $\pi$, refine the integration and matching radius, and test a known exact potential. A wrong real phase can still satisfy elastic unitarity exactly.

A packet near a sharp feature. A packet spans a narrow transmission resonance. Why can evaluating the stationary transmission only at its mean momentum be misleading?

Solution

The transmitted probability weights the full momentum distribution by the momentum-dependent transmission. The mean can lie on a tall narrow peak while much of the packet lies off it. Compute the weighted average and compare only after incident and outgoing packets are sufficiently separated and numerical boundary effects are controlled.

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 propagation of scattering solutions.
• Taylor, J. R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions. Dover (2006; reprint of the 1972 edition). Asymptotic matching and wave-packet interpretation.