Phase-Shift Extraction

Numerical phase extraction converts a computed radial solution into an exterior scattering phase without depending on its arbitrary amplitude. The matching derivation and a potential-sweep workflow are given in the scattering notebook. This page concentrates on stable evaluation, branch continuity, and low-energy conditioning.

Required background. Phase shifts defines the convention, and partial-wave numerics supplies the radial solution.

Wronskian extraction without dividing by a node

For a real short-range elastic problem, write the exterior reduced solution as $u=C(J\cos\delta-N\sin\delta)$, with $J=\hat j_\ell(kr)$ and $N=\hat n_\ell(kr)$. Use radial derivatives $J_r,N_r$, including the factor of $k$. Define

$$a=uJ_r-u'J,\qquad b=uN_r-u'N.$$

The reference-function Wronskian is $JN_r-J_rN=k$; for example, $J=\sin kr$ and $N=-\cos kr$ for the $s$ wave. Substituting the exterior solution therefore gives $a=Ck\sin\delta$ and $b=Ck\cos\delta$, so $\tan\delta=a/b$. Unlike an expression involving $u'/u$, these combinations remain defined when $u$ crosses zero. A common rescaling of $u,u'$ cancels from the result. Do not rescale the function and derivative independently.

Compute a phase modulo $\pi$ with $\operatorname{atan2}(a,b)$, or compute the branch-invariant elastic matrix directly:

$$S_\ell=\frac{b+ia}{b-ia}.$$

For real $a,b$, this has unit modulus in exact arithmetic. Very small $a$ relative to $b$ demands an absolute phase tolerance rather than a misleading relative error.

Branches and energy sweeps

The physical $S_\ell=e^{2i\delta_\ell}$ is unchanged by $\delta_\ell\mapsto\delta_\ell+\pi$. To build a continuous phase curve, unwrap $\arg S_\ell$ with period $2\pi$ and divide by two. Fix its additive branch using a declared reference, such as continuity from zero interaction or an appropriate high-energy limit.

An energy step too large to resolve a rapid phase rotation can fool unwrapping. Refine the energy mesh around large changes and compare the $S$ values directly. Never repair a discontinuity by choosing a branch that conceals an unresolved resonance.

If $u$ changes sign from an independent normalization choice at each energy, $a,b$ both change sign. The raw atan2 phase changes by $\pi$, while the displayed $S$ ratio remains invariant.

Threshold extraction and scattering length

For a sufficiently short-range $s$-wave problem, fit the effective-range quantity,

$$k\cot\delta_0=-\frac1{a_s}+\frac12r_ek^2+\cdots.$$

Near zero phase, computing $\cot\delta$ from a rounded phase loses information. Use $k\,b/a$ directly when its conditioning is acceptable. Near $a_s=0$, that representation is itself singular; choose the quantity appropriate to the regime.

An independent zero-energy integration gives $u(r)\propto r-a_s$ outside the potential, hence $a_s=R-u(R)/u'(R)$. Vary the matching radius and integration step. Near a threshold bound state, $u'$ becomes small and the scattering length is extremely sensitive; report that sensitivity instead of claiming many stable digits from one match.

Benchmarks that isolate conventions

For $V=0$, the regular solution gives $a=0$ and $S_\ell=1$. For a hard sphere of radius $R_0$, the $s$-wave exterior solution is $\sin[k(r-R_0)]$, so $\delta_0=-kR_0$ modulo $\pi$ and $a_s=R_0$. These tests distinguish a physical minus sign from a branch convention.

For a general potential, compare extraction at several radii in the known exterior region. A plateau under grid refinement but not under radius variation indicates unresolved tail matching. Coulomb tails require Coulomb reference functions and their derivatives.

Exercises

If the radial derivative of a reference function is mistakenly replaced by its derivative with respect to $z=kr$, what changes?

Solution

The factor $k$ is lost. The matching combinations no longer compare quantities with consistent dimensions, and even free scattering generally acquires a spurious phase except at accidental parameter values. Testing several momenta, rather than only $k=1$ in chosen units, reveals the mistake.

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.