Eikonal Approximation

Every heavy-ion collision at RHIC and the LHC is scored by this article's mathematics: "Glauber Monte Carlo" codes draw nucleon positions, assign each baryon a straight-line path, and convert nucleon–nucleon cross sections into participant counts and centrality classes — the eikonal premise, run millions of times per data set. Proton–nucleus total cross sections are still unfolded with eikonal profile functions, and the same $e^{i\chi(b)}$ formalism now computes graviton-exchange amplitudes in studies of Planckian scattering and post-Minkowskian orbital dynamics, where resumming the infinite tower of ladder diagrams is precisely the $e^{i\chi}$ trick again. Optics returned the favor too: adaptive-optics wavefront correction is the inverse eikonal problem — measure the corrugated $\chi(b)$, then undo it.

Think of the incident wave as a bundle of parallel rays, one per impact parameter $\mathbf b$. A ray that passes far from the target feels nothing and emerges as it entered. A ray that pierces the potential travels the same straight line — at $ka\gg1$ the deflection inside the target is negligible — but its phase clock runs at a different rate while $V\ne0$: slower over a barrier, faster through a well, by exactly $-V\,\mathrm{d}t/\hbar = -V\,\mathrm{d}z/\hbar v$ per element of path. The emerging wavefront is no longer flat; it carries a position-dependent phase $\chi(b)$, and a corrugated wavefront is a scattered wave. Everything downstream — the amplitude, the cross section, the diffraction lobes — is the far-field interference of this corrugation, computed exactly as a lens designer would.

The dictionary with partial waves is exact bookkeeping, not analogy. A ray at impact parameter $b$ carries angular momentum $\hbar kb$, so it is the partial wave $\ell\approx kb$; at high energy thousands of $\ell$ contribute and the discrete ladder blurs into the continuous variable $b$. The eikonal phase at that $b$ plays the role of twice the phase shift — the identification we verify numerically below. Where the WKB machinery of The WKB Approximation accumulates phase along a one-dimensional coordinate between turning points, the eikonal accumulates it along the undeflected beam direction: the same semiclassical idea, aimed forward.

The construction begins with an ansatz that separates the fast oscillation from the slow imprint. Write the scattering state as a plane wave times a modulation, $\psi(\mathbf r)=e^{ikz}\phi(\mathbf r)$, insert it into the time-independent Schrödinger equation at energy $E=\hbar^2k^2/2m$, and the $k^2$ terms cancel identically (verified symbolically), leaving

where: $\phi$ — the modulation, equal to 1 far upstream; $U$ — the potential in scattering units (dimension 1/length²). Nothing is approximate yet. The eikonal step is dropping $\nabla^2\phi$ against $2ik\,\partial_z\phi$: the modulation varies over the potential's range $a$, so $|\nabla^2\phi|\sim\phi/a^2$ while $|2k\partial_z\phi|\sim k\phi/a$ — the neglected term is smaller by $1/ka$. Reading (1) without its Laplacian gives a first-order equation that integrates along each straight line of fixed transverse position $\mathbf b$.

The exit wavefront must now be converted into a detector-side amplitude, and the honest route runs through the closed formula this chapter already owns. Green's Function for Scattering derived, exactly, $f(\mathbf k',\mathbf k)=-\frac{1}{4\pi}\int e^{-i\mathbf k'\cdot\mathbf r'}U(\mathbf r')\psi(\mathbf r')\,\mathrm d^3r'$ — valid for the true $\psi$. The Born approximation of Born Approximation inserts a plane wave for $\psi$; the eikonal inserts something better: the phase-imprinted wave of Derivation A.

The convention repair card — surfaced because the base export tripped on it. The exported tldr defined $\chi$ with the full-line integral and used the profile $e^{2i\chi}-1$: linearizing that pair overshoots the Born amplitude by exactly a factor 2 (machine-proven, sympy). Its own equation (4), with a half-line integral, was internally consistent — the two statements simply belonged to different dialects. The same export's Yukawa derivation also misprinted the Coulomb coupling as $n=2mZ_1Z_2e^2/\hbar^2k$; the Rutherford modulus $|f|=(n/2k)\csc^2(\theta/2)$ requires the Sommerfeld parameter $n=mZ_1Z_2e^2/\hbar^2k$, and the printed value would quadruple $\mathrm d\sigma/\mathrm d\Omega$ — also machine-proven against the exact Rutherford form. Both repairs are folded into the formulas above and flagged in the cheat sheet.

Comparing the eikonal amplitude with the Legendre-sum amplitude of Partial-Wave Expansion term by term, using the large-$\ell$ asymptotic $P_\ell(\cos\theta)\approx J_0\bigl((\ell+\tfrac12)\theta\bigr)$ and replacing $\sum_\ell\to k\!\int\!\mathrm db$, the two expressions coincide exactly when

where: the factor 2 is the round-trip bookkeeping of the S-matrix element $e^{2i\delta_\ell}$ — the profile $e^{i\chi}$ is $S_\ell$ read at continuous angular momentum. This is the statement that $\chi(b)$ is not merely analogous to a phase shift; it is the phase-shift function, sampled where the classical ray sits. We tested equation (2) against an exact solve. For a Gaussian well $U=-2.5\,e^{-r^2/a^2}$ with $k=5$, $a=3$ ($ka=15$, $\hbar=2m=1$), the radial equation was integrated by Numerov's method for $\ell=0\ldots45$ and the exact $\delta_\ell$ extracted from the asymptotic match; the eikonal prediction $\chi\bigl((\ell+\tfrac12)/k\bigr)/2$ tracks them to 1.7% median deviation ($\delta_0$: 0.653 exact vs 0.664 eikonal; $\ell=20$: 0.1043 vs 0.1027). This is the regime where the well is not weak — $\chi(0)=1.33$ rad — and the payoff shows in the cross section:

At $\theta=0.05$ rad the exact answer is 629.0 (script units); the eikonal gives 618.1 (−1.7%) while first Born gives 675.3 (+7.4%), degrading to 56% across the forward cone as the eikonal stays within 12%. Born's failure is structural, not numerical: it is first order in $V$, while $e^{i\chi}-1$ contains all powers — the resummation that Born Series and Higher-Order Corrections organizes term by term, obtained here in one closed profile.

An approximation earns trust when it respects exact theorems uninvited. For real $\chi$ (no absorption), the identity $|e^{i\chi}-1|^2=2(1-\cos\chi)$ turns the small-angle Parseval relation for the Fourier–Bessel transform into a chain of equalities:

where: the last member is precisely the statement of the Optical Theorem, whose derivation belongs to that article — here it emerges as an internal identity of the approximation, and the verify script confirms all three members agree to machine precision ($23.213506$ in the Gaussian-well lab, deviation $10^{-16}$). Contrast Born: for a real potential the first-order amplitude is real, $\mathrm{Im}\,f_{\mathrm{Born}}(0)=0$ exactly, so the optical theorem fails at first order and is only rescued at second — one more face of the resummation. Because $e^{i\chi}$ is a legitimate S-matrix element ($|e^{i\chi}|\le1$, with equality for real $\chi$), the eikonal can also swallow absorption whole: give $\chi$ a positive imaginary part and $\eta(b)=|e^{i\chi}|<1$ reproduces the inelasticity bookkeeping of Inelastic Scattering and Multichannel Theory ray by ray. That connection pays immediately. Inelastic Scattering and Multichannel Theory computed the black disk's cross-section totals — $\sigma_{\mathrm{el}}=\sigma_{\mathrm{inel}}=\pi R^2$ — and deferred the angular shape here. Set $e^{i\chi}=0$ for $b<R$ (total absorption) and 1 outside: the amplitude integral truncates and evaluates in closed form,

where: the Bessel identity $\int_0^R J_0(qb)\,b\,\mathrm db=(R/q)J_1(qR)$ is verified symbolically, and $J_1$ makes this Fraunhofer diffraction — the same lobes as light behind an opaque screen. As $q\to0$, $f(0)=ikR^2/2$, so $(4\pi/k)\,\mathrm{Im}f(0)=2\pi R^2$: the factor-of-two total, recovered from the wave side. The first dark ring sits at $qR=3.8317$ ($J_1$'s first zero), and the forward lobe's width $\theta\sim1/kR$ is the shadow cone of the multichannel article, now with its exact profile.

For the Yukawa potential $V=(g/r)e^{-\mu r}$ of Scattering by a Screened Coulomb Potential, the line integral is a textbook Bessel representation: $\int_{-\infty}^{\infty}e^{-\mu\sqrt{b^2+z^2}}/\sqrt{b^2+z^2}\;\mathrm dz=2K_0(\mu b)$ (checked numerically to $10^{-12}$), so

where: $n$ — the dimensionless Sommerfeld-type coupling (the repaired value; the base's extra factor 2 is the error proven in the verify script). In the Born limit the Hankel identity $\int_0^\infty J_0(qb)K_0(\mu b)\,b\,\mathrm db=1/(q^2+\mu^2)$ reproduces the screened-Coulomb Born amplitude exactly — consistency with the owner article held symbolically. As $\mu\to0$, $K_0(\mu b)\to-\ln(\mu b/2)-\gamma_E$ and the phase diverges logarithmically at large $b$: the pure Coulomb potential has no eikonal (or any other) scattering theory without screening, the same infinite-range honesty flagged across this chapter. The remarkable accident, quoted with its source (Sakurai §6.5; Newton Ch. 18): carrying the screened calculation through and then removing the screen reproduces the Rutherford modulus $|f|=(n/2k)\csc^2(\theta/2)$ exactly — high-energy approximation notwithstanding — because for Coulomb, Born, eikonal, and the exact answer all share one modulus, differing only in phase. We verify the modulus identity symbolically; the phase statement belongs to the owner article.

Example 1 — a 1 GeV proton through a nuclear well: is the eikonal licensed, and is Born? A proton with kinetic energy $T=1$ GeV traverses a Gaussian optical well of depth 40 MeV and range $a=4$ fm.

Step 1. Kinematics (relativistic, honestly): $pc=\sqrt{T(T+2m_pc^2)}=\sqrt{1000\times2877}=1696$ MeV, so $k=pc/\hbar c=1696/197.3=8.60\ \mathrm{fm^{-1}}$ and $v=pc^2/E_{\mathrm{tot}}=0.875c$.

Step 2. License check: $ka=34.4\gg1$ ✓ and $V_0/T=0.04\ll1$ ✓ — both eikonal conditions hold comfortably.

Step 3. The imprint: $|\chi(0)|=\frac{V_0\sqrt\pi a}{\hbar v}=\frac{40\times1.772\times4}{0.875\times197.3}=\mathbf{1.64\ rad}$. The phase is order one: first Born, which needs $|\chi|\ll1$, is not licensed even though $V\ll E$ — the two smallness conditions are genuinely different, and this beam sits exactly in the eikonal's exclusive territory.

Example 2 — diffraction minima of 14 MeV neutrons on lead. Treat the Pb nucleus ($R\approx7.6$ fm) as a black disk for 14 MeV neutrons.

Step 1. $k=\sqrt{2m_nE}/\hbar=0.822\ \mathrm{fm^{-1}}$, so $kR=6.2$ — marginal for the asymptotic machinery, which is worth stating rather than hiding; the lobe positions are nonetheless robust.

Step 2. First dark ring at $qR=3.8317$: $q=0.504\ \mathrm{fm^{-1}}$, and $\theta=2\arcsin(q/2k)=\mathbf{36^\circ}$.

Step 3. Boundary check: measured 14 MeV neutron elastic distributions on heavy nuclei show their first deep minimum in the 30–40° window — the $J_1$ structure of equation (4), fuzzed by the nuclear surface. All numbers recomputed in the verify script.

References. Sakurai & Napolitano, Modern Quantum Mechanics, §6.5 (Eikonal Approximation) · Glauber, in Lectures in Theoretical Physics, Vol. I (Boulder 1959) · Newton, Scattering Theory of Waves and Particles, Ch. 18 · Shankar, Principles of Quantum Mechanics, Ch. 19 · Cohen-Tannoudji, Diu & Laloë, Quantum Mechanics, Vol. II, Ch. VIII · Miller & Amsler (PDG reviews) for Glauber-model centrality practice.

Provenance. Equations derived from the high-energy reduction of the Schrödinger / Lippmann–Schwinger equation following Sakurai & Napolitano §6.5, Shankar Ch. 19, and Newton Ch. 18. A verifier should: (1) reproduce the transport equation by substituting $\psi=e^{ikz}\phi$ and dropping $\partial_z^2\phi$; (2) confirm the Yukawa eikonal phase via the $K_0$ integral (Gradshteyn–Ryzhik 3.387); (3) check the black-disk amplitude $f=ikR\,J_1(qR)/q$ and $\sigma_{\mathrm{tot}}=2\pi R^2$; (4) verify the partial-wave correspondence $b=(\ell+\tfrac12)/k$, $\delta_\ell\to\chi(b)$. All special-function identities are standard; the Rutherford modulus and Coulomb phase follow the screened-Coulomb limit. No non-textbook numbers are quoted. — Base repairs (machine-proven, verify script sections 2–3): the export's tldr paired a full-line phase integral with the $e^{2i\chi}$ profile, overshooting its own Born limit by exactly 2 (its eq. (4) half-line definition was self-consistent; the dialects were mixed); and its Yukawa derivation printed the Coulomb coupling $n=2mZ_1Z_2e^2/\hbar^2k$ — the extra 2 quadruples Rutherford. Corrected throughout; trap lines in the cheat sheet.

Builds on: Scattering Amplitude · Born Approximation · Partial-Wave Expansion · Phase Shifts · Lippmann-Schwinger Equation. Leads to: Scattering by a Screened Coulomb Potential · Optical Theorem · Born Series and Higher-Order Corrections · The S-Matrix and Unitarity in Scattering.