Modern Physics · Upper Division

Scattering Theory

Scattering is how we probe the structure of matter — from Rutherford discovering the nucleus to the LHC discovering the Higgs. The formal theory of quantum scattering connects asymptotic states to the S-matrix, and produces Feynman diagrams in the relativistic limit.

Prerequisites

Quantum mechanics (Ch. 20) \cdot Green's functions (Ch. GF) \cdot Angular momentum (Ch. SP

Learning Goals

$Define the scattering amplitude f(\theta) and relate it to the differential cross section d\sigma/d\Omega$ = | $f|^{2}$
Apply the Born approximation to compute scattering amplitudes as Fourier transforms of the potential.
$Expand the scattering amplitude in partial waves, extract phase shifts \delta_{l}, and apply the o$ ptical theorem.
$Derive the Breit-Wigner resonance formula and explain why nuclear resonances can have \sigma ≫$ $\piR^{2}$
Write down the Lippmann-Schwinger equation for the T-matrix and relate its Born expansion to Feynman diagrams.

SC.1 Scattering Formalism

A particle of momentum ℏk is scattered by a potential V(r), localized near the origin. Far from the target, the wavefunction is:

\psi(r) \to e^{ik\cdotr} + f(k, k') \times e^{ikr}/r \qquad (asymptotic scattered wave)

(SC.1)

where f(k, k') is the scattering amplitude(k' = k r̂). The differential cross section is:

d\sigma/d\Omega = |f(\theta, \phi)|^{2} \qquad (differential cross section)

(SC.2)

Total cross section: σ_total = ∫|f|² dΩ. The optical theorem relates σ_total to the forward scattering amplitude:

\sigma_total = (4\pi/k) Im f(\theta=0) \qquad (optical theorem)

(SC.3)

This is exact — it follows from unitarity of the S-matrix: probability is conserved. Physically: the forward scattered wave must interfere destructively with the incident wave to remove probability from the forward beam (shadow scattering).

SC.2 Born Approximation

For a weak potential V ≪ typical kinetic energies, the Born approximation gives:

f(q) = -m/(2\pi\hbar^{2}) \int e^{-iq\cdotr} V(r) d^{3}r = -m/(2\pi\hbar^{2}) Ṽ(q)

(SC.4)

where q = k − k' is the momentum transfer (|q| = 2k sin(θ/2)). The scattering amplitude is the Fourier transform of V at the momentum transfer.

For Coulomb potential V = Ze²/(4πε₀r): Ṽ(q) = Ze²/(ε₀q²). Born cross section: dσ/dΩ = (2mZe²/(4πε₀ℏ²))² / (2k sin(θ/2))⁴ = (Ze²/(4E))² / sin⁴(θ/2). This is the Rutherford formula — and it is exact for Coulomb scattering (higher-order Born terms are zero due to the long range of Coulomb).

SC.3 Partial Waves

For a spherically symmetric potential V(r), expand in angular momentum eigenstates. The scattering amplitude:

f(\theta) = (1/k) \sum_{l} (2l+1) e^{i\delta_{l}} sin \delta_{l} \times P_{l}(cos\theta) \qquad (partial wave expansion)

(SC.5)

where δₗ is the phase shift of the l-th partial wave — the phase acquired relative to free propagation. The partial wave cross section:

\sigma_{l} = (4\pi/k^{2})(2l+1) sin^{2} \delta_{l} \le (4\pi/k^{2})(2l+1) \qquad (unitarity bound)

(SC.6)

Resonances: when δₗ passes through π/2, the partial wave cross section reaches its maximum (unitarity limit). A Breit-Wigner resonance at energy E_r with width Γ:

\sigma_{l}(E) \approx (4\pi/k^{2})(2l+1) \times (Γ^{2}/4)/((E-E_{r})^{2} + Γ^{2}/4) \qquad (Breit-Wigner resonance)

(SC.7)

Example SC.1 — S-wave Scattering from a Hard Sphere

$Find the s-wave phase shift and cross section for a hard sphere of radius a (V = \infty for r <$ a, 0 for r > a) at low energy ka ≪ 1.

Radial equation:

For l=0, the radial equation gives u(r) = r \psi_0(r). Outside: u = A sin(kr + \delta_{0}). At r=a: u

(

a) = 0 \to ka + \delta_{0} = n\pi \to \delta_{0} = -ka (for small ka

Scattering length:

Define the scattering length a_{s} = -lim(k\to0) \delta_{0}/k = a (positive for hard sphere). Low-ener

gy limit: f \approx -a_{s} = -a. \sigma_total = 4\pia^{2} (at low energy, four times the geometric cross sec

tion!).

Physical:The factor of 4 arises because qua

ntum diffraction occurs in all directions, not just the shadow. At ka ≫ 1 (high energy): \sigma

\pia^{2} from diffraction ring

Universality:

All short-range potentials with the same scattering length a_{s} have the same low-energy cr

oss section \sigma \to 4\pia_s^{2}. The scattering length fully characterizes low-energy scattering —

this is why ultracold atoms can be described by just a_{s} regardless of the microscopic pot

ntial details.

SC.4 The S-Matrix

The S-matrix (scattering matrix) maps incoming asymptotic states to outgoing asymptotic states: |out⟩ = S|in⟩. For elastic scattering: S_l = e^(2iδₗ) — a pure phase (unitarity: S†S = 1 → |S_l| = 1).

The T-matrix: S = 1 + 2iT. The cross section: dσ/dΩ = |⟨k'|T|k⟩|². Lippmann-Schwinger equation:

T = V + V G_{0}(E+i\varepsilon) T \qquad (Lippmann-Schwinger for the T-matrix)

(SC.8)

where G₀ = 1/(E − H₀ + iε) is the free-particle Green's function. Iterating: T = V + VG₀V + VG₀VG₀V + ... — the Born series. Each term is a Feynman diagram in position-space (non-relativistic).

SC.5 Inelastic Scattering and Form Factors

For scattering from an extended object (nucleus, atom), the cross section involves the form factor F(q) — the Fourier transform of the charge distribution:

d\sigma/d\Omega = (d\sigma/d\Omega)_point \times |F(q)|^{2} \qquad (form factor)

(SC.9)

At small q: F(q) ≈ 1 − q²⟨r²⟩/6 + ... where ⟨r²⟩ is the mean-square radius. The proton charge radius was measured by electron-proton scattering (Hofstadter, 1961): r_p ≈ 0.84 fm. This became the proton radius puzzle when muonic hydrogen measurements gave r_p = 0.84087 fm vs ordinary hydrogen spectroscopy giving 0.8775 fm — a 7σ discrepancy (since resolved to ~0.84 fm by 2019 CODATA).

Definition SC.1 — Common Traps

Cross sections are probabilities per flux: they are not geometric areas except in special limits.
Phase shifts contain the interaction: different partial waves contribute differently by energy.
Resonances have widths: lifetime and energy uncertainty are linked.
Born approximation is weak-scattering: strong potentials require nonperturbative treatment.

Exercises — SC.1–SC.5 Scattering Theory

Verify the optical theorem \sigma = (4\pi/k) Im f(0) for s-wave scattering from a hard sphere of

radius a at low energy.

× πa²

Straightforward

Fin

d the differential and total cross section for Born scattering from a Yukawa potential V =

e^{-\mur}/r. How does \sigma behave at low and high energies

Intermediate

3.Describe s-wave resonance scattering from an attractive square well. Derive

the Breit-Wigner form for the cross section and explain why nuclear resonances can have \sigma

Intermediate

4.Derive the eikonal approximation for scattering at high energy (ka ≫ 1). Show how it reduces to the Born approximation for weak potentials, and to Rutherford for Coulomb.

Challenging

Key Takeaways

$Scattering amplitude f(\theta): d\sigma/d\Omega = |f|^{2}. Optical theorem: \sigma = (4\pi/k) Im f(0) — from unitar$ ity.
$Born approximation: f = -m/(2\pi\hbar^{2}) Ṽ(q). Cross section = |FT of potential|^{2}. Rutherford = e$ xact Born.
$Partial waves: f = (1/k)\sum(2l+1) e^{i\delta_{l}} sin\delta_{l} P_{l}(cos\theta). Phase shifts encode all scattering$ information.
$S-wave at low energy: \sigma \to 4\pia_s^{2} (a_{s} = scattering length). Hard sphere: \sigma = 4\pia^{2} (4\times geom$ etric).
$Resonances: \delta_{l} \to \pi/2, \sigma_{l} \to unitarity limit 4\pi(2l+1)/k^{2}. Breit-Wigner: Lorentzian in energy$ .
$S-matrix: S = e^{2i\delta}. T = V + VG_{0}T (Lippmann-Schwinger). Born series = Feynman diagrams$ .