Scattering Theory

Most of what we learn about microscopic physics comes from scattering: fire a beam at a target, measure how it deflects. Scattering theory extracts the interaction potential (or, in QFT, the fundamental couplings) from the angular and energy distribution of the outgoing particles. This page develops the non-relativistic framework — cross sections, the scattering amplitude, partial waves and phase shifts, the optical theorem, and the Born approximation — which is the direct ancestor of the relativistic $S$-matrix.

1. Cross Section and Scattering Amplitude

For a particle of energy $E={\hslash }^2{k}^2/2m$ incident on a localized potential $V(\mathbf{r})$, seek stationary solutions with the asymptotic form of an incident plane wave plus an outgoing spherical wave:

$$\psi (\mathbf{r})\stackrel{r\to \infty }{\to }{e}^{ikz}+f(\theta ,\varphi )\frac{e^{ikr}}{r}.$$

The scattering amplitude $f(\theta ,\varphi )$ encodes all the physics. The experimentally measured differential cross section — flux scattered into solid angle $d\Omega$ per unit incident flux — is its modulus squared:

$$\boxed{\frac{d\sigma }{d\Omega }=|f(\theta ,\varphi ){|}^2,}$$

and the total cross section is $\sigma =\int |f{|}^{2}d\Omega$. The cross section has units of area — the effective "size" the target presents to the beam.

2. Partial Waves and Phase Shifts

For a central potential $V(r)$, angular momentum is conserved and the problem separates into partial waves of definite $\ell$. Far from the potential each radial wave is a free spherical wave shifted in phase; the entire effect of the potential on the $\ell$-th wave is a single real number, the phase shift ${\delta }_{\ell }(k)$. The amplitude becomes

$$f(\theta )=\frac{1}{k}\sum _{\ell =0}^{\infty }(2\ell +1)e^{i{\delta }_{\ell }}\sin {\delta }_{\ell }{P}_{\ell }(\cos \theta ),$$

and the total cross section is a sum of partial contributions,

$$\sigma =\frac{4\pi }{k^2}\sum _{\ell }(2\ell +1){\sin }^2{\delta }_{\ell }.$$

At low energy only $\ell =0$ ($s$-wave) survives, since the centrifugal barrier keeps higher partial waves away from a short-range potential — so cold scattering is characterized by a single number, the scattering length $a_s=-{\lim }_{k\to 0}{\delta }_0/k$, central to ultracold-atom physics.

3. Resonances

When a phase shift passes rapidly through $\pi /2$ (so ${\sin }^2{\delta }_{\ell }=1$) as energy varies, that partial wave's cross section hits its maximum — a resonance, signaling a quasi-bound state at energy $E_R$ with width $\Gamma$. Near it the cross section takes the Breit–Wigner form

$${\sigma }_{\ell }\propto \frac{{\Gamma }^2/4}{(E-E_R{)}^2+{\Gamma }^2/4},$$

the same Lorentzian lineshape as an unstable state's energy width, with lifetime $\tau =\hslash /\Gamma$. Resonances are how unstable particles and metastable compound states appear in scattering data.

4. The Optical Theorem

Conservation of probability (unitarity) ties the total cross section to the forward scattering amplitude:

$$\boxed{{\sigma }_{\text{tot}}=\frac{4\pi }{k}\mathrm{Im}f(\theta =0).}$$

Physically, whatever is scattered out of the forward beam must be removed from it, so the forward amplitude (which interferes with the unscattered wave) determines the total depletion. The theorem holds for elastic and inelastic channels combined, and survives into relativistic QFT as a consequence of $S$-matrix unitarity.

5. The Born Approximation

When the potential is weak, treat it in first-order perturbation theory (the Lippmann–Schwinger equation to leading order). The first Born approximation gives the amplitude as the Fourier transform of the potential with respect to the momentum transfer $\mathbf{q}={\mathbf{k}}^{\prime }-\mathbf{k}$:

$$f(\theta )\approx -\frac{m}{2\pi {\hslash }^2}\int d^{3}r{e}^{-i\mathbf{q}\cdot \mathbf{r}}V(\mathbf{r})=-\frac{2m}{{\hslash }^{2}q}{\int }_0^{\infty }rV(r)\sin (qr)dr,q=2k\sin \frac{\theta }{2}.$$

Measuring $d\sigma /d\Omega$ vs. angle thus maps out the potential — the principle behind form-factor measurements of nuclear and nucleon charge distributions. The exact integral equation it approximates is the Lippmann–Schwinger equation $|\psi ⟩=|\varphi ⟩+{\hat{G}}_0\hat{V}|\psi ⟩$, whose iteration is the Born series — the non-relativistic prototype of the perturbative expansion of the QFT $S$-matrix.

See also

• Orbital angular momentum — partial-wave decomposition.
• Time-dependent perturbation theory — the Born series and golden rule.
• Piecewise potentials — 1D reflection/transmission.
• QFT: Cross Sections — the relativistic generalization.