Scattering Theory

Intuition [Beginner]

Imagine throwing a tennis ball at a tree. The ball bounces off at some angle. If you throw many balls from many directions and record where they go, you build up a picture of the tree's shape and size without ever touching it directly. This is scattering: you probe a target by firing particles at it and studying what comes out.

In quantum mechanics, scattering is not just a convenient technique -- it is essentially the only way to study the subatomic world. We cannot put an electron under a microscope. Instead, we fire beams of particles (photons, electrons, protons, neutrinos) at targets and measure the angular distribution of what emerges. The Large Hadron Collider is a scattering machine: two proton beams collide, and detectors surrounding the collision point record the debris. The data are scattering cross-sections.

A cross-section $\sigma$ has units of area and measures the effective size of the target as seen by the incoming beam. The differential cross-section $d\sigma /d\Omega$ tells you the probability of scattering into a particular solid angle -- it is the angular pattern of scattered particles. Intuitively, $d\sigma /d\Omega$ at angle $\theta$ is proportional to the fraction of incoming particles deflected into a detector placed at that angle.

Two complementary approaches exist for calculating cross-sections:

Partial wave analysis expands the scattering wavefunction in angular momentum eigenstates (spherical harmonics). Each partial wave picks up a phase shift ${\delta }_{\ell }$ due to the potential. The cross-section becomes a sum over partial waves:

$$\frac{d\sigma }{d\Omega }={\left|\frac{1}{2ik}\sum _{\ell =0}^{\infty }(2\ell +1)\left(e^{2i{\delta }_{\ell }}-1\right)P_{\ell }(\cos \theta )\right|}^2$$

This works best when the potential is spherically symmetric and the incoming wavelength is comparable to or larger than the target (low energy).

The Born approximation treats scattering as a perturbation: the potential is weak enough that the scattered wave is small. The first Born approximation gives the scattering amplitude as the Fourier transform of the potential:

$$f({\mathbf{k}}^{\prime },\mathbf{k})=-\frac{m}{2\pi {\hslash }^2}\int V(\mathbf{r})e^{i(\mathbf{k}-{\mathbf{k}}^{\prime })\cdot \mathbf{r}}{d}^{3}r$$

This works best for weak potentials and high energies.

Visual [Beginner]

SCATTERING GEOMETRY
====================

         scattered particle
              \
               \  theta (scattering angle)
                \
    incoming     |       target
    beam   ----->O-------> (potential V(r))
    (plane       |
     wave)       |
                /
               /
              /
         scattered particle

    Omega = solid angle element at (theta, phi)
    d(sigma)/d(Omega) = scattered flux per solid angle
                        / incident flux per area


PARTIAL WAVE PICTURE
=====================

  Free wave:             Scattered wave:

      ~ cos(kr)            ~ cos(kr + delta_l)
                         
  phase shift delta_l measures how much the
  potential "delays" or "advances" each partial wave

  l=0 (s-wave): spherically symmetric  ( )
  l=1 (p-wave): dumbbell              ( O )
  l=2 (d-wave): cloverleaf          (* * * *)

Quantity	Symbol	Meaning
Total cross-section	${\sigma }_{\text{tot}}$	Total scattering probability (area)
Differential cross-section	$d\sigma /d\Omega$	Scattering probability per solid angle
Scattering amplitude	$f(\theta ,\varphi )$	Angular part of scattered wave; $|f{|}^2=d\sigma /d\Omega$
Phase shift	${\delta }_{\ell }$	Phase shift of $\ell$-th partial wave
Momentum transfer	$\mathbf{q}=\mathbf{k}-{\mathbf{k}}^{\prime }$	Change in momentum vector

Worked example [Beginner]

Problem: A particle of mass $m$ scatters off a hard sphere of radius $a$ (the potential is $V(r)=\infty$ for $r<a$ and $V(r)=0$ for $r>a$). Calculate the s-wave ($\ell =0$) phase shift and the s-wave contribution to the total cross-section at low energy ($ka\ll 1$).

Solution:

Step 1: Outside the sphere ($r>a$), the radial wavefunction for $\ell =0$ satisfies the free-particle equation. The general solution is:

$$u_0(r)=A\sin (kr)+B\cos (kr)$$

where $u_0=r{R}_0$.

Step 2: The boundary condition at $r=a$ is $u_0(a)=0$ (wavefunction vanishes inside the hard sphere). Applying this to the asymptotic form $u_0(r)\sim \sin (kr+{\delta }_0)$:

$$\sin (ka+{\delta }_0)=0⟹ka+{\delta }_0=n\pi$$

Choosing $n=0$: ${\delta }_0=-ka$.

Step 3: The s-wave contribution to the total cross-section is:

$${\sigma }_0=\frac{4\pi }{k^2}{\sin }^2{\delta }_0=\frac{4\pi }{k^2}{\sin }^2(ka)$$

Step 4: For $ka\ll 1$ (low energy), $\sin (ka)\approx ka$:

$${\sigma }_0\approx \frac{4\pi }{k^2}(ka{)}^2=4\pi a^2$$

At low energy, the cross-section is four times the geometric cross-section $\pi a^2$. This enhancement is a wave interference effect: the incoming wave "sees" the sphere as larger than its geometrical size because of diffraction.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Scattering states. In potential scattering with a localized potential $V(\mathbf{r})$ decaying faster than $1/r$, the scattering solutions to the Schrodinger equation have the asymptotic form:

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

The first term is the incident plane wave propagating along $z$; the second is the outgoing spherical wave modulated by the scattering amplitude $f(\theta ,\varphi )$. The differential and total cross-sections are:

$$\frac{d\sigma }{d\Omega }=|f(\theta ,\varphi ){|}^2,{\sigma }_{\text{tot}}=\int |f(\theta ,\varphi ){|}^{2}d\Omega$$

Partial wave expansion. For a spherically symmetric potential $V(r)$, the scattering amplitude can be expanded in partial waves:

$$f(\theta )=\frac{1}{2ik}\sum _{\ell =0}^{\infty }(2\ell +1)\left(S_{\ell }-1\right)P_{\ell }(\cos \theta )$$

where $S_{\ell }=e^{2i{\delta }_{\ell }}$ is the S-matrix element and ${\delta }_{\ell }$ is the phase shift. The total cross-section is:

$${\sigma }_{\text{tot}}=\frac{4\pi }{k^2}\sum _{\ell =0}^{\infty }(2\ell +1){\sin }^2{\delta }_{\ell }$$

Optical theorem. The total cross-section is related to the forward scattering amplitude:

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

This is a consequence of probability conservation (unitarity). It connects a measurable (total cross-section) to a single value of the scattering amplitude.

Born series. The scattering amplitude can be expanded in powers of the potential:

$$f({\mathbf{k}}^{\prime },\mathbf{k})=f^{(1)}+f^{(2)}+\cdots$$

where the first Born approximation is:

$$f^{(1)}({\mathbf{k}}^{\prime },\mathbf{k})=-\frac{m}{2\pi {\hslash }^2}\int e^{-i{\mathbf{k}}^{\prime }\cdot \mathbf{r}}V(\mathbf{r})e^{i\mathbf{k}\cdot \mathbf{r}}{d}^{3}r=-\frac{m}{2\pi {\hslash }^2}\tilde{V}(\mathbf{q})$$

with $\mathbf{q}=\mathbf{k}-{\mathbf{k}}^{\prime }$ the momentum transfer. Higher-order terms involve multiple scattering (integrals over intermediate momentum states).

Lippmann-Schwinger equation. The formal scattering solution satisfies:

$$|{\psi }^{(+)}⟩=|\varphi ⟩+\frac{1}{E-H_0+iϵ}V|{\psi }^{(+)}⟩$$

where $|\varphi ⟩$ is the incident plane wave, $H_0$ is the free Hamiltonian, and the $+iϵ$ prescription selects outgoing boundary conditions. This integral equation encodes all orders of perturbation theory.

Resonance scattering. When the energy matches a quasi-bound state, the phase shift passes rapidly through $\pi /2$ (modulo $\pi$). Near a resonance at energy $E_r$ with width $\Gamma$:

$${\delta }_{\ell }(E)\approx {\delta }_{\ell ,\text{bg}}+\arctan \frac{\Gamma /2}{E_r-E}$$

The cross-section displays a Breit-Wigner peak:

$${\sigma }_{\ell }\approx \frac{4\pi (2\ell +1)}{k^2}\frac{(\Gamma /2{)}^2}{(E-E_r{)}^2+(\Gamma /2{)}^2}$$

Key results [Intermediate+]

1. Unitarity bound. For each partial wave, ${\sin }^2{\delta }_{\ell }\le 1$ implies ${\sigma }_{\ell }\le \frac{4\pi (2\ell +1)}{k^2}$. The maximum scattering for each angular momentum channel is bounded by unitarity.

2. Effective range expansion. At low energy, the s-wave phase shift can be parameterized:

$$k\cot {\delta }_0=-\frac{1}{a_s}+\frac{1}{2}{r}_e{k}^2+O(k^4)$$

where $a_s$ is the scattering length and $r_e$ is the effective range. The scattering length determines low-energy scattering completely and can be much larger than the range of the potential (important for cold atom physics).

Advanced treatment [Master]

Formal scattering theory. The full machinery of scattering theory is built on the Moller operators (wave operators):

$${\Omega }^{(\pm )}=\underset{t\to \mp \infty }{\lim }{e}^{iHt}{e}^{-i{H}_{0}t}$$

which map free states to interacting states: $|{\psi }^{(\pm )}⟩={\Omega }^{(\pm )}|\varphi ⟩$. The S-matrix is:

$$S={\Omega }^{(-)†}{\Omega }^{(+)}=\underset{t\to +\infty ,t_0\to -\infty }{\lim }{e}^{i{H}_{0}t}{e}^{-iH(t-t_0)}{e}^{-i{H}_0{t}_0}$$

Unitarity of $S$ ($S^{†}S=1$) encodes probability conservation.

Analyticity and dispersion relations. The scattering amplitude is an analytic function of the energy (or momentum) in the complex plane, with branch cuts along the real axis corresponding to physical scattering thresholds and poles corresponding to bound states (negative imaginary part) and resonances (complex energy with positive imaginary part). Dispersion relations exploit this analyticity:

$$\mathrm{Re}f(\omega )=\frac{1}{\pi }\mathcal{P}\int \frac{\mathrm{Im}f({\omega }^{\prime })}{{\omega }^{\prime }-\omega }d{\omega }^{\prime }+\text{subtractions}$$

where $\mathcal{P}$ denotes the Cauchy principal value. These relations, combined with crossing symmetry and unitarity, form the basis of the S-matrix program.

Regge theory. In the complex angular momentum plane, the scattering amplitude can be analytically continued to complex $\ell$. The poles in this plane -- Regge poles at $\ell =\alpha (s)$ -- describe families of resonances with increasing spin. A Regge trajectory $\alpha (s)$ relates the spin to the mass-squared of a particle family (e.g., the $\rho$ trajectory). Regge behavior controls the high-energy, small-angle scattering cross-section:

$$\frac{d\sigma }{dt}\sim s^{2\alpha (t)-2}$$

where $s$ and $t$ are Mandelstam variables.

Multichannel scattering. When multiple final states are available (e.g., elastic scattering, inelastic scattering, particle production), the S-matrix becomes a matrix in channel space. The unitarity condition generalizes to $S^{†}S=1$ (matrix equation). The optical theorem generalizes to include inelastic contributions:

$${\sigma }_{\text{tot}}={\sigma }_{\text{el}}+{\sigma }_{\text{inel}}=\frac{4\pi }{k}\mathrm{Im}{f}_{\mathrm{el}}(0)$$

Time-dependent scattering. The time-dependent formulation uses wave packets. The incident wave packet is prepared in the distant past as a free state; after interacting with the potential, it separates into a transmitted wave and a scattered wave. The Moller operators connect the asymptotic free states to the full interacting states, and their existence and completeness are proven for a large class of short-range potentials (Kuroda, Agmon-Kato-Kuroda theorems).

Inverse scattering. Given the phase shifts ${\delta }_{\ell }(k)$ for all $\ell$ and $k$, one can reconstruct the potential $V(r)$ (the Marchenko or Gel'fand-Levitan equations). This has practical applications in seismology (reconstructing Earth's density profile from seismic data) and in quantum mechanics (soliton potentials related to integrable systems).

Relativistic scattering. In relativistic quantum field theory, scattering amplitudes are computed via the LSZ reduction formula from time-ordered correlation functions, which are computed perturbatively using Feynman diagrams. The S-matrix elements $⟨f|S|i⟩$ give the scattering amplitude, and the cross-section is:

$$d\sigma =\frac{(2\pi {)}^4}{4\sqrt{(p_1\cdot p_2{)}^2-m_1^2{m}_2^2}}|\mathcal{M}{|}^{2}d{\Phi }_n$$

where $\mathcal{M}$ is the invariant amplitude and $d{\Phi }_n$ is the Lorentz-invariant phase space for $n$ final-state particles.

Connections [Master]

• 12.07.01 -- Time-dependent perturbation theory provides the framework for the Born series.
• 12.07.02 -- Fermi's golden rule gives the transition rate for scattering into a continuum of states.
• 12.09.01 -- Identical-particle statistics modify the cross-section (symmetrization/antisymmetrization of the amplitude).
• 12.12.01 -- Canonical QFT provides the full machinery for relativistic scattering amplitudes via the LSZ formula and Feynman diagrams.
• 12.13.01 -- Quantum electrodynamics scattering processes (Compton scattering, Mott scattering) are direct applications.
• 13.06.01 -- Black hole scattering and quasi-normal modes are scattering problems in curved spacetime.

Bibliography [Master]

1. Taylor, J. R. Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006. A clear and thorough introduction to nonrelativistic scattering theory.

2. Sakurai, J. J. and Napolitano, J. Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020. Chapter 7 provides an excellent treatment of scattering theory at the intermediate level.

3. Newton, R. G. Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982. The definitive reference for formal scattering theory, including multichannel and inverse scattering.

4. Goldberger, M. L. and Watson, K. M. Collision Theory, Dover, 2004. Advanced treatment of formal scattering theory with emphasis on the S-matrix and dispersion relations.

5. Srednicki, M. Quantum Field Theory, Cambridge University Press, 2007. Chapter 5 covers the LSZ reduction formula and the connection between correlation functions and scattering amplitudes.

6. Joachain, C. J. Quantum Collision Theory, North-Holland, 1975. Comprehensive reference covering partial wave analysis, Born approximation, and formal scattering theory.