Scattering Theory

Recall prompts

Attempt these from memory before reading.

What is a cross-section, and what are its dimensions?
Write the partial wave expansion of the scattering amplitude.
What is a phase shift, and what does $\delta_l = 0$ mean physically?
State the Born approximation and the condition for it to be valid.
What is the difference between a bound state and a resonance?

Check your answers against the text. Prompt 1 is the one that connects this page to experiment: everything on it exists to predict a number someone can measure.

Motivation: scattering is how anything is known

Nobody has ever seen an atomic nucleus. What Rutherford saw in 1911 was that alpha particles occasionally came backwards, and he inferred from the angular distribution that the positive charge was concentrated in a region $10^5$ times smaller than the atom.

That is the pattern for the whole subject, and it has not changed. The electron’s magnetic moment, the proton’s internal structure, the existence of the up and down quarks — every one of these was inferred from how something scattered off something else. Scattering theory is not one topic among many; it is the interface between theory and experiment at every scale.

The central object is the scattering amplitude $f(\theta)$ . Far from the target the wave function is a plane wave plus an outgoing spherical wave:

\psi \sim e^{ikz} + f(\theta)\,\frac{e^{ikr}}{r},

and $f(\theta)$ contains everything. The differential cross-section is $|f(\theta)|^2$ , the total cross-section is its integral, and resonances show up as rapid variation of its phase. Solving a scattering problem means finding one complex-valued function of one angle.

Why expand in partial waves. A central potential is spherically symmetric, so angular momentum is conserved and each $l$ evolves independently. That decomposes a problem in three dimensions into a stack of one-dimensional problems — one radial equation per $l$ — each with a single number to describe it, the phase shift $\delta_l$ . The amplitude becomes

f(\theta) = \frac{1}{k}\sum_l (2l+1)\,e^{i\delta_l}\sin\delta_l\,P_l(\cos\theta),

and the physics of a complicated 3D collision is now a list of angles. This is the same move as diagonalising an operator: the basis in which the problem decouples is the one worth using.

13.1 Partial Wave Analysis

For a spherically symmetric potential $V(r)$ The scattering amplitude can be expanded in partial waves:

f(\theta) = \frac{1}{2ik}\sum_{l=0}^{\infty}(2l + 1)(e^{2i\delta_l} - 1)P_l(\cos\theta) = \frac{1}{k}\sum_{l=0}^{\infty}(2l + 1)e^{i\delta_l}\sin\delta_l\,P_l(\cos\theta)

Where $\delta_l$ is the phase shift for partial wave $l$ .

Optical theorem:

\sigma_{\text{total} = \frac{4\pi}{k}\,\text{Im}\,f(0) = \frac{4\pi}{k^2}\sum_{l=0}^{\infty}(2l+1)\sin^2\delta_l}

Partial wave unitarity bound: $\sin^2\delta_l \leq 1$ So the maximum contribution of partial wave $l$ to the cross section is:

\sigma_l^{\max} = \frac{4\pi}{k^2}(2l + 1)

13.2 The Born Approximation

For a weak potential, the scattering amplitude to first order is:

f(\theta, \phi) \approx -\frac{m}{2\pi\hbar^2}\int e^{-i\mathbf{k}"\cdot\mathbf{r}'}V(\mathbf{r}')\,d^3r'

Where $\mathbf{k}'$ is the scattered wave vector and $\mathbf{q} = \mathbf{k}' - \mathbf{k}$ is the momentum transfer.

For the Yukawa potential $V(r) = (V_0/r)e^{-\mu r}$ :

f(\theta) = -\frac{2m V_0}{\hbar^2(\mu^2 + q^2)}, \quad q = 2k\sin(\theta/2)

Setting $\mu = 0$ (Coulomb potential), this reproduces the Rutherford scattering formula.

13.3 Resonance Scattering

When the scattering energy is near a quasi-bound state, the phase shift passes through $\pi/2$ (resonance):

\delta_l(E) \approx \delta_{\text{bg} + \arctan\!\left(\frac{\Gamma/2}{E_R - E}\right)}

Where $E_R$ is the resonance energy and $\Gamma$ is the width. The cross section has the Breit—Wigner form:

\sigma_l(E) = \frac{4\pi}{k^2}(2l+1)\frac{(\Gamma/2)^2}{(E - E_R)^2 + (\Gamma/2)^2}

At resonance ( $E = E_R$ ): $\sigma_l^{\max} = \frac{4\pi}{k^2}(2l+1)$ (unitarity limit).

Worked Example 12.1: Low-Energy Scattering and Scattering Length

For $s$ -wave scattering ( $l = 0$ ) at low energy ( $ka \ll 1$ ), only the $l = 0$ phase shift contributes:

\sigma \approx \frac{4\pi}{k^2}\sin^2\delta_0 \approx 4\pi a_s^2

Where the scattering length $a_s$ is defined by $k\cot\delta_0 \to -1/a_s$ as $k \to 0$ .

For a hard sphere of radius $a$ : $\delta_0 = -ka$ (exact), giving $a_s = a$ and $\sigma = 4\pi a^2$ (four times the geometric cross section $\pi a^2$ --- a purely quantum result).

For the $^3$ He— $^4$ He system: $a_s \approx 1.4$ nm (positive, indicating a repulsive effective potential). For the neutron—proton system (triplet): $a_s \approx 5.4$ fm (positive, with a bound state --- the deuteron). For singlet: $a_s \approx -23.7$ fm (negative, indicating a virtual state).

Worked Example 12.2: Born Approximation for a Gaussian Potential

Consider $V(r) = V_0\,e^{-r^2/(2a^2)}$ .

f(\theta) = -\frac{m}{2\pi\hbar^2}\int e^{-i\mathbf{q}\cdot\mathbf{r}}V_0 e^{-r^2/(2a^2)}\,d^3r

= -\frac{m V_0}{2\pi\hbar^2}(2\pi a^2)^{3/2}e^{-q^2 a^2/2} = -\frac{m V_0}{\hbar^2}(2\pi)^{1/2}a^3\,e^{-2k^2a^2\sin^2(\theta/2)}

The total cross section:

\sigma = \int |f|^2\,d\Omega = 2\pi\int_0^\pi |f|^2\sin\theta\,d\theta

At low energy ( $ka \ll 1$ ): $f \approx -\frac{mV_0}{\hbar^2}(2\pi)^{1/2}a^3$ (independent of $\theta$ ), giving:

\sigma \approx 4\pi\left(\frac{mV_0}{\hbar^2}\right)^2 2\pi\,a^6 = \frac{8\pi^2 m^2 V_0^2 a^6}{\hbar^4}

The Born approximation is valid when $|V_0| \ll \hbar^2/(ma^2)$ I.e., the potential is weak compared to the kinetic energy associated with the length scale $a$ .

Common Pitfalls (Additional)

Symmetrisation applies to the full wavefunction: For fermions, the overall wavefunction (spatial $\otimes$ spin $\otimes$ any other degrees of freedom) must be antisymmetric. A symmetric spatial part requires an antisymmetric spin part (singlet), and vice versa. Do not apply (anti)symmetrisation to spatial and spin parts separately without ensuring the correct combined symmetry.
The variational principle gives an upper bound: $E_{\text{trial} \geq E_0}$ always. If you obtain a variational energy lower than the known exact ground state energy, you have made an error in the calculation (wrong normalisation, incorrect matrix element, or the trial function is not in the correct Hilbert space).
Fermi’s Golden Rule applies to transitions to a continuum: For transitions to discrete states, use the Rabi formula instead. The density of states $\rho(E_f)$ is essential --- if it is zero, the transition rate is zero regardless of the matrix element.
The Born approximation assumes a weak potential: The condition is $m|V_0|a/\hbar^2 \ll 1$ where $a$ is the range of the potential. For strong potentials (like the nuclear potential or hard spheres), the Born approximation gives qualitatively wrong results. Use partial wave analysis instead.
Resonances require careful treatment: Near a resonance, perturbation theory breaks down. The Breit—Wigner formula is non-perturbative in the width $\Gamma$ . The scattering length can be much larger than the range of the potential near a resonance (the unitarity limit).

Problems (Additional)

Problem 19: Exchange Energy in Lithium

Lithium ( $Z = 3$ ) has the electron configuration $1s^2 2s^1$ . Using the variational method with $Z_{\text{eff}}$ for the $1s$ electrons:

(a) Calculate $Z_{\text{eff}}$ for the $1s$ electrons, treating the $2s$ electron as a perturbation.

(b) Calculate the ionisation energy (removing the $2s$ electron) and compare with the experimental value of 5.39 eV.

Solution:

(a) For the $1s$ electrons, the effective charge is reduced from $Z = 3$ by screening from the other $1s$ electron and partial penetration of the $2s$ electron. The $1s$ electrons screen each other partially: using the helium result, $Z_{\text{eff}(1s) \approx Z - 5/16 = 3 - 0.3125 = 2.69}$ .

(b) The $2s$ electron sees an effective nuclear charge of $Z_{\text{eff}(2s) \approx 3 - 2 \times 0.85 = 1.3}$ (Slater’s rules). The energy:

E_{2s} = -\frac{Z_{\text{eff}^2}{n^2}\times 13.6\ \text{eV} = -\frac{1.3^2}{4}\times 13.6 = -\frac{1.69}{4}\times 13.6 = -5.75\ \text{eV}}

The ionisation energy is $|E_{2s}| = 5.75$ eV, close to the experimental 5.39 eV. The discrepancy reflects the crudeness of the Slater screening constants.

(c) The $2s$ electron has significant radial extent beyond the $1s$ core, so it sees a nearly bare nuclear charge at small $r$ but is screened by both $1s$ electrons at large $r$ . The effective charge $Z_{\text{eff} \approx 1.26}$ (using Hartree—Fock) represents this average screening.

Problem 20: Partial Wave Analysis for Square Well

Consider scattering from the attractive square well $V(r) = -V_0$ for $r < a$ and $V(r) = 0$ for $r > a$ .

(a) Show that the $s$ -wave phase shift satisfies:

\delta_0 = -ka + \arctan\!\left(\frac{k}{\kappa}\tan(\kappa a)\right)

Where $\kappa = \sqrt{2m(V_0 + E)}/\hbar$ and $k = \sqrt{2mE}/\hbar$ .

(b) Show that a bound state exists at energy $E = -|E|$ when $\kappa_0 a = \pi/2$ where $\kappa_0 = \sqrt{2m(V_0 - |E|)}/\hbar$ .

Solution:

(a) Inside the well ( $r < a$ ), the radial wavefunction for $l = 0$ is $u(r) = A\sin(\kappa r)$ . Outside ( $r > a$ ), $u(r) = B\sin(kr + \delta_0)$ .

Matching $u$ and $u'$ at $r = a$ :

\kappa\cos(\kappa a) = k\cos(ka + \delta_0)/\sin(ka + \delta_0)\cdot k

Wait: $\kappa\cot(\kappa a) = k\cot(ka + \delta_0)$ .

\cot(ka + \delta_0) = \frac{\kappa}{k}\cot(\kappa a)

ka + \delta_0 = \arccot\!\left(\frac{\kappa}{k}\cot(\kappa a)\right) = \arctan\!\left(\frac{k}{\kappa}\tan(\kappa a)\right)

\delta_0 = -ka + \arctan\!\left(\frac{k}{\kappa}\tan(\kappa a)\right)

(b) A bound state has $E < 0$ So $k = i\kappa'$ where $\kappa' = \sqrt{2m|E|}/\hbar$ . The bound state condition is that the exterior solution decays exponentially: $u(r) = Be^{-\kappa' r}$ . Matching:

\kappa\cot(\kappa a) = -\kappa'

As $|E| \to 0$ : $\kappa' \to 0$ So $\kappa\cot(\kappa a) \to 0$ Giving $\kappa a = \pi/2$ (the threshold for the first bound state).

(c) The scattering length $a_s = -\lim_{k \to 0}(\delta_0/k)$ . As $\kappa a \to \pi/2$ , $\tan(\kappa a) \to \infty$ So:

\delta_0 \approx -ka + \arctan(\infty) = -ka + \pi/2

a_s = -\frac{1}{k}\left(-ka + \frac{\pi}{2}\right) = a - \frac{\pi}{2k}

As $k \to 0$ : $a_s \to \pm\infty$ (diverges), changing sign as the bound state appears.

Intuition

Scattering theory extracts information about forces by firing particles at targets and analyzing how they deflect. The differential cross-section tells you the probability of scattering into a given solid angle, and it is directly measurable. For a central potential, the scattering amplitude is decomposed into partial waves, each corresponding to a specific angular momentum. The scattering length characterizes low-energy scattering and diverges when a new bound state appears, signaling a resonance. The optical theorem connects the total cross-section to the forward scattering amplitude, reflecting the fact that particles removed from the forward beam must go somewhere. Partial wave analysis turns a complex three-dimensional scattering problem into a sum of one-dimensional problems.

Counterexamples worth knowing by name

Claim	True?	Counterexample
The quantum cross-section of a hard sphere matches the classical one	no	low-energy $s$ -wave scattering gives $\sigma = 4\pi a^2$ , four times the classical $\pi a^2$ , because the wave diffracts around the sphere
A resonance is a bound state	no	it is a quasi-bound state: metastable, with a finite lifetime, appearing as a pole of the $S$ -matrix on the unphysical sheet rather than on the real axis
The Born approximation needs a weak potential	incomplete	it needs $\lvert V\rvert$ small compared with the kinetic energy and the potential to vary slowly on the scale of the wavelength
Higher partial waves matter at low energy	no	the centrifugal barrier suppresses them, and only the $l = 0$ wave contributes; that is why low-energy cross-sections are isotropic
Phase shifts are small for weak scattering	not necessarily	a resonance makes $\delta_l$ swing through $\pi$ rapidly even for a weak potential
Cross-sections cannot exceed the geometric size	no	the hard-sphere result above is larger, and a resonance can exceed it by orders of magnitude
Scattering conserves probability, so $S$ is unitary	yes	and this is the constraint that makes the phase shifts real for real potentials

The first row is the one that most strongly marks the difference between classical and quantum intuition. A classical ball bearing thrown at a basketball is shadowed by it; a long-wavelength quantum particle is diffracted by it, and the forward diffraction adds exactly the extra $3\pi a^2$ .

The third row is the one that catches people applying the Born approximation to a Coulomb field, where it fails not because the potential is strong but because it is long-ranged, and the amplitude has to be treated separately.

Where the research frontier is

The inverse scattering problem. Given the scattering data, can the potential be reconstructed? In one dimension the answer is yes, exactly, via the Gel’fand-Levitan equation — and that result is the mathematical foundation of soliton theory.
Few-body scattering. Three bodies do not reduce to pairwise problems, and the Faddeev equations are the framework; Efimov states, where three bodies bind even though no pair does, are the celebrated prediction.
Scattering in quantum field theory. The $S$ -matrix is the central object and its analyticity properties are what make dispersion relations possible; the modern amplitudes programme tries to compute it without a Hamiltonian at all.
Cold atoms and Feshbach resonances. A magnetic field tunes a resonance through threshold, letting experimentalists dial the interaction strength between two atoms from zero to infinity. That control is what makes precision many-body physics possible.
Scattering and quantum chaos. How chaotic classical dynamics shows up in scattering data — through the statistics of resonance spacings — connects to random matrix theory.

Interleaving problems

Mix these with one-dimensional problems, angular momentum, approximation methods, Symplectic Structure and KAM Theory and Ordinary Differential Equations.

(Angular momentum.) Derive the partial wave expansion by expanding a plane wave in spherical harmonics and identifying each $l$ term as an incoming and an outgoing wave. Show that only the phase of each can change, and that this is why one number per $l$ suffices.
(Operators.) Prove that the $S$ -matrix is unitary for a real potential, and explain why unitarity is what makes the phase shifts real. What goes wrong with absorption?
(One-dimensional problems.) Show that the 1D transmission and reflection coefficients of the barrier problem are the $l = 0$ partial-wave result in disguise, and identify what the higher $l$ waves add in three dimensions.
(Approximation methods.) Derive the Born approximation as first-order perturbation theory in $\hat{V}$ , and show that it is the Fourier transform of the potential evaluated at the momentum transfer. Then explain why that makes electron diffraction a way of photographing potentials.
(Classical mechanics.) Derive the Rutherford cross-section classically from the Coulomb force, and show that quantum mechanics gives the same answer. That agreement is unusual — identify what makes it work.
(Research.) State the Gel’fand-Levitan equation and explain in what sense it solves the inverse problem exactly. Why does the reconstruction need the bound state data as well as the scattering data?

Cross-References

Wave Functions and the Schrodinger Equation: The Schrodinger equation provides the foundation for scattering theory and partial wave analysis.
Angular Momentum and the Hydrogen Atom: Angular momentum theory provides the partial wave expansion for scattering amplitudes.
Time-Dependent Perturbation Theory: Fermi’s Golden Rule from time-dependent perturbation theory underlies transition rates in scattering.
Calculus
Linear Algebra
Vector Calculus
Quantum Computing

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