Quantum Mechanics · Lesson 10

Chapter practice

Bring the chapter together: cross sections, the scattering amplitude, the integral equation and the Born approximation, Coulomb scattering, partial waves and phase shifts, the optical theorem, and scattering at low energy with its resonances. No worked example sits above the answers.

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\frac{d σ}{d Ω} = ∣ f (θ, ϕ) ∣^{2}

ψ \sim e^{ik z} + f (θ) \frac{e ^{ik r}}{r}

S = I + i T

σ = \int \frac{d σ}{d Ω} d Ω

Chapter navigation

Complete Low-energy scattering and resonances first.

Intuition

Bring the chapter together: cross sections, the scattering amplitude, the integral equation and the Born approximation, Coulomb scattering, partial waves and phase shifts, the optical theorem, and scattering at low energy with its resonances. No worked example sits above the answers.

A bat maps a dark cave from the echoes of its calls, never touching the walls. Scattering maps the inside of atoms and nuclei the same way, from what comes back.

Before you practise

$d\sigma/d\Omega=|f|^{2}$ ; $f_{B}=-\frac{m}{2\pi\hbar^{2}}\int e^{-i\mathbf{q}\cdot\mathbf{r}}V\,d^{3}r$ with $q=2k\sin\frac{\theta}{2}$ ; $f=\frac1k\sum(2\ell+1)e^{i\delta_{\ell}}\sin\delta_{\ell}P_{\ell}$ .

\sigma=\frac{4\pi}{k^{2}}\sum_{\ell}(2\ell+1)\sin^{2}\delta_{\ell}=\frac{4\pi}{k}\operatorname{Im}f(0)

Reminders

Rutherford: $(\beta/4E)^{2}/\sin^{4}(\theta/2)$ .
Low energy: $\sigma\to4\pi a^{2}$ ; a hard sphere has $a=R$ .
Unitarity: $\sigma_{\ell}\le4\pi(2\ell+1)/k^{2}$ .

Applications

Practice

Try it

The $s$ -wave phase shift is $90^{\circ}$ at $k=0.5$ fm $^{-1}$ , and the other partial waves do not scatter. What is $\sigma$ , in fm²? Give two decimal places.