Quantum Mechanics Course

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← Part IV/3D Scattering Theory

7. 3D Scattering Theory

Reading time: ~35 minutes | Pages: 9

Theory of particle collisions and scattering from potentials in three dimensions.

Scattering Setup

Incident plane wave + outgoing spherical wave:

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

$f(\theta,\phi)$ is the scattering amplitude

Differential Cross Section

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

Probability per unit solid angle of scattering into direction $(\theta,\phi)$

Total Cross Section

$$\sigma_{\text{tot}} = \int\frac{d\sigma}{d\Omega}d\Omega = \int_0^{2\pi}\int_0^\pi |f(\theta,\phi)|^2\sin\theta\,d\theta\,d\phi$$

Partial Wave Expansion

For spherically symmetric potential $V(r)$:

$$f(\theta) = \sum_{\ell=0}^\infty (2\ell+1)f_\ell P_\ell(\cos\theta)$$

where $f_\ell$ is the partial wave amplitude for angular momentum $\ell$

Phase Shifts

Asymptotic radial wave function:

$$R_\ell(r) \xrightarrow{r\to\infty} \frac{1}{kr}\sin(kr - \ell\pi/2 + \delta_\ell)$$

$\delta_\ell$ is the phase shift produced by the potential

Partial wave amplitude:

$$f_\ell = \frac{1}{k}\frac{e^{2i\delta_\ell} - 1}{2i} = \frac{e^{i\delta_\ell}\sin\delta_\ell}{k}$$

Optical Theorem

Relates forward scattering to total cross section:

$$\sigma_{\text{tot}} = \frac{4\pi}{k}\text{Im}[f(0)]$$

Consequence of probability conservation

Partial Wave Cross Sections

$$\sigma_\ell = \frac{4\pi}{k^2}(2\ell+1)\sin^2\delta_\ell$$

Total cross section:

$$\sigma_{\text{tot}} = \sum_{\ell=0}^\infty\sigma_\ell = \frac{4\pi}{k^2}\sum_{\ell=0}^\infty(2\ell+1)\sin^2\delta_\ell$$

Low Energy (s-wave) Scattering

For $ka \ll 1$ (long wavelength), only $\ell = 0$ contributes:

$$f \approx f_0 = -a_s$$

$a_s$ is the scattering length

Cross section:

$$\sigma = 4\pi a_s^2$$

Resonance Scattering

When $\delta_\ell \approx \pi/2$, resonance occurs:

$$\sigma_\ell \approx \frac{4\pi}{k^2}(2\ell+1)$$

Maximum possible cross section for partial wave $\ell$

Occurs near quasi-bound states of the potential

Born Approximation

First-order perturbation theory:

$$f(\theta) = -\frac{m}{2\pi\hbar^2}\int V(\vec{r})e^{i\vec{q}\cdot\vec{r}}d^3r$$

where $\vec{q} = \vec{k}_f - \vec{k}_i$ is momentum transfer, $|\vec{q}| = 2k\sin(\theta/2)$

Valid when $|V| \ll E$

Rutherford Scattering

Coulomb potential $V(r) = k_e/r$:

$$\frac{d\sigma}{d\Omega} = \left(\frac{k_e}{2E}\right)^2\frac{1}{\sin^4(\theta/2)}$$

Same as classical result! Forward scattering diverges

Applications

  • Nuclear physics: Proton-proton, neutron-nucleus scattering
  • Atomic physics: Electron-atom collisions
  • Particle physics: High-energy scattering experiments
  • Condensed matter: Electron-phonon, electron-impurity scattering
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