Quantum Mechanics/Quantum Scattering

Stationary scattering wave

The (elastic) scattering stationary state (azimuthally symmetric) is described by a wave-function with the following asymptotic, Template:NumBlk where $e^{i\overrightarrow{k}\overrightarrow{r}}$ is the incident plane wave of projectiles with momentum $\overrightarrow{k}$, $f(\theta )\frac{e^{ikr}}{r}$ is the scattered spherical wave, and $f(\theta )$ is the scattering amplitude.

Cross-section

Consider a detector with the window $d\mathrm{\Omega }$ positioned at the angle $\theta$ at the distance $r$ from the scattering center. The count rate of the detector, $dN/dt$, is given by the radial flux density of particles, $j_r$ through the detector window,

$\frac{dN}{dt}=j_r{r}^{2}d\mathrm{\Omega }.$

The radial flux from the stationary wave (Template:EquationNote) is given as Template:NumBlk The cross-section $d\sigma$ is defined as the count rate of the detector divided by the flux density of the incident beam, Template:NumBlk