Quantum Mechanics/Quantum Scattering

Stationary scattering wave

The (elastic) scattering stationary state (azimuthally symmetric) is described by a wave-function with the following asymptotic,

$\psi ({\vec {r}}){\underset {r\to \infty }{\longrightarrow }}e^{i{\vec {k}}{\vec {r}}}+f(\theta ){\frac {e^{ikr}}{r}}\;,$

(1)

where $e^{i{\vec {k}}{\vec {r}}}$ is the incident plane wave of projectiles with momentum ${\vec {k}}$, $f(\theta ){\frac {e^{ikr}}{r}}$ is the scattered spherical wave, and $f(\theta )$ is the scattering amplitude.

Cross-section

Consider a detector with the window $d\Omega$ 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,

${\frac {dN}{dt}}=j_{r}r^{2}d\Omega \,.$

The radial flux from the stationary wave (1) is given as

$j_{r}={\frac {\hbar }{2mi}}\left(\psi ^{*}{\frac {\partial \psi }{\partial r}}-{\frac {\partial \psi ^{*}}{\partial r}}\psi \right)=|f(\theta )|^{2}{\frac {1}{r^{2}}}{\frac {\hbar k}{m}}\,.$

(2)

The cross-section $d\sigma$ is defined as the count rate of the detector divided by the flux density of the incident beam,

$d\sigma ={\frac {1}{|j_{i}|}}{\frac {dN}{dt}}\,.$

(3)