Quantum Mechanics/Quantum Scattering

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

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

(1)

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

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

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

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

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

(2)

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

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

(3)