Quantum Mechanics/Quantum Scattering

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

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 \,. $$ The radial flux from the stationary wave ($$) is given as

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