[QUE/QFT-15011] Rutherford Scattering in Second quantized Schrodinger equation

 Write the Lagrangian for a nonrelativistic particle moving in  a potential  \(V(r)\). Using the second quantized formalism compute the scattering cross  section in the lowest order in the potential. Show that this result is same as the first Born approximation expression in nonrelativistic, first quantized formulation of quantum mechanics. Taking V(r) as Coulomb potential obtain the Rutherford formula:\[\left(\frac{d\sigma}{d\Omega}\right)_\text{R} = \frac{Ze^2}{64\pi m^2v^4 \sin^4(\theta/2)}\]