Delta Potential Born Approximation

In the Born approximation, the scattering amplitude is

$f_{\text{Born}}(\theta )=-{\frac {m}{2\pi \hbar ^{2}}}\int d^{3}\mathbf {r'} \,V(\mathbf {r'} )e^{-i\mathbf {q} \cdot \mathbf {r} '},$

where $\mathbf {q} =\mathbf {k} '-\mathbf {k}$ and $\mathbf {k}$ and $\mathbf {k} '$ are the wave vectors of the incident and scattered waves, respectively. Substituting in the delta function potential, we get

$f_{B}(\theta )=-{\frac {m}{2\pi \hbar ^{2}}}\int d^{3}\mathbf {r'} \,g\delta ^{3}(\mathbf {r} ')e^{-i\mathbf {q} \cdot \mathbf {r} '}=-{\frac {mg}{2\pi \hbar ^{2}}},$

and therefore the differential cross section is

$\sigma (\theta )=|f_{\text{Born}}(\theta )|^{2}={\frac {m^{2}g^{2}}{4\pi ^{2}\hbar ^{4}}}.$

As the distribution is isotropic, the total cross section is

$\sigma _{t}=4\pi \sigma ={\frac {m^{2}g^{2}}{\pi \hbar ^{4}}}$.

Back to Differential Cross Section and the Green's Function Formulation of Scattering