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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d\sigma}{d\Omega}=|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

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

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