Delta Potential Born Approximation

Problem

Calculate the Born approximation to the differential and total cross sections for a particle of mass m off the $\delta$-function potential $V(\mathbf {r} )=g\delta ^{3}(\mathbf {r} )$.

Solution:

In Born approximation,

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

where $\mathbf {q} =\mathbf {k} '-\mathbf {k}$ with $\mathbf {k}$ and $\mathbf {k} '$ are the wave vectors of the incident and scattered waves, respectively. Then

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

and the differential cross section is

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

As the distribution is isotropic, the total cross section is

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