12.7: Example of Born Approximation

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Consider scattering of particles interacting via a 3D square well potential: \(V (r < a) = V_0; V (r > a) = 0\).

The integral required here is then (with \(\chi = 2k \sin \frac{\theta}{2}\)):

\[ \int_{0}^{a} r V_{0} \sin (\chi r) d r = \left[\frac{\sin (\chi r)-\chi r \cos (\chi r)}{\chi^{2}} \right]_{0}^{a} \nonumber\]

whence:

\[\frac{d \sigma}{d \Omega} = \left[\frac{2 \mu V_{0}}{\chi \hbar^{2}} \right]^{2} \left[ \frac{\sin (\chi a)-\chi a \cos (\chi a)} {\chi^{2}} \right]^{2} \nonumber\]

Using a Maclaurin expansion, the low energy limit is:

\[\frac{d \sigma}{d \Omega} = \left[\frac{2 \mu V_{0}}{\chi \hbar^{2}}\right]^{2} \frac{1}{9}\left[1-\frac{1}{5} \chi^{2} a^{2}\right] \nonumber\]

From integrating over \(\theta\) and \(\phi\) the low and high energy limits for the total cross section are

\[\sigma(E \rightarrow \infty) = 2 \pi \left[\frac{\mu}{\hbar^{2}}\right]^{2} \left[\frac{V_{0} a^{3}}{k a} \right]^{2} \\ \sigma(E \rightarrow 0) = 2 \pi \left[\frac{\mu}{\hbar^{2}} \right]^{2} \left[\frac{V_{0} a^{3}}{k a} \right]^{2} \frac{8}{9} \left(k^{2} a^{2}-\frac{2}{5} k^{4} a^{4}+\ldots \right) \nonumber\]