Past exam of the theoretical physics II course of the University of Cambridge / 2019 / 2 / c

The radial S wave vanishes at the hard surface, so outside the sphere $u_0(r)=A\sin [k(r-R)]$. Comparison with $A\sin (kr+{\delta }_0)$ givesThe hard-sphere S-wave scattering amplitude and cross section are thereforeThe three prominent features are the finite low-energy limit ${\sigma }_0\to 4\pi R^2$, the oscillations with exact zeros at $x=n\pi$ for positive integers $n$, and the decreasing envelope ${\sigma }_0\le 4\pi /k^2$. The zero-energy value is four times the geometric area $\pi R^2$. Near zero the normalized cross section is $1-x^2/3+O(x^4)$. The large-$k$ fall here concerns the S wave alone; summing all partial waves gives a different high-energy total.