Consider two particles whose interaction is governed by the following rectangular-well potential:
$$
\begin{array}{lll}
U(r)=0 & \text { for } & r>a \\
U(r)=-U_0 & \text { for } & r \leq a
\end{array}
$$
(a) Calculate the differential scattering cross section $\sigma(\chi)$ and show that it is given by (considering $b<a$ )
$$
\sigma(\chi)=\frac{p^2 a^2[p \cos (\chi / 2)-1][p-\cos (\chi / 2)]}{4 \cos (\chi / 2)\left[1-2 p \cos (\chi / 2)+p^2\right]^2}
$$
where
$$
p=\left(1+\frac{2 U_0}{\mu g^2}\right)^{1 / 2}
$$
(b) Show that the total scattering cross section is given by
$$
\sigma_t=2 \pi \int_0^a b d b=\pi a^2
$$