[QUE/CM-06004]

 A central force potential frequently encountered in nuclear physics  is the so called rectangular well, defined by the potential    $$ V(r) = \left\{ \begin{array}{cr} 0 & r\ge a \qquad \\ -V_0 & r\le a  \end{array} \right.$$
      Show that the scattering produced by such a potential in classical  mechanics is identical with the refraction of light rays by a sphere  of  radius $a$ and relative index of refraction  $$ n= \sqrt{E+V_0 \over E } $$  Show that the differential cross section is  $$ \sigma(\Theta) ={ n^2 a^2 \over 4\cos^2} {\Theta \over 2} \times  { \left(n \cos{\Theta \over 2}-1 \right)  \left(n - \cos{\Theta \over 2}  \right)\over  \left( 1+n^2 -2n \cos{\Theta \over 2} \right)^2}$$

Goldstein