Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 35E (Applications of Quantum Mechanics)

Insert the incident plane wave for $\psi$ in the integral. At large $r$, $|\mathbf{r}-{\mathbf{r}}^{\prime }|\simeq r-\hat{\mathbf{r}}\cdot {\mathbf{r}}^{\prime }$, so the outgoing amplitude isThus Born approximation gives the stated differential cross section.

For $V(r)\sim r^{-n}$, the small-$q$ transform scales as $q^{n-3}$. Since $d\Omega$ contributes $qdq$ near forward scattering, the total Born cross section behaves as ${\int }_0{q}^{2n-5}dq$, finite exactly for $n>2$. Coulomb decay $n=1$ gives the familiar divergent forward Rutherford cross section and requires long-range treatment.

The Gaussian transform isThereforeA sufficient weak-scattering regime is kinetic energy much larger than $|V_0|$ (more intrinsically $m|V_0|r_0^2/{\hslash }^2\ll 1$).