 Solution

The differential scattering cross-section is scattered rate per unit solid angle divided by incident flux. For the outgoing scattering solution, the incident plane wave has current magnitude $\hslash k/m$, while the outgoing spherical wave has radial current $(\hslash k/m)|f(\hat{x}){|}^2/r^2$. ThereforeThe incoming plane-wave normalization and outgoing radiation condition select the physical scattering state.

The time-independent Schrödinger equation is $[-{\hslash }^2{\nabla }^2/(2m)+V]\psi =E\psi$, equivalently $({\nabla }^2+k^2)\psi =(2m/{\hslash }^2)V\psi$. The given outgoing Helmholtz fundamental solution has source $-4\pi \delta$. Multiplying it by the source and adding the incident homogeneous solution givesApplying ${\nabla }^2+k^2$ verifies its equation and sign. At large $r$, $|x-x^{\prime }|=r-\hat{x}\cdot x^{\prime }+O(r^{-1})$ and $|x-x^{\prime }{|}^{-1}=r^{-1}+O(r^{-2})$ over the potential's support. ConsequentlyThe Born approximation replaces $\psi (x^{\prime })$ by the incident wave, givingIt requires weak wave distortion. A sufficient smallness condition from the integral equation is $(m/2\pi {\hslash }^2){\sup }_x\int |V(x^{\prime })|/|x-x^{\prime }|d^3{x}^{\prime }\ll 1$; for a bounded potential of range $R$ this is of order $m|V_0|R^2/{\hslash }^2\ll 1$. At high energy a small accumulated phase and weak deflection can also justify the approximation, but small potential relative to energy alone is not sufficient for an arbitrarily long interaction range.

For the exponentially screened potential, angular integration in its Fourier transform givesIt is not compactly supported, but for $\mu >0$ its decay gives the same scattering asymptotics and convergent integrals. Since $q=2k\sin (\theta /2)$,Thus the Born amplitude in this limit is independent of $\hslash$ when expressed using $E$ and angle; its squared magnitude is the Rutherford cross-section. This is a fixed-nonzero-angle limit of the Born formula. It does not establish uniform validity in the forward direction or preserve the short-range outgoing-wave asymptotics for the unscreened Coulomb potential.