the Born approximation
/ born /
When you fire a particle at a target and watch how it deflects, you are doing scattering, and you want the scattering amplitude -- the quantity whose square tells you the probability of deflection into each angle. The Born approximation is the simplest useful answer: if the scattering potential is weak, the outgoing wave is essentially the incoming wave scattered just once, and the amplitude becomes a single, computable integral over the potential. It is the entry point to all of scattering theory.
The physical picture is single scattering. The incident plane wave passes through the potential largely undisturbed, and each point of the potential launches a small spherical wavelet; add them up, ignoring the tiny chance of scattering twice, and you get the first Born approximation for the scattering amplitude: f(theta) is approximately -(m/(2 pi hbar^2)) times the integral of exp(i q dot r) V(r) d^3r, where q = k_out - k_in is the momentum transfer (with magnitude q = 2k sin(theta/2)). The punchline is beautiful: the scattering amplitude is, up to constants, the Fourier transform of the potential, with the momentum transfer q as the conjugate variable. Scatter at large angles (large q) and you probe the potential's fine, short-distance structure; scatter at small angles and you see its gross, long-range shape.
This Fourier-transform relationship is why scattering is the primary tool for seeing structure too small to image directly. X-ray and neutron diffraction read crystal structure off the scattering pattern; electron scattering off nuclei revealed their finite size and charge distribution; and Rutherford's famous alpha-scattering formula for a Coulomb potential drops straight out of the Born approximation. The honest limitation is right there in the name: it is the first term of the Born series, valid only when the potential is weak or the energy high enough that a single scattering dominates -- for a strong or resonant potential you need higher Born terms or a full partial-wave treatment.
Applied to a screened Coulomb (Yukawa) potential V(r) = (A/r) exp(-r/a), the Born integral gives f(theta) proportional to 1/(q^2 + 1/a^2). Letting the screening length a go to infinity recovers the pure Coulomb case, and |f|^2 then reproduces exactly the Rutherford scattering cross-section proportional to 1/sin^4(theta/2) -- the quantum calculation matching Rutherford's classical result.
The Born approximation for a screened Coulomb potential reproduces Rutherford's 1/sin^4(theta/2) cross-section.
The first Born approximation assumes a single, weak scattering event; it fails for strong potentials, low energies, and near resonances, where multiple scattering matters and you need higher Born terms or partial waves. That it reproduces the exact Rutherford result is a happy coincidence of the 1/r potential, not a sign the approximation is generally exact.