Born approximation

The Born approximation is a way to estimate how a particle scatters off a target when the interaction between them is weak. It treats the scattering as a single, gentle nudge: the incoming wave passes through the target's potential, gets slightly deflected once, and travels on. The strong, repeated re-scattering that a strong potential would cause is simply neglected, which is exactly why the approximation is restricted to weak interactions.

Mechanically, it is the first term of a perturbation series for scattering. The pattern of where particles are deflected — the differential cross-section, telling us how likely each scattering angle is — comes out as essentially the strength of the potential measured at the particular change in momentum the particle undergoes. In clean language, the scattering pattern is a kind of transform of the potential's shape, so measuring how particles scatter at different angles maps out the structure of the target.

This makes the Born approximation a workhorse of scattering physics, from electrons probing atoms to neutrons mapping crystals to the analysis of particle collisions. It is honest about its domain: it holds when the scattering is weak — fast particles or feeble potentials — and it grows unreliable for slow particles, strong potentials, or resonances, where multiple scattering and the higher terms of the series can no longer be ignored.