Born approximation and Fourier potentials

Take the incident wavevector $\mathbf k$ and outgoing wavevector $\mathbf k'$ to have equal magnitude in elastic scattering. Define the transferred wavevector as outgoing minus incoming. It has inverse-length units; the transferred momentum is reduced Planck constant times that vector. Choose a Fourier transform with a negative exponential and no normalization factors inside its spatial integral. A different transform convention can move numerical factors, so it must be stated before comparing amplitudes from different references.

The scattered field is generated by the potential multiplied by the exact wave in the interaction region. At first order replace that exact wave by the incident plane wave, and use the far-field outgoing Green function. The incident phase and outgoing observation phase combine into the Fourier exponential. The resulting minus sign and mass factor follow from the Schrödinger Green function, not from whether the potential is attractive or repulsive. For a real central potential the resulting first-order amplitude is real.

For a central potential, the angular part of the Fourier integral can be performed by aligning its polar axis with the transfer vector. The remaining radial integral contains a sine divided by its argument. Small scattering angles correspond to small transfer, which probes the spatially integrated potential. Large transfer produces stronger cancellation among separated regions of a smooth potential. Consequently a smooth long-range but screened interaction often concentrates scattering toward forward angles without making the scattered state a classical trajectory.

A screened Coulomb or Yukawa potential supplies a soluble example. Its radial transform is a rational function of the squared transfer and inverse screening length. The screening scale regularizes the forward amplitude. Sending the screening length to infinity is not an innocuous way to recover every short-range assumption: the Coulomb limit changes the asymptotic wave and creates a forward singularity. Although its Born intensity has the Rutherford form, that special agreement does not prove the unmodified short-range wavefunction or perturbation argument is valid.

The approximation is controlled when the rescattered correction is small within the interaction region, not merely when the far-field amplitude is small at a detector angle. For a bounded finite-range potential of characteristic strength and range, a small ratio of potential energy to confinement-scale kinetic energy is a useful sufficient low-energy estimate. At high incident energy, small accumulated scattering phase can support a different regime. Neither estimate is a universal necessary-and-sufficient criterion for arbitrary singular or resonant potentials.

Near a bound-state threshold or a narrow resonance, repeated encounters can strongly distort the internal wave even when a naive coupling comparison looks moderate. The first Born term then fails to capture the large scattering length or the unitary resonant limit. Its apparent optical-theorem mismatch also requires order-consistent interpretation rather than arbitrary repair. The schematic scattering lab uses a fourth power of its coupling control; a physical first Born amplitude is linear in a potential-strength parameter and its leading cross section is quadratic, so those are deliberately different scaling laws.