According to the Born approximation, if a scattering potential V(r) is sharply peaked and localized in a tiny region of space, what angular distribution of scattered particles is predicted?
ANearly all scattering occurs at small forward angles, because the potential is weak
BNo scattering occurs, because a very small potential has negligible effect
CScattering is relatively isotropic, because a sharply localized potential has significant Fourier components at large momentum transfer q
DScattering is concentrated at exactly 90°, by symmetry of the spherical potential
The Born approximation gives f(θ) ∝ ∫ e^{iq·r'} V(r') d³r' — the Fourier transform of V evaluated at momentum transfer q = k_f − k_i, with magnitude q = 2k sin(θ/2). A sharply localized potential (like a delta function) has a flat Fourier transform — all spatial frequencies (all q values) are equally present. Large q corresponds to large scattering angles, so a sharply peaked potential scatters in all directions roughly equally. Contrast this with the Coulomb potential, which is long-range and has a Fourier transform concentrated at small q, producing predominantly forward scattering.
Question 2 Multiple Choice
The Born approximation formula f(θ) ≈ −(m/2πℏ²) ∫ e^{iq·r'} V(r') d³r' reveals a deep structural connection between scattering and another mathematical operation. What is it?
AThe scattering amplitude is the Laplace transform of the potential, evaluated at the imaginary frequency corresponding to energy
BThe scattering amplitude is proportional to the Fourier transform of the potential, evaluated at the momentum transfer vector q = k_f − k_i
CThe scattering amplitude is the convolution of the potential with the incoming plane wave
DThe scattering amplitude equals the matrix element of V in the energy eigenbasis
The integral ∫ e^{iq·r'} V(r') d³r' is exactly the three-dimensional Fourier transform of V(r) evaluated at wavevector q. This is the defining structure of the Born approximation and has deep physical content: the scattering pattern encodes the 'spatial frequency content' of the potential. Long-range potentials have Fourier transforms peaked at small q (small scattering angles); short-range potentials have significant components at large q (wide-angle scattering). The same Fourier connection appears in optical diffraction, where the far-field pattern is the Fourier transform of the aperture — scattering and diffraction are mathematically the same phenomenon.
Question 3 True / False
The Born approximation is most accurate when the scattering potential is strong and the incident particle energy is low.
TTrue
FFalse
Answer: False
The Born approximation is a first-order perturbation theory in the potential V: it treats the scattered wave as a small correction to the incident plane wave. This is valid when the perturbation is small — either because |V| is intrinsically weak relative to the kinetic energy, or because the incident energy ℏ²k²/2m is large (making the kinetic energy dominate over the potential). Strong potentials or low energies lead to multiple scattering events, where the particle bounces off the potential many times. These higher-order contributions are neglected in Born, causing the approximation to fail. The criterion is roughly |V_typical| ≪ ℏ²k²/2m.
Question 4 True / False
For the Coulomb potential V(r) = Ze²/r, the Born approximation gives the same differential cross section as Rutherford's classical calculation.
TTrue
FFalse
Answer: True
The Fourier transform of the Coulomb potential V(r) = Ze²/r is proportional to 1/q², and with q = 2k sin(θ/2), this gives the differential cross section dσ/dΩ ∝ 1/sin⁴(θ/2) — exactly Rutherford's formula. This agreement between a first-order quantum calculation and the classical result (for the exact same potential) is not accidental: for the pure Coulomb potential, all higher-order Born terms vanish due to the special properties of 1/r. The Rutherford formula's experimental success in 1911 was one of the first triumphs of quantum scattering theory — even though Rutherford derived it classically.
Question 5 Short Answer
Why does the Born approximation have the mathematical form of a Fourier transform, and what physical insight does this provide about the relationship between a potential's spatial structure and its scattering pattern?
Think about your answer, then reveal below.
Model answer: In the Born approximation, the incoming particle travels as a plane wave e^{ik_i·r} and barely deviates. At each point r' in the potential, the interaction re-radiates a small spherical wave weighted by V(r'). Each re-radiated contribution carries a phase factor e^{iq·r'} recording the path-length difference between the incoming wave and the outgoing wave in direction k_f, where q = k_f − k_i is the momentum transfer. The total scattered amplitude is the coherent sum of all these contributions — an integral of the form ∫ e^{iq·r'} V(r') d³r', which is exactly the Fourier transform of V evaluated at q. The physical insight: the scattering pattern is a 'Fourier portrait' of the potential. Long-range potentials have Fourier transforms concentrated at small q (scatter mainly forward); sharply localized potentials have flat Fourier transforms (scatter isotropically). This is the quantum analog of optical Fraunhofer diffraction.