Introduction to Quantum Mechanics · Problem 10.15
Scattering — David J. Griffiths
Physics — Quantum Mechanics · second-born-approximation-soft-sphere
Problem
Find the scattering amplitude for low-energy soft-sphere scattering in the second Born approximation. Answer:Intuitive explanation
In low-energy scattering, only the -wave (zero angular momentum) contributes, so you can treat the potential as a simple spherical obstacle. The Born approximation expands the scattering amplitude as a perturbative series in the potential strength; the second-order term captures how the particle can scatter twice inside or near the sphere. The key is computing the forward-scattering amplitude from the potential form factor and then adding the second-order correction, which involves an integral over intermediate momentum transfers—this correction is typically smaller and proportional to the potential strength squared.
Intuitive explanation
In low-energy scattering, only the -wave (zero angular momentum) contributes, so you can treat the potential as a simple spherical obstacle. The Born approximation expands the scattering amplitude as a perturbative series in the potential strength; the second-order term captures how the particle can scatter twice inside or near the sphere. The key is computing the forward-scattering amplitude from the potential form factor and then adding the second-order correction, which involves an integral over intermediate momentum transfers—this correction is typically smaller and proportional to the potential strength squared.