Scattering length and effective range of microscopic two-body potentials

Scattering processes are a fundamental way of experimentally probing distributions and properties systems in several areas physics. Considering two-body scattering at low energies, when the de Broglie wavelength is larger than range potential, partial waves with high angular momentum typically unimportant. The dominant contribution comes from $l=0$ waves, commonly known as $s$-wave scattering. This situation very relevant atomic physics, e.g. cold gases, nuclear structure matter. manuscript intended pedagogical introduction to topic while covering numerical approach compute desired quantities. We introduce low-energy particular attention concepts length effective range. These two quantities appear effective-range approximation, which universally describes processes. outline procedure for calculating spherically symmetric potentials. As examples, we apply method spherical well, modified P\"oschl-Teller, Gaussian, Lennard-Jones hope provide tools so students can implement similar calculations extend them other