Up to now we have mainly studied bound states, except for the brief mention of one-dimensional scattering in Section 9.4. However, essential information on interactions between particles, atoms, molecules, etc., as well as on the structure of composite objects, can be obtained from scattering experiments. Bound states – when they exist, which is not always the case – give only partial information on such interactions, whereas it is nearly always possible to perform scattering experiments. In this chapter we shall limit ourselves to potential scattering, which can be used to describe elastic collisions of two particles of masses m1 and m2. Indeed, in the center-of-mass frame the problem is reduced to that of a particle of mass m = (m1m2)/(m1 + m2) in a potential (Exercise 8.5.6).

In Sections 12.1 and 12.2 we develop the elementary formalism of elastic scattering theory with emphasis on the low-energy limit, which plays an extremely important role in practice. In Section 12.3 we generalize the formalism to the inelastic case; more precisely, we examine the effect of inelastic channels on elastic scattering. Finally, Section 12.4 is devoted to some more formal aspects of scattering theory.

The cross section and scattering amplitude

The differential and total cross sections

A scattering experiment is shown schematically in Fig. 12.1. A beam of particles of mass m1 and well-defined momentum moving along the z axis collides with a target composed of particles of mass m2. To simplify the discussion, we assume that m1 ≪ m2 and we neglect the recoil of the target in the collision. In general, it is necessary to go from the laboratory frame to the center-of-mass frame via a simple kinematic transformation (Exercise 8.5.6).