The three-body and, more generally, the few-body problem sometimes has the bad reputation of being a jungle where non-experts are quickly discouraged, whereas specialists enjoy endless debates on technical improvements concerning scattering or bound-state equations, with much attention paid to the computational consequences and very little concern about the underlying physics. This is, of course, a wrong impression. To give just one example, many aspects of the nuclear forces have been discovered from the few-nucleon systems such as tritium or helium, and one of the most striking challenges in strong interaction physics remains the understanding of the three-nucleon binding energy and form factor [1].
In the present review, I shall try to convince the reader that the non-relativistic quantum three-body problem, as it appears in baryon spectroscopy, is reasonably easy to handle, involves amazing pieces of mathematics, and provides crucial tests of quark dynamics. In fact, the three-body problem which is needed for the non-relativistic models of baryons is relatively simple compared to most other three-body problems encountered in atomic or nuclear physics: in the He atom (αe−e−), or in the positronium ion (e+e−e−), asymmetry occurs already in the potential energy; in 3H or 3He nuclei, one should account, at the very beginning, for the complicated spin and isospin dependence of the internucleon potential. Three quarks in a baryon have an antisymmetric colour wave function and thus behave as bosons bound by a symmetric potential which does not depend much on spin. Such a simple situation occurs only for molecular clusters like (4He)3 with, however, a more sharply varying potential [2].
The quark model is a mine of exercises for those who have the chance of teaching quantum mechanics, once they have exhausted the charm of the Stark effect and other examples borrowed from atomic physics. The baryon sector is particularly rich. For instance, it illustrates how antisymmetrization is important, in a situation intermediate between the trivial two-body case and the limit of a large number of constituents, where second quantization techniques are applied. It is also amazing to show that, if (qqq) is bound by a pairwise potential, V = ∑ v(rij), whose strength is half of that of the qq potential, i.e., v(r) = 1 2V qq (r), then the energies or masses of ground-state mesons and baryons fulfil the inequality 2(qqq) ≥ 3(qq ). This is a simple consequence of the variational principle, as shown in Chapter 9.