Consider the simplest atom, Hydrogen. To get a detailed account of the energy levels, the Schrodinger and Dirac equations are not good enough, but Quantum Electrodynamics works fine. Suppose we replace the electron with a positron? Now the system is unbound, no comparison. So supposed instead we replace the proton with a positron. Now we have a system as simple as Hydrogen, and as before we need Quantum Electrodynamics to get all the fine details right. So this caused physicists in the 1970s to study quark-antiquark systems, in other words mesons, with the simplest possible approach that could give realistic results.  QCD could not be used for such calculations, in practice it is always approximated, so physicists thought quarkonium would be an excellent test case for various approximations.

Here, the first term is the “Coulomb” energy, with the strong coupling constant α_s instead of α of electrodynamics, the second term is the energy term linear in distance, which enforces confinement, and the third term contains spin-dependent terms to model the fine structure of the spectrum. The constant κ is the the so-called string tension. In the spectrum shown in the figure, the best fit to the experimental data with this spectrum is shown in red. It corresponds to an effective  mass of the charm quark, m_c = 1.46 GeV/c², a coupling constant α_s = 0.55, and a string tension (called "b" in the plot) of κ = 0.72 GeV/fm --- meaning that an energy of 0.72 GeV is needed to separate the quark-antiquark pair by 10^-15 meters.

The charm quark mass is about 1.29 GeV, the bottom quark mass about 4.18 GeV, and the top quark mass about 173 GeV. So “toponium” is beyond the reach of current accelerators, but both charmonium and bottomonium have been extensively studied.

Phenomenological fits to the meson masses use a simple idea such as M = m_q + m_anti-q + Δ, where Δ is mainly a spin-spin interaction term.

Restricting strongly interacting particles just to states of three valence quarks, or a valence quark-antiquark pair, feels as if there is a lack of nature taking advantage of its own possibilities, given that we are dealing with the strongest force in nature. Thus, experimentalists have searched for more exotic states, predicted by some theorists, with larger numbers of valence quarks and antiquarks. Evidence of a few such states has gradually emerged.  The surprisingly close similarity between the atom-atom potential and the nucleon-nucleon potential would suggest naively that hadronic equivalents of "diatomic molecules" should exist, albeit for a very, very short time.