The modifications to the program are minor. The two BC’s at x max are the same as in the pure V$_{s}$ case, Eq. (16). The BC’s at x = ϵ are slightly modified from that of Eq. (13) because here it is V$_{v}$ that dominates. For l = 0, as in the hydrogen atom case, but with α$_{s}$ generally much larger than α . Running the code for α$_{s}$ = 0 . 4 gives the wave functions ψ$_{a}$ and ψ$_{b}$ shown in Fig. 18. For comparison, the wave functions for case of α$_{s}$ = 0 (i.e., only using the scalar potential) are also shown. The eigenenergies for these two cases are 0.306 GeV ( α$_{s}$ = 0) and 0.251 GeV ( α$_{s}$ = 0 . 4). That is, adding the Coulomb attraction lowers the ground state energy. The sharp rise in ψ$_{a}$ ( x ) and rapid falloff in ψ$_{b}$ ( x ) near the origin is another effect of the point Coulomb attraction. This behavior is unphysical, as the motions of the quark and anti-quark in the q ¯ q meson will smear out the point Coulomb potential somewhat. One might think that a Coulomb divergence remains in the relative coordinate, but this is a non-relativistic prejudice. Relativistic retardation effects and pair creation and annihilation both act to ameliorate this divergence, significantly so in this case, as α$_{s}$ ≫ α . This will give a potential V$_{v}$ ( x ) that is similar to the potential used to describe the nuclear size effect in muonic atoms, Eq. (47). We will come back to such a calculation presently. Before doing that, however, we present the results of running the massless quarks code for a variety of choices for α$_{s}$ . The 1 s ground state energies and the inital slope parameters a 0 are shown in the second and third columns of Table II. As the Coulomb attraction increases, the eigenenergy falls off fairly rapidly and eventually goes through zero at a value near α$_{s}$ = 1. At this point, like the Klein paradox situation discussed in Sec. III E, it is possible to produce q ¯ q pairs at liberty. In fact, what is called a “quark condensate” forms, a modification of the vacuum state.[31] To take into account the smearing of the point Coulomb potential by the motions of the quarks in the meson, we again use the potential V$_{v}$ ( x ) given in Eq. (47), but with α replaced by α$_{s}$ and R taken as one half of the electromagnetic radius of the pion, i.e., R = 1 2 × 0 . 672 = 0 . 336 fm.[26] We choose this fraction on the basis that the root-mean-square electromagnetic radius represents the average separation of the quark and antiquark, but since they are oppositely correlated, each should be at about half of this distance from the center of mass of the meson. 21 g ( ϵ ) = a 0 ϵ, f ( ϵ ) = − a 0 ( V$_{v}$ ( ϵ ) / 3) ϵ 2 = a 0 α$_{s}$ ϵ/ 3 , (49)