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## G. Solutions for Massive Quarks 

The cases for the linear scalar potential V$_{s}$ ( r ) of Eq. (9) when the quarks are massive are computed straightforwardly using the program described above. For example, using masses appropriate [22] for the Q ¯ q mesonic states, where q stands for a massless non-strange quark ( u and d ) and Q for the massive strange ( s ), charmed ( c ), and bottom ( b ) quarks, we find the following eigenenergies: 

<img src='content_image/345179.jpg'>

<img src='content_image/345180.jpg'>

<img src='content_image/345181.jpg'>

<img src='content_image/345182.jpg'>

by requiring a match to the appropriately weighted average of the lowest pseudoscalar and vector states.[26] Note that for the heaviest quarks the states are conventionally bound, E$_{Q}$ < m$_{Q}$ , and confinement need not be invoked to understand the stability of the state, in contrast to the situation for the light quarks. 

Figure 7 compares the results for the u , s , c , and b quarks for the upper components, ψ$_{a}$ ( x ), and Fig. 8 does the same for the lower components, − ψ$_{b}$ ( x ). Note that as the mass m$_{q}$ increases, the Q ¯ q system becomes more and more non-relativistic, i.e., the lower component wave function ψ$_{b}$ gets smaller, relative to the upper component ψ$_{a}$ . Also, as m$_{q}$ gets larger, the wave functions are more and more concentrated near the origin. 

## III. VECTOR POTENTIAL – HYDROGEN-LIKE ATOMS 

In contrast with the case of massless quarks, an electron bound in a Coulomb potential, as in the hydrogen atom, is at the opposite extreme, i.e., is to a very good approximation a completely non-relativistic system. The binding energy of the ground state, Ry = 13.6 eV, is very small compared to the mass of the electron, m$_{e}$ = 0 . 511 MeV. (In this section we will use MeV instead of GeV.) The Schr¨ odinger equation for this problem predicts its energy levels basically correctly, missing only the fine-structure splitting, a spin-orbit effect about 10 − 4 times smaller than the binding of the n = 2 levels.[27]. 

