content_image/331034.jpg content_image/331032.jpg content_image/331033.jpg content_image/331035.jpg content_image/331036.jpg content_image/331031.jpg
where l ′ = 2 j − l and the Y l jm are the spin-orbital angular momentum wave functions, 

<img src='content_image/331034.jpg'>

where < jm | l 1 $_{2}$, m − m$_{s}$ m$_{s}$ > is a Clebsch-Gordan coefficient and χ is a Pauli spinor. 

We consider here the case of bound-state wave functions for a central potential with energy eigenvalue E . For example, for hydrogen-like atoms the potential is V$_{v}$ ( r ) = − Zα/r . With appropriate boundary conditions, the radial wave functions g ( r ) and f ( r ), which can be taken as real functions, are solutions of the following coupled first-order ordinary differential equations (ODEs), [9, 12] 

<img src='content_image/331032.jpg'>

<img src='content_image/331033.jpg'>

Here the integer k is determined by the angular momentum quantum numbers according to 

<img src='content_image/331035.jpg'>

<img src='content_image/331036.jpg'>

The reason for the subscript on the potential V$_{v}$ in those equations is to indicate that it is the fourth component of a Lorentz four-vector, such as the Coulomb potential. However, our motivation is not so much in atomic physics applications as it is for treating mesons as q ¯ q states in a relativistic quark model. The non -relativistic quark model often assumes the confining potential for quarks to be the so-called “Cornell potential,” [6] 

<img src='content_image/331031.jpg'>

where α$_{s}$ = g 2 s / 4 π , with g$_{s}$ being the (running) quark-gluon coupling constant, and κ 2 (or σ ) is the string tension. It is the linear term in V ( r ) that confines the quarks, corresponding to the effective potential in the relativistic Bag Model.[2] 

For a relativistic quark model, the two pieces of that non-relativistic potential have dif- ferent Lorentz transformation properties. The color Coulomb potential, α$_{s}$/r , is the fourth component of a Lorentz vector, while the confining linear potential transforms as a Lorentz scalar, which we will write as V$_{s}$ ( r ). Thus, in solving the relativistic radial Dirac equations, 

