content_image/352688.jpg content_image/352689.jpg content_image/352690.jpg content_image/352691.jpg
are given by Eqs. (13) and (14), but otherwise the coding is very similar to (basically can be copied from) the 1 s 1 2 and 2 s 1 2 cases. Starting this time with 

<img src='content_image/352688.jpg'>

the gaps close in four iterations, with final parameters ˜ E = 2 . 23003 (i.e., 0.940 GeV) and a 0 = 1 . 57784. The 2 p 3 2 wave functions ψ$_{a}$ ( x ) and − ψ$_{b}$ ( x ) are shown in Fig. 5. In contrast to the 2 s case, here both ψ$_{a}$ ( x ) and ψ$_{b}$ ( x ) have two nodes, reflecting a higher energy eigenvalue. 

The other p -wave state, 2 p 1 $_{2}$, is qualitatively different. This is the first case for which j = l − 1 / 2, and k is now a positive integer. In a sense, this switches the roles of g ( x ) and f ( x ). The boundary conditions for the outward integration when j = l − 1 2 are different from those of Eq. (13): 

Substituting g ( ϵ ) into the first ODE allows us to determine this b 0 , so 

<img src='content_image/352689.jpg'>

<img src='content_image/352690.jpg'>

That is, if a 0 > 0, both g ( x ) and f ( x ) start out from the origin with positive slopes. 

Starting for this 2 p 1 2 case with initial parameters 

<img src='content_image/352691.jpg'>

the iterative procedure converges on the fifth iteration, with final parameter values ˜ E = 2 . 37846 (i.e., 1.002 GeV) and a 0 = 0 . 171483. The ψ ’s are plotted in Fig. 6. Here also both component wave functions have two nodes. 

We note in passing that the higher spin state of this pair has the lower energy, contrary to the well-known case for the Coulomb potential. This is a common feature for nuclear states (even ground states) and reflects the presence of an effective scalar potential as found in many nuclear potential models.[23] 

The energy difference between the 2 p 3 2 and 2 p 1 2 states, 62 MeV, is due to a spin- orbit interaction. However, there is evidence in the meson spectrum that the spin-orbit interaction is suppressed.[24] Page, Goldman, and Ginocchio (PGG) [25] claim this reflects a cancellation between a scalar potential V$_{s}$ and a vector potential, V$_{v}$ , having the same linear slope at large distances. We will return to this point in Sec. IV. 

