C. The 1 s Ground State The natural unit of length for the hydrogen atom problem is the Bohr radius, a$_{B}$ = ¯ h $^{2}$/m$_{e}$ α = 0 . 529 ˚ A = 0 . 529 × 10 5 fm. Thus a natural scale for x is in units of the dimensionless Bohr radius, x$_{B}$ = a$_{B}$m$_{e}$/ ¯ hc which is 1 /α in ¯ h = c = 1 units. In view of the slow exponential decay in Eq. (38) (i.e., the smallness of µ ), for our calculations we chose x max = 7 x$_{B}$ , along with x match = 0 . 5 x$_{B}$ . Fixing a 1 = 10 (recall, it is arbitrary before normalization) and the starting values of the parameters to be tuned, B = 12 × 10 − 6 MeV (less than the expected 13.6 eV) and a 0 = 0 . 015, we found the 2 × 2 matrix of partial derivatives to be This shows a big sensitivity of the hydrogen atom wave functions to the choice of binding energy. Nonetheless, the iterative process of refining ˜ B and a 0 proceeds nicely to a solution with B = 13 . 6059 eV and a 0 = 0 . 0104306. The wave functions g ( x ) and f ( x ) are displayed as the solid curves in Figs. 9 and 10. As expected, f ( x ) is much smaller than g ( x ). Their shapes differ from those shown for the scalar linear potential shown in Fig. 2. Figures 9 and 10 also display, as dashed curves, the corresponding ψ$_{a}$ ( x ) and ψ$_{b}$ ( x ) radial wave functions. These are, in fact, both purely decaying exponentials proportional to e − $^{µx}$. This is not surprising for the upper component, ψ$_{a}$ ( x ), since that is the just what the Schr¨ odinger equation predicts for the hydrogen atom ground state. It may not be so obvious, however, that the lower component, ψ$_{b}$ ( x ), has the same form. That it has no node at the origin (in contrast to Fig. 3) is a consequence of the Zα/x potential changing the presumed x l $^{+2}$-dependence of f ( x ) near the origin to an x l $^{+1}$-dependence. The n = 2 excited states of hydrogen are done in a very similar manner to the 1 s ground state calculation. The major difference is that, from the Balmer formula [27], we expect the binding energy B for these states to be about 1 /n 2 = 1 / 4 times the ground state binding energy. Also, based on Eq. (39), we expect the wave functions will extend outward about twice as far, so, for our calculations we chose x max = 16 a$_{B}$ instead of 7 a$_{B}$ . D. Some Excited States 17 M =    − 108877 107 . 348 − 320810 107 . 494    . (44)