content_image/328873.jpg content_image/328872.jpg content_image/328871.jpg content_image/328870.jpg content_image/328869.jpg
in the exponential decay is small: 

<img src='content_image/328873.jpg'>

The Coulomb wave functions are quite long-ranged, and are all the more so the smaller the binding energy. Equation (38) also shows that, asymptotically, f ( x ) is very much smaller than g ( x ). The smallness of the lower component is an indication of how non-relativistic the hydrogen atom is. 

## B. Boundary Conditions for Outward Integration 

As before, to avoid the singularity at x = 0, we integrate outwards from x = ϵ , where ϵ takes an appropriately small value. In this case, one might even consider taking ϵ as the radius of the nucleus providing the Coulomb potential (but see Sec. III F for a better approach). For the boundary conditions for integrating outwards from x = ϵ when j = l + 1 $_{2}$, i.e., l ′ = l + 1 and k = − ( l + 1), we can again assume 

Substituting these in the second ODE, we can solve for b 0 , finding 

<img src='content_image/328872.jpg'>

<img src='content_image/328871.jpg'>

Noticeably different is the lowering of the power behavior of f ( x ) due to the 1 /x behavior of the potential. Note that, because of the factor of α ≈ 1 / 137, the magnitude of f ( x ) is much smaller than that of g ( x ), also here near the origin. 

For the j = l − 1 2 case, i.e., l ′ = l − 1 and k = l , we let f ( x ) determine the nature of the boundary conditions [as in Eq. (28)], i.e., 

Substituting these in the first ODE, we again solve for b 0 and find 

<img src='content_image/328870.jpg'>

<img src='content_image/328869.jpg'>

Again, for this case, g ( x ) and f ( x ) start off with the same slope, as in Eq. (29). And again, there is a lower power behavior of g ( x ) due to the 1 /x in the potential, as in Eq. (41). 

