Reply to “Comment on ‘Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit’ ”

This is a response to the comment we received on our recent paper “Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit.” In that paper, we introduced a computational algorithm that is appropriate for solving stiff initial value problems, and which we applied to the one-dimensional time-independent Schrödinger equation with a soft Coulomb potential. We solved for the eigenpairs using a shooting method and hence turned it into an initial value problem. In particular, we examined the behavior of the eigenpairs as the softening parameter approached zero (hard Coulomb limit). The commenters question the existence of the ground state of the hard Coulomb potential, which we inferred by extrapolation of the softening parameter to zero. A key distinction between the commenters' approach and ours is that they consider only the half-line while we considered the entire $x$ axis. Based on mathematical considerations, the commenters consider only a vanishing solution function at the origin, and they question our conclusion that the ground state of the hard Coulomb potential exists. The ground state we inferred resembles a $\delta \left(x\right)$, and hence it cannot even be addressed based on their argument. For the excited states, there is agreement with the fact that the particle is always excluded from the origin. Our discussion with regard to the symmetry of the excited states is an extrapolation of the soft Coulomb case and is further explained herein.

Figure 1

Comparison of states ${\Psi }_1^1$ and ${\Psi }_1^0$ for the shown values of $\beta$.

Figure 2

${\Psi }_1^0$ near the origin. The area under the curve gets smaller with $\beta$.

Figure 3

Comparison of states ${\Psi }_2^1$ and ${\Psi }_2^0$ for the shown values of $\beta$.

Figure 4

The even ground state for different values of $\beta$.