Calculations for the one-dimensional soft Coulomb problem and the hard Coulomb limit

An efficient way of evolving a solution to an ordinary differential equation is presented. A finite element method is used where we expand in a convenient local basis set of functions that enforce both function and first derivative continuity across the boundaries of each element. We also implement an adaptive step-size choice for each element that is based on a Taylor series expansion. This algorithm is used to solve for the eigenpairs corresponding to the one-dimensional soft Coulomb potential, $1/\sqrt{x^2+{\beta }^2}$, which becomes numerically intractable (because of extreme stiffness) as the softening parameter ($\beta$) approaches zero. We are able to maintain near machine accuracy for $\beta$ as low as $\beta ={10}^{-8}$ using 16-digit precision calculations. Our numerical results provide insight into the controversial one-dimensional hydrogen atom, which is a limiting case of the soft Coulomb problem as $\beta \to 0$.

Figure 1

For $\beta ={10}^{-8}$ a plot of odd solution at $x_{\max }=23$ vs energy has a root at the eigenvalue $E_1^1$ (Table 2).

Figure 2

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

Figure 3

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

Figure 4

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

Figure 5

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

Figure 6

Log scale plot of $h_i$ used in evolving states with $(n,\beta ,\sigma )$ values as shown. The time steps are as low as of the same order of magnitude as $\beta$ near the origin. Their size varies as shown, automatically adapting to the stiffness of the DE at every element.