Local approximation to the critical parameters of quantum wells

TL;DR

This paper introduces a modified Riccati-Padé method to accurately compute critical parameters of quantum wells, improving convergence especially for s states, and compares it with WKB and perturbation approaches.

Contribution

It proposes a simple modification to the Riccati-Padé method that significantly enhances convergence for quantum well critical parameters, especially for s states.

Findings

Modified RPM yields highly accurate results for s states.

Convergence improves with higher angular momentum quantum number l.

Comparison with WKB shows good agreement for large l.

Abstract

We calculate the critical parameters for some simple quantum wells by means of the Riccati-Pad\'{e} method. The original approach converges reasonably well for nonzero angular-momentum quantum number $l$ but rather too slowly for the s states. We therefore propose a simple modification that yields remarkably accurate results for the latter case. The rate of convergence of both methods increases with $l$ and decreases with the radial quantum number $n$. We compare RPM results with WKB ones for sufficiently large values of $l$. As illustrative examples we choose the one-dimensional and central-field Gaussian wells as well as the Yukawa potential. The application of perturbation theory by means of the RPM to a class of rational potentials yields interesting and baffling unphysical results.