Extending the class of solvable potentials: II. Screened Coulomb potential with a barrier

TL;DR

This paper extends the class of solvable quantum potentials by analyzing a screened Coulomb potential with a barrier, providing new solutions and insights into electron-molecule interactions.

Contribution

It introduces a method to solve a new class of potentials using a tridiagonal matrix representation, expanding solvable models in quantum mechanics.

Findings

Derived S-wave solutions for a three-parameter potential with a barrier

Analyzed the energy spectrum and resonance structure of the potential

Modeled electron scattering with molecules using the new potential parameters

Abstract

This is the second article in a series where we succeed in enlarging the class of solvable problems in one and three dimensions. We do that by working in a complete square integrable basis that carries a tridiagonal matrix representation of the wave operator. Consequently, the wave equation becomes equivalent to a three-term recursion relation for the expansion coefficients of the wavefunction in the basis. Finding solutions of the recursion relation is equivalent to solving the original problem. This method gives a larger class of solvable potentials. The usual diagonal representation constraint results in a reduction to the conventional class of solvable potentials. However, the tridiagonal requirement allows only very few and special potentials to be added to the solvability class. In the present work, we obtain S-wave solutions for a three-parameter 1/r singular but short-range potential with a non-orbital barrier and study its energy spectrum. We argue that it could be used as a more appropriate model for the screened Coulomb interaction of an electron with extended molecules. We give also its resonance structure for non-zero angular momentum. Additionally, we plot the phase shift for an electron scattering off a molecule modeled by a set of values of the potential parameters.