Closed-form Second Solutions to the Coulombic Schr\"odinger Equation

TL;DR

This paper derives exact closed-form second irregular solutions to the Coulombic Schrödinger equation, which are useful for numerical analysis, complex energy studies, and boundary matching in quantum systems.

Contribution

It introduces a novel approach using the Nikiforov-Uvorov method to obtain closed-form second solutions for Coulomb bound states, previously unreported.

Findings

Derived exact second irregular solutions for Coulomb bound states

Provided a new tool for controlling numerical contamination in differential equations

Facilitated analysis of Coulomb wave functions in complex energy and angular momentum planes

Abstract

The regular solutions to the Sch\"ordinger equation in the case of an electron experiencing a Coulomb force are well known. Being that the radial part of the differential equation to be solved is second order in derivatives, it will have two independent solutions, with the second irregular solutions being ill-behaved at the origin and unbound at infinity. For this reason, second solutions are dropped for bound electrons. However, these second solutions are still of academic interest for several reasons. One reason is to help devise schemes to control numerical contamination by second solutions to more general second-order differential equations when attempting to calculate the first solutions through recursion relations. Another is to study the analytic behavior of the electron Coulomb wave functions as the electron energy and angular momentum are each extended into the complex plane, important in investigations of bound-state poles and of Regge trajectories. In addition, toy problems having a finite radial region with a Coulombic interaction, with interior and exterior regions non Coulombic, require both regular and irregular solutions to match solutions across the boundaries. In this presentation, exact and closed form second irregular solutions are derived for the Coulomb bound states, using a hither-to unnoticed trick in the Nikiforov-Uvorov method.