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New Horizons in Mathematical Physics
NHMP > Volume 3, Number 4, December 2019

A Review on the Bound-State Solutions of the Stationary Schrödinger Equation for General Pseudo-Coulomb Potential

Download PDF  (397.2 KB)PP. 105-117,  Pub. Date:December 31, 2019
DOI: 10.22606/nhmp.2019.34001

Author(s)
Spiros Konstantogiannis
Affiliation(s)
4 Antigonis Street, Nikaia 18454, Athens, Greece
Abstract
Considering the stationary Schrödinger equation for a general pseudo-Coulomb potential as the normal form of the associated Laguerre equation, we review, in one and three dimensions, the bound-state solutions for the potential, when the inverse-square-term coupling is not less than a negative critical value. We show that, as a consequence of the inverse-square-term coupling being a two-to-one mapping for all but one of the allowed negative values of its parameter, an additional sequence of bound-state energies emerges for each of the respective potentials. In this framework, the slightest relaxation of the boundary condition for the radial wave function at the origin results in minus-infinity ground-state energy for the Coulomb potential, rendering the hydrogen atom unstable.
Keywords
associated Laguerre equation; pseudo-Coulomb potential; inverse square potential; Kratzer potential; valence electron model; additional energies; instability.
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