Bound state solutions of the Schrödinger equation for the atomic systems interacting with the radial screened Coulomb potential: analytical approximation methods

Research Question and Approach

This paper addresses the challenge of calculating energy levels for atomic systems embedded in plasma, where the interaction potential deviates from the standard Coulomb form. Specifically, the authors investigate the radial screened Coulomb potential (RSCP), which is non-singular at the origin and maintains a proper long-range Coulomb tail. To solve the Schrödinger equation for this potential, the authors employ three complementary analytical techniques: expectation values using Coulomb and Kratzer reference states, a variational method with a scaled Kratzer basis, and the Hellmann-Feynman theorem. Unlike standard perturbation theory, which truncates the potential, these methods evaluate the full, exact RSCP Hamiltonian using optimized trial wavefunctions.

The study demonstrates that the Kratzer reference basis provides a superior starting point compared to the standard Coulomb basis because it incorporates the leading-order $1/r^2$ screening correction into the effective centrifugal barrier. The expectation-value approach using this Kratzer basis achieves relative errors as low as 0.63% for the first ten s-states at a screening parameter of $c=0.1$. The variational method further refines these results by optimizing the spatial extent of the wavefunctions. The authors show that the formalism is general, extending naturally to systems like Positronium, and provides a robust framework for error estimation in plasma-embedded atomic systems.

Accurate atomic energy levels are essential for plasma diagnostics, such as interpreting spectral line shifts and broadening in fusion experiments and astrophysical plasmas. While high-precision numerical methods like generalized pseudospectral (GPS) calculations exist, they are computationally intensive. The closed-form analytical expressions derived in this paper offer a significant advantage for rapid parametric scaling studies and provide direct physical insight into how screening parameters influence atomic structure. This makes them highly efficient tools for researchers needing to model complex plasma environments without the overhead of full numerical simulations.