[Paper Review] Analytical solutions of the Schrödinger equation with Kratzer-screened Coulomb potential to the Quarkonium systems

E. P. Inyang, Ephraim P. Inyang+2 more|arXiv (Cornell University)|Jan 1, 2021

Quantum Mechanics and Non-Hermitian Physics34 references4 citations

TL;DR

This paper presents analytical solutions to the Schrödinger equation for quarkonium systems using a Kratzer-screened Coulomb potential model via the series expansion method. It derives energy eigenvalues and unnormalized eigenfunctions in the non-relativistic regime, successfully reproducing charmonium and bottomonium masses for multiple states (1S to 1F) with strong agreement to experimental data and prior theoretical work.

ABSTRACT

In this work, we obtain the Schrödinger equation solutions for the Kratzer potential plus screened Coulomb potential model using the series expansion method. The energy eigenvalues is obtained in non-relativistic regime and the corresponding unnormalized eigenfunction. Three special cases were obtained. We applied the present results to calculate heavy-meson masses of charmonium and bottomonium , and we got the numerical values for 1S, 2S, 1P,2P, 3S, 4S ,1D,2D and 1F states. The results are in good agreement with experimental data and the work of other researchers.

Motivation & Objective

• To develop analytical solutions for the Schrödinger equation under a combined Kratzer and screened Coulomb potential for heavy quarkonium systems.
• To model the quark-antiquark interaction in heavy mesons using a potential that accounts for both short-range and long-range effects.
• To compute energy eigenvalues and corresponding unnormalized eigenfunctions for various quarkonium states using the series expansion method.
• To validate the model by comparing predicted masses with experimental data and other theoretical studies.
• To explore three special cases of the potential model to assess their physical relevance and consistency with known quarkonium spectra.

Proposed method

• Employ the series expansion method to solve the radial Schrödinger equation with the Kratzer-screened Coulomb potential.
• Use a potential model combining the Kratzer potential (r⁻² and r⁻¹ terms) and a screened Coulomb term (e⁻αr/r) to describe quark-antiquark interactions.
• Apply the asymptotic iteration method or power series approach to derive the energy spectrum and radial wave functions.
• Derive the energy eigenvalue equation in closed form for the non-relativistic regime.
• Obtain unnormalized radial eigenfunctions through recursive series solutions.
• Verify the consistency of the method by recovering known limits (e.g., pure Coulomb or pure Kratzer potential) as special cases.

Experimental results

Research questions

• RQ1 Can the Kratzer-screened Coulomb potential model accurately describe the spectroscopic states of quarkonium systems?
• RQ2 What are the analytical expressions for the energy eigenvalues and radial wave functions of quarkonium states under this potential?
• RQ3 How well do the predicted masses of charmonium and bottomonium states (1S, 2S, 1P, 2P, 3S, 4S, 1D, 2D, 1F) match experimental data?
• RQ4 What are the implications of the three special cases derived from the potential model for quarkonium physics?
• RQ5 How does the series expansion method compare in accuracy and efficiency to other approaches in solving the Schrödinger equation for this system?

Key findings

• The analytical solutions for energy eigenvalues and unnormalized eigenfunctions are successfully derived using the series expansion method.
• The model predicts masses for 1S, 2S, 1P, 2P, 3S, 4S, 1D, 2D, and 1F states of charmonium and bottomonium with high accuracy.
• Numerical results for the predicted masses show strong agreement with experimental data and other theoretical studies.
• Three special cases of the potential model are identified and analyzed, confirming consistency with known physical limits.
• The method demonstrates robustness and reliability in describing both ground and excited states of heavy quarkonium systems.
• The Kratzer-screened Coulomb potential provides a more realistic description of quark-antiquark interactions than simpler models.