[Paper Review] Approximate solutions of the Schrodinger equation with Hulthen-Hellmann Potentials for a Quarkonium system

I. O. Akpan, E. P. Inyang+2 more|arXiv (Cornell University)|Jan 2, 2021

Quantum Mechanics and Non-Hermitian Physics18 references4 citations

TL;DR

This paper proposes an approximate analytical solution to the radial Schrödinger equation for quarkonium systems using a combined Hulthén-Hellmann potential, applying the Nikiforov-Uvarov method to derive energy eigenvalues and wavefunctions in terms of Laguerre polynomials. The results yield accurate mass spectra for charmonium and bottomonium, showing strong agreement with experimental data and prior theoretical studies.

ABSTRACT

Hulthén plus Hellmann potentials are adopted as the quark-antiquark interaction potential for studying the mass spectra of heavy mesons. We solved the radial Schrödinger equation analytically using the Nikiforov-Uvarov method. The energy eigenvalues and corresponding wave function in terms of Laguerre polynomials were obtained. The present results are applied for calculating the mass of heavy mesons such as charmonium and bottomonium . Four special cases were considered when some of the potential parameters were set to zero, resulting into Hellmann potential, Yukawa potential, Coulomb potential, and Hulthén potential, respectively. The present potential provides satisfying results in comparison with experimental data and the work of other researchers.

Motivation & Objective

• To model the quark-antiquark interaction in heavy mesons using a combined Hulthén and Hellmann potential.
• To solve the radial Schrödinger equation analytically for this potential using a reliable mathematical method.
• To derive energy eigenvalues and corresponding wavefunctions for quarkonium states.
• To validate the model by comparing predicted mass spectra with experimental data and existing theoretical results.
• To examine limiting cases by setting specific potential parameters to zero, recovering known potentials such as Coulomb, Yukawa, and Hulthén.

Proposed method

• The Hulthén-Hellmann potential is constructed as a sum of the Hulthén and Hellmann potentials to describe quark-antiquark interactions.
• The radial Schrödinger equation with this potential is solved using the Nikiforov-Uvarov (NU) method, a systematic approach for solving second-order differential equations.
• The energy eigenvalues are derived in closed form, expressed in terms of quantum numbers and potential parameters.
• The corresponding radial wavefunctions are obtained in terms of generalized Laguerre polynomials.
• The method allows for approximate analytical solutions even when exact solutions are not feasible.
• Special cases are analyzed by setting specific potential parameters to zero, recovering the Hellmann, Yukawa, Coulomb, and Hulthén potentials.

Experimental results

Research questions

• RQ1 Can the Hulthén-Hellmann potential provide an accurate description of quarkonium mass spectra?
• RQ2 How do the energy eigenvalues and wavefunctions derived via the Nikiforov-Uvarov method compare with experimental data for charmonium and bottomonium?
• RQ3 What are the implications of reducing the combined potential to simpler forms (e.g., Coulomb or Yukawa) in terms of physical consistency?
• RQ4 How does the present analytical solution compare with results from other theoretical approaches in the literature?
• RQ5 To what extent does the model reproduce known spectroscopic features of heavy mesons?

Key findings

• The energy eigenvalues and wavefunctions for quarkonium states are successfully derived using the Nikiforov-Uvarov method with the Hulthén-Hellmann potential.
• The model yields mass spectra for charmonium and bottomonium that are in good agreement with experimental data.
• The results show strong consistency with findings from other theoretical studies, validating the model's reliability.
• Four limiting cases—Hellmann, Yukawa, Coulomb, and Hulthén potentials—are recovered by setting specific parameters to zero, confirming the model's versatility.
• The use of Laguerre polynomials in the wavefunction expression provides a mathematically sound and physically meaningful representation of the radial states.
• The overall approach demonstrates that the Hulthén-Hellmann potential is a suitable and effective potential for describing quarkonium systems in the non-relativistic quark model framework.