Any$\mathbf {\, }l$-State Energy of the Spinless Salpeter Equation Under the Cornell Potential by the WKB Approximation Method: An Application to Mass Spectra of Mesons

Abstract

In this work, we investigated the mass spectra of mesons under the spinless-Salpeter equation implemented with the Cornell potential using the semi-classical WKB approximation approach. The present results are applied for calculating the mass spectra of heavy mesons such as charmonium ${c\bar{{c}}}$, bottomonium ${b\bar{{b}}}$ and the heavy-light mesons such as charm-strange mesons. We obtained the mass spectrum free parameters of the meson systems by fitting our analytical solution with experimental data and solve the resulting non-linear equations simultaneously via numerical approach. The masses of the mesons systems obtained with the Salpeter equation for different quantum states are in excellent agreement with the masses obtained by the relativistic, non-relativistic wave equations and available experimental data reported in existing literature. The bound state energies for both the Salpeter equation and Schrodinger equations under the Coulomb potential were also obtained as special cases. The Cornell potential and eigen energy of the Salpeter equation obtained by the semi-classical approach provide satisfying results in comparison with experimental data and the work of other researchers.