Generalised Cornell Potential using Hellmann Theorem: Expectation values for Quark Confinement.

In this research work, we explored non-relativistic quantum mechanics by
employing the Nikiforov-Uvarov (NU) method alongside the Schrödinger wave
equation, utilizing the generalized Cornell potential model. Through this
approach, we derived the bound-state energy spectra and formulated the mass
spectra equation. The results were applied to charmonium and bottomonium
mesons to provide phenomenological predictions for their mass spectra.
Additionally, the Hellmann-Feynman Theorem was utilized to determine the
expectation values of the square of the inverse of position 〈r−2〉, the inverse of
position 〈r−1〉, the square of momentum, 〈p2〉 and the kinetic energy 〈T〉 for the
quantum states of charmonium and bottomonium. The phenomenological
predicted results were compared with experimental data, yielding a percentage
error of approximately 0.99% for charmonium and 0.44% for bottomonium. The expectation values showed
similar characteristics between the charmonium and bottomonium mesons for the same quantum states, with the
charmonium meson leading only in the expectation values for, but lagging behind the bottomonium meson in the
expectation values for 〈r−2〉 , 〈r−1〉, and 〈p2〉.