Heavy Mesons and Diatomic Molecules with Improved Eckart--Hellmann Potential Model in a Deformation Space-Space Background: New Bound States and The Effect on Thermodynamic Properties

Abstract: This study examines the 3D deformed Klien-Gordon and Schrödinger equations (DKGE and DSE), taking into account the effect of non-commutativity space-space in the 3D-relativistic/non-relativistic non-commutative quantum mechanics (3D-(R/NR)NCQS) regime. The investigation is done using the improved Eckart-Hellmann potential (IEHP) model. The DKGE and DSE in the 3D-(R/NR) NCQS regime for this consideration are solved using the well-known Bopp's shifts method and standard perturbation theory. For the homogeneous (I2, N2, H2) and heterogeneous (CO, NO, VH, TiH, NiC, TiC, and CuLi) diatomic molecules, the new relativistic and non-relativistic energy equations under the IEHP in the presence of deformation space-space are obtained to be sensitive to the discrete quantum numbers (j,l,s,m), the mixed potential depths (U_0,U_1,U_2,U_3), the screening parameter α, and the non-commutativity parameters (Φ,χ,ζ). The non-relativistic limit of new energy spectra is analyzed. We examine the obtained new bound state eigenvalues of the DKGE and deformed Schrödinger equation with the IEHP in 3D-(R/NR) NCQS symmetries by suitable adjustment of the combined potential parameters and get the new modified Hellmann potential, the new modified Eckart potential, the new modified Coulomb potential, and the new modified Yukawa potential. The homogeneous and heterogeneous composite systems under the IEHP model are investigated in the context of the 3D-NRNCQS regime. Under the IEHP model in 3D-NRNCQS symmetries, the influence of space-space deformation on the spin-averaged mass spectra of the heavy mesons, such as charmonium and bottomonium, is examined. Furthermore, the thermal properties such as partition function, mean energy, free energy, specific heat, and entropy of the IEHP are duly investigated in both 3D-NRQM and 3D-NRNCQS symmetries. The present research finds many applications in various fields, such as molecular and atomic physics.