Abstract: In this work, we have obtained analytically the bound state solution for both the relativistic modified Klein-Gordon equation MKG and non-relativistic modified Schrödinger equation for the modified unequal mixture of scalar and time-like vector Cornell (MUSVC) potentials in the relativistic noncommutative three-dimensional real space (RNC: 3D-RS) symmetries. The unequal mixture of scalar and time-like vector Cornell potentials is extended by including new radial terms. Also, MUSVC potentials are proposed as a quark-antiquark interaction potential for studying the masses of heavy and heavy-light mesons in (RNC: 3D-RSP) symmetries. The ordinary Bopp’s shift method and perturbation theory are surveyed to get generalized excited states’ energy as a function of shift energy and the energy of USVC potentials in the relativistic quantum mechanics RQM and NRQM. Furthermore, the obtained preservative solutions of discrete spectrum depended on the parabolic cylinder function, the gamma function, the ordinary discrete atomic quantum numbers, as well as the potential parameters and the two infinitesimal parameters (θ and σ) which are generated with the effect of (space-space) noncommutativity properties. We have also applied our obtained results for bosonic particles, like the charmonium cc ¯ and bottomonium bb ¯ mesons (that have quark and antiquark flavour) and cs ¯ mesons with spin-(0 and 1) and shown that MKG equation under MUSVC potentials becomes similar to the Duffin–Kemmer equation. We have shown that the degeneracy of the initial spectral under USVC potentials in RQM is changed radically and replaced by the newly triplet degeneracy of energy levels under the MUSVC potentials; this gives more precision in measurement and better results compared to the results of ordinary RQM under USVC potentials.
Keywords: Klein-Gordon equation, Schrödinger equation, Unequal mixture of scalar and time-like vector Cornell potentials, Noncommutative quantum mechanics, Star product, Bopp’s shift method, Heavy–light mesons.
PACS Nos.: 03.65.Ta; 03.65.Ca; 03.65.Ge.
Justification of the discrete nonlinear Schrödinger equation from a parametrically driven damped nonlinear Klein–Gordon equation and numerical comparisons
