The Klein–Gordon Equation with Modified Coulomb Potential Plus Inverse-Square–Root Potential in Three-Dimensional Noncommutative Space

SUMMARY

In present work, the three-dimensional modified Klein-Gordon equation (MKGE) is analytically solved under modified Coulomb potential plus inverse-square–root potential, in the symmetries of noncommutative quantum mechanics (NCQM), using the generalized Bopp’s shift method. The new energy shift (ground state, first excited state and excited state) is obtained via first order perturbation theory in the 3-dimensional noncommutative real space (NC: 3D-RS) symmetries instead of solving MKGE with the Weyl Moyal star product. It is found that the perturbative solutions of discrete spectrum for studied potential depended on the parabolic cylinder functions, the Gamma function, the discreet atomic quantum numbers  and the potential parameters (and), in addition to noncommutativity parameters (and).