Relativistic Treatment of Quantum Mechanical Gravitational-Harmonic Oscillator Potential

The solutions of the Klein- Gordon equation for the quantum mechanical gravitational plus harmonic oscillator potential with equal scalar and vector potential have been presented using the parametric Nikiforov-Uvarov method. The energy eigenvalues were obtained in relativistic and non-relativistic regime and the corresponding un-normalized eigenfunctions in terms of Laguerre polynomials. The numerical values for the S – wave bound state were obtained.

Approximate Solutions to the Klein-Fock-Gordon Equation for the Sum of Coulomb and Ring-Shaped-Like Potentials

We consider the quantum mechanical problem of the motion of a spinless charged relativistic particle with mass M, described by the Klein-Fock-Gordon equation with equal scalar Sr→ and vector Vr→ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at E<Mc2 and a continuous at E>Mc2 energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group SU1,1 for the radial wave equation of motion is constructed. The algebra of generators of this group makes it possible to find energy spectra in a purely algebraic way. It is also shown that relativistic expressions for wave functions, energy spectra, and group generators in the limit c⟶∞ go over into the corresponding expressions for the nonrelativistic problem.

The Klein–Gordon equation with modified Coulomb plus inverse-square potential in the noncommutative three-dimensional space

The Klein–Gordon equation with equal scalar and vector potentials [Formula: see text] describing the dynamics of a three-dimensional under the modified Coulomb plus inverse-square potential is considered, in the symmetries of noncommutative quantum mechanics (NCQM), using Bopp’s shift method. The new energy of [Formula: see text]th excited state [Formula: see text] is obtained as a function of the shift energy [Formula: see text] and [Formula: see text] is obtained via first-order perturbation theory in the three-dimensional noncommutative real space (NC: 3D-RS) symmetries instead of solving modified Klein–Gordon equation (MKGE) with the Weyl–Moyal star product. It is found that the perturbative solutions of discrete spectrum depended by the Gamma function, the discreet atomic quantum numbers [Formula: see text] and the potential parameters (A and B), in addition to noncommutativity parameters ([Formula: see text] and [Formula: see text]), which are induced with the effect of (space–space) noncommutativity properties.

Exact treatment of the relativistic double ring-shaped Kratzer potential using the quantum Hamilton–Jacobi formalism

We study the Klein–Gordon equation with noncentral and separable potential under the condition of equal scalar and vector potentials and we obtain the corresponding relativistic quantum Hamilton–Jacobi equation. The application of the quantum Hamilton–Jacobi formalism to the double ring-shaped Kratzer potential leads to its relativistic energy spectrum as well as the corresponding eigenfunctions.

Scattering and Bound State Solutions of the Yukawa Potential within the Dirac Equation

In the presence of spin symmetry case, we obtain bound and scattering states solutions of the Dirac equation for the equal scalar and vector Yukawa potentials for any spin-orbit quantum numberκ. The approximate analytical solutions are presented for the bound and scattering states and scattering phase shifts.

Bound-State Solutions of the Klein-Gordon Equation with -Deformed Equal Scalar and Vector Eckart Potential Using a Newly Improved Approximation Scheme

We present the analytical solutions of the Klein-Gordon equation for q-deformed equal vector and scalar Eckart potential for arbitrary -state. We obtain the energy spectrum and the corresponding unnormalized wave function expressed in terms of the Jacobi polynomial. We also discussed the special cases of the potential.