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Register a new userApproximate Solutions to the Klein-Fock-Gordon Equation for the sum of Coulomb and Ring-Shaped like potentials
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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 $S(vec{r})$ and vector $V(vec{r})$ Coulomb plus ring-shaped potentials. It is shown that the system under consideration has both a discrete at $left|Eright|<Mc^{2} $ and a continuous at $left|Eright|>Mc^{2} $ energy spectra. We find the analytical expressions for the corresponding complete wave functions. A dynamical symmetry group $SU(1,1)$ 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 $cto infty $ go over into the corresponding expressions for the nonrelativistic problem.
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We point out a misleading treatment in the literature regarding to bound-state solutions for the $s$-wave Klein-Gordon equation with exponential scalar and vector potentials. Following the appropriate procedure for an arbitrary mixing of scalar and vector couplings, we generalize earlier works and present the correct solution to bound states and additionally we address the issue of scattering states. Moreover, we present a new effect related to the polarization of the charge density in the presence of weak short-range exponential scalar and vector potentials.
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