Semiclassical methods to the Klein–Gordon equation with the unequal scalar and vector potentials

Abstract

When the scalar potential is larger than the vector potential there are very few exactly solvable Klein–Gordon equations. Based on a general transformation between the unequal scalar and vector potential, in this paper, we employ two semiclassical methods to determine the bound state energy spectrum of the Klein–Gordon equation. To illustrate this procedure, the scalar potentials are chosen as the linear, exponential and linear plus Coulomb potentials and the corresponding energy spectra are analytically obtained. It is shown that the energy spectrum can be obtained by a simple algebraic method and our proposal methods can be extended to discuss the quasi-exactly solvable cases.

Relativistic Bound States of Spinless Particle by the Cornell Potential Model in External Fields

“…The bound state solution exists for the relativistic spin-0 particle only when ( ) > ( ). This is obvious from (29) where the KG possesses bound state spectrum when | V | ≤ and | V | ≤ or when the scalar potential is larger than the vector potential there are very few exactly solvable KG equations [59]. The relativistic bound state solutions can also be obtained in this work for any mixture with ( ) = ( ), where | | ≤ 1.…”

Exact Klein-Gordon equation with spatially dependent masses for unequal scalar-vector Coulomb-like potentials

“…where V 0 and β being arbitrary constants of certain proportions have to be chosen after solving the problem under consideration [19][20][21]. It is interesting to note that, this restriction includes the case where V (r) = 0, when both constants vanish, the situation where the potentials are equal in magnitude and sign V (r) = S(r) or equal in magnitude but opposite in sign V (r) = −S(r) (i.e., V 0 = 0; β = ±1), and also the case where the potentials are proportional when V 0 = 0 [21].…”

“…Furthermore, the exact results for the scattering states of the KG equation with Coulomb-like scalar plus vector potentials have been investigated in an arbitrary dimension [18]. This equation has been exactly solved for a larger class of linear, exponential and linear plus Coulomb potentials to determine the bound state energy spectrum using two semiclassical methods [19]. Many authors have considered a more general transformation between the unequal vector and scalar potentials given by…”

Exact analytical solution to the relativistic Klein-Gordon equation with noncentral equal scalar and vector potentials

“…For the case S( r) = ±V ( r), the solution of the Klein-Gordon equation has been studied recently [1,2]. The exact solutions of these equations are possible only for certain potentials such as Coulomb, Morse, Pöschl-Teller, Hulthen and harmonic oscillator etc.…”