On the Role of Differentiation Parameter in a Bound State Solution of the Klein-Gordon Equation

Abstract

Recently, the bound state solutions of a confined Klein-Gordon particle under the mixed scalarvector generalized symmetric Woods-Saxon potential in one spatial dimension have been investigated. The obtained results reveal that in the spin symmetric limit discrete spectrum exists, while in the pseudo-spin symmetric limit it does not. In this manuscript, new insights and information are given by employing an analogy of the variational principle. The role of the difference of the magnitudes of the vector and scalar potential energies, namely the differentiation parameter, on the energy spectrum is examined. It is observed that the differentiation parameter determines the measure of the energy spectrum density by modifying the confined particle's mass-energy in addition to narrowing the spectrum interval length. PACS numbers: 03.65.Ge, 03.65.Pm Keywords: Klein-Gordon equation, generalized symmetric Woods-Saxon potential, bound state spectrum, spin symmetry limit, analytic solutions. In nuclear physics, the spin symmetry (SS) and pseudopsin symmetry (PSS) concepts, originally postulated by Smith et al. [1] and Bell et al. [2], were widely used to explain the nuclear structure dynamical phenomena [3-7]. The basics of these symmetries were explored comprehensively and the conclusion was that they depended on the existence of vector, V v , and scalar, V s , potential energies [8-10]. In 1997, Ginocchio revealed that PSS and SS occur with an attractive scalar and a repulsive vector potential energies that satisfy V v + gV s = ε − , V v − gV s = ε + conditions, respectively [10]. The Dirac equation (DE) was investigated by using various potential energies in the SS and PSS limits. For instance in the SS limit, Wei et al. obtained an approximate analytic bound state solution of the DE by employing the Manning-Rosen potential [11],the deformed generalized Pöschl-Teller potential [12] energies. Furthermore, in the same limit, they proposed a novel algebraic method to obtain the bound state solution of the DE with the second Pöschl-Teller potential energy [13]. In their other work, they examined the symmetrical well potential energy solutions in the DE within the exact SS limit [14].They contributed the field with important papers in the PSS limit too. For instance, they discussed an algebraic approach in the DE for the modified Pöschl-Teller potential energy [15]. Moreover, they applied the Pekeris-type approximation to the pseudo-centrifugal term and investigated the bound state solutions in the DE for Manning-Rosen potential [16] and modified Rosen-Morse potential [17] energies. Another relativistic equation, Klein-Gordon equation (KGE), also has been the subject of many scientific investigations in the SS and PSS limits. Ma et al. studied the D-dimensional KGE with a Coulomb potential in addition to a scalar potential [18]. Dong et al. obtained the exact bound state solution of the KGE with a ring-shaped potential in the SS limit [19]. Hassanabadi et al. studied the radial KGE for an Eckart and modified Hylleraas potential...

A Statistical Mechanical Analysis on the Bound State Solution of an Energy-Dependent Deformed Hulthén Potential Energy*

“…In 2019, one of the authors of the present paper, Ikot, with his collaborators studied the thermodynamic properties of a q-deformed quantum oscillator in the scale of minimal length [64]. The other author of the present paper, Lütfüoglu, also contributed to the field by the studies via the investigation of the generalized symmetric WS potential energy in non relativistic [65] and relativistic equations [66]. With the non relativistic results, they obtained the thermodynamic properties of a nucleon in relatively small [67] and big radius nuclei [68].…”

“…[64] The other author of the present paper, Lütfüoglu, also contributed to the field by the studies via the investigation of the generalized symmetric WS potential energy in non relativistic [65] and relativistic equations. [66] With the non relativistic results, they obtained the thermodynamic properties of a nucleon in relatively small [67] and big radius nuclei. [68] Then, he compared the thermodynamic functions with excluding and including the surface effects in non-relativistic, [69] and relativistic regimes.…”

Thermal properties of a two-dimensional Duffin-Kemmer-Petiau oscillator under an external magnetic field in the presence of a minimal length

“…Two well-known differential equations, namely the Klein-Gordon(KG) and the Dirac equations, are being frequently studied in the relativistic quantum mechanics with the Heisenberg Uncertainty Principle [18][19][20][21][22][23] and with the GUP. For example, in the presence of the ML, Jana et al obtained an exact solutions to the KG equation by employing a linear and vector potential energy [24].…”