Advances in High Energy Physics

1. Introduction

A yet open challenge in the annals of wave mechanics is the equivalence or nonequivalence of Duffin-Kemmer-Petiau (DKP) and Klein-Gordon (KG) equations. While the latter solely describes relativistic spin-zero bosons, the former can investigate spin-one particles as well. This equation was introduced 1930s in search for a linear relativistic wave equation similar to Dirac equation for relativistic bosons [1–4]. In its present format, the DKP equation is normally represented in two five and ten-dimensional versions that respectively work for spin-zero and spin-one bosons. At the present, to our best knowledge, the equivalence of KG and spin-zero DKP equations is doubted [5–14]. This is also true for the other counterpart of DKP equation, that is the Proca equation which is a relativistic framework to study spin-one bosons. There are papers that investigate related problems via these equations and report motivating data [15–20]. Here, our focus is on the spin-zero version of the equation, which has many applications [21–25].

In solving wave equations of mechanics, various analytical techniques including supersymmetry quantum mechanics (SUSYQM), Nikiforov-Uvarov (NU), quantization rule, Lie algebras, WKB approximation, point canonical transformation (PCT), and series expansion have been applied [26–40]. Nevertheless, we face situations in which these methodologies do not work very well. A very successful approach in such cases is proposing a physical ansatz solution that is on the one hand consistent with the requirements of quantum mechanics and on the other hand satisfies the equation. This is in many cases a definitely cumbersome, and in some cases an impossible task. Here we follow this rich quasianalytical technique to solve the DKP equation under the Deng-Fan potential [41]. There are, however, a few papers which discuss this attractive potential. This is partially due to the complex structure of this interaction. As far as we know, the published works include references [42–46] that consider the potential under either relativistic or nonrelativistic equations. The structure of the present paper is as follows. In Section 2, we briefly introduce the DKP equation and a Pekeris-type approximation. In the next part, we introduce some transformations as well as an attractive ansatz by which the problem is solved in a quasianalytical manner.

2. The DKP Equation

The DKP Hamiltonian for scalar   and vector   interactions is where and the upper and lower components, respectively, are

The engaged matrices are with , and , respectively, being   zero matrices

The equation, in (3+0)-dimensions, is written as [1–4] where  . In (2.4a) and (2.4b), is a simultaneous eigenfunction of   and  , that is, and the general solution is considered as where spherical harmonics are of order , are the normalized vector spherical harmonics, and ,  and   represent the radial wavefunctions. The above equations yield the following coupled differential equations [30–38]: which give [39–41] Where   and  . When , we recover the well-known formula [30–32]

We consider the Deng-Fan vector and scalar potentials [42–46] By a change of variable of the form

(2.9) is written as where

From (2.11), (2.13) can be written as

Here, we use the following approximations for the centrifugal term [47]:

Equation (2.16a) is a quite logical alternative for   (see Figure 1), and (2.16a) and (2.16b) brings (2.15) into the form

Figure 1: and its approximation for different value of .

By introducing   and making the transformation  , we obtain which after decomposition of fractions gives with