Approximate Complete Solutions of DKP Equation under a Vector Exponential Interaction via a Pekeris-Type Approximation

Abstract

The DKP equation for an exponential potential is exactly solved via an appropriate approximation using the methodology of supersymmetric quantum mechanics. We see that the solutions are already known without any cumbersome algebra we face in any numerical or analytical approach. Closed forms of eigenfunctions and eigenvalues are reported.

1. Introduction

Working on thebasis of Dirac and Klein-Gordon equations, spin-0 and spin-1/2 particles have been extensively discussed via many analytical and numerical techniques. For the spin-1 particles, however, there are only few investigations. The main reason for this lack of literature is probably the mathematical complexity of Proca equation describing the spin-1 particles. The Duffin-Kemmer-Petiau (DKP) [1–4] equation, however, provides us with a theoretical basis for describing both spin-0 and spin-1 particles on a relatively easier background. For many years, the DKP equation was thought to be exactly equivalent to Klein-Gordon (KG) and Proca equations and consequently was in the shadow of them. Now, we know that the equations are not exactly the same and the equivalence does not hold generally [5–14]. In addition, the DKP equation is richer in the investigation of interactions and is even closer to some experimental data in comparison with KG or Proca equations [15–20]. Moreover, besides cosmology and gravity, this equation has been tested in many branches of physics including particle and nuclear physics [21–25].

As usual, the most appealing case studies are Coulomb and quadratic terms [26–28], and other ones including the woods-Saxon and Hulthen are investigated by different approaches as well [29–32]. Within the present work, we first review the DKP equation. Next, an introductory section is included on supersymmetry (SUSY) quantum mechanics. In the last part, we obtain the approximate analytical solutions of the problem.

2. DKP Equation

For the sake of briefness, our starting square is 𝛽⋅^→𝑝𝑐+𝑚𝑐²+𝑈_𝑠+𝛽⁰𝑈^𝑜_𝑣𝜓^→𝑟=𝛽⁰𝐸𝜓^→𝑟, (2.1) where 𝜓^→𝑟=𝜓_upper𝑖𝜓_lower, (2.2) with the upper and lower components, respectively, being 𝜓_upper≡𝜙𝜑,𝜓_lower≡𝐴₁𝐴₂𝐴₃, (2.3)𝛽⁰ is the usual 5×5 matrix, and 𝑈_𝑠,𝑈^𝑜_𝑣 denote the scalar and vector interactions. The equation, in (3+0)-dimensions, is therefore written as 𝑚𝑐²+𝑈_𝑠𝜙=𝐸−𝑈^𝑜_𝑣𝜑+ℏ𝑐^→∇⋅𝐴^→,_→∇𝜙=𝑚𝑐²+𝑈𝐴_𝑠^→,𝑚𝑐²+𝑈_𝑠𝜑=𝐸−𝑈^𝑜_𝑣𝜙, (2.4) where 𝐴^→=(𝐴₁,𝐴₂,𝐴₃). It should be noted that 𝜓 is a simultaneous eigenfunction of 𝐽² and 𝐽₃, that is 𝐽²𝜓_upper𝜓_lower=𝐿²𝜓_upper(𝐿+𝑆)²𝜓_lower()𝜓=𝐽𝐽+1^upper𝜓_lower,𝐽₃𝜓_upper𝜓_lower=𝐿₃𝜓_upper𝐿₃+𝑠₃𝜓_lower𝜓=𝑀^upper𝜓_lower, (2.5) and the general solution is 𝜓_𝐽𝑀(𝑟)=𝑓_𝑛𝐽(𝑟)𝑌_𝐽𝑀(Ω)𝑔_𝑛𝐽(𝑟)𝑌_𝐽𝑀(Ω)𝑖∑_𝐿ℎ_𝑛𝐽𝐿(𝑟)𝑌^𝑀_𝐽𝐿1(Ω), (2.6) where the spherical harmonics 𝑌_𝐽𝑀(Ω) are of order 𝐽, 𝑌^𝑀_𝐽𝐿1(Ω) are the normalized vector spherical harmonics, and 𝑓_𝑛𝐽,  𝑔_𝑛𝐽, and ℎ_𝑛𝐽𝐿 denote the radial wave functions. It is shown that the above equations result in the coupled differential equations [26, 33]: 𝐸−𝑈⁰_𝑣𝐹(𝑟)=𝑚𝑐²+𝑈_𝑠𝐺(𝑟),𝑑𝐹(𝑟)−𝑑𝑟𝐽+1𝑟𝐹(𝑟)1=−𝛼_𝐽𝑚𝑐²+𝑈_𝑠𝐻₁(𝑟),𝑑𝐹(𝑟)+𝐽𝑑𝑟𝑟𝐹(𝑟)=1𝜁_𝐽𝑚𝑐²+𝑈_𝑠𝐻₋₁(𝑟),−𝛼_𝐽𝑑𝐻₁(𝑟)+𝑑𝑟𝐽+1𝑟𝐻₁(𝑟)+𝜁𝑑𝐻₋₁(𝑟)−𝐽𝑑𝑟𝑟𝐻₋₁(𝑟)=1ℏ𝑐𝑚𝑐²+𝑈_𝑠𝐹(𝑟)−𝐸−𝑈⁰_𝑣𝐺(𝑟), (2.7) which give 𝑑²𝐹(𝑟)𝑑𝑟²𝜁1+^2𝐽𝛼²_𝐽−𝑑𝐹(𝑟)𝑈𝑑𝑟𝑚+^′𝑠𝑈_𝑠𝜁1+^2𝐽𝛼²_𝐽(𝑟−𝐽()+𝐹)𝐽+1𝑟²𝜁1+^2𝐽𝛼²_𝐽+𝑈^′_𝑠𝑚+𝑈_𝑠𝐽+1𝑟−𝜁²_𝐽𝛼²_𝐽𝐽𝑟−1𝛼²_𝐽𝑚+𝑈_𝑠²−𝐸−𝑈⁰_𝑣²=0, (2.8) where 𝛼_𝐽=(𝐽+1)/(2𝐽+1) and 𝜁_𝐽=𝐽/(2𝐽+1). In the case of 𝑈_𝑠=0, we recover the well-known formula 𝑑²𝑑𝑟²−𝐽(𝐽+1)𝑟²+𝐸−𝑈⁰_𝑣²−𝑚²𝐹(𝑟)=0. (2.9) Within the next section, we give a brief introduction to the SUSY method.

2.1. SUSY and Shape Invariance

The basic idea of the SUSY approach is based on finding the solutions of the Riccati equation 𝑉_∓=Φ²∓Φ^′, (2.10) with 𝑉 being the potential of Schrödinger equation. If the condition 𝑉₊𝑎₀,𝑥=𝑉₋𝑎₁,𝑥+𝑅𝑎₁ (2.11) is satisfied, we call the partner Hamiltonians shape invariant. In the latter relation, 𝑎₁ is a new set of parameters uniquely determined from the old set 𝑎₀ via the mapping 𝐹∶𝑎₀↦𝑎₁=𝐹(𝑎₀) and the residual term 𝑅(𝑎₁) does not include 𝑥. Actually, the shape invariance implies that the partner potential, apart from some constant terms, is interpreted as a new partner potential 𝑉₋(𝑎₁,𝑥) associated with a new SUSY potential Φ(𝑎₁,𝑥) [34]. In such a case, the problem is simplified to a high degree and everything of interest is calculated via the elegant idea of [34–36]: 𝐸_𝑛=^𝑛_𝑠=1𝑅𝑎_𝑠,𝜙⁻_𝑛𝑎₀,𝑥=^𝑛−1_𝑠=0𝐴^†𝑎_𝑠𝐸_𝑛−𝐸_𝑠^1/2𝜙⁻₀𝑎_𝑛,𝜙,𝑥₋₀𝑎_𝑛,𝑥=𝐶exp−^𝑥₀𝑑𝑧Φ𝑎_𝑛,𝑧. (2.12)(2.13)(2.14) That is, the energy eigenvalues as well as the corresponding eigenfunctions are obtained without cumbersome algebra, the 𝐴^†_𝑠𝜕=−𝑎𝜕𝑥+Φ_𝑠,𝑥. (2.15) Hence, shape invariance gives the complete and exact information about the spectrum of the bound states of the following Hamiltonians: 𝐻_𝑠𝜕=−²𝜕𝑥²+𝑉₋𝑎_𝑠,𝑥+𝐸_𝑠. (2.16) It should be noted that the energy eigenfunctions 𝐻_𝑠𝜙⁻_𝑛−𝑠𝑎_𝑠,𝑥=𝐸_𝑛𝜙⁻_𝑛−𝑠𝑎_𝑠,𝑥,𝑛≥𝑠, (2.17) of this family of Hamiltonians are related by [3, 4] 𝜙⁻_𝑛−𝑠𝑎_𝑠=𝐴,𝑥^†𝐸_𝑛−𝐸_𝑠^1/2𝜙⁻_{()𝑛−𝑠+1}𝑎_𝑠+1,𝑥. (2.18) In other words, everything is obtained without any cumbersome algebra provided that the superpotential is simply found and the shape invariance exists.

2.2. A Famous SUSY Example

Within this section, we review an SUSY example. The results can be found in [34–36]. Choosing the superpotential as Φ(𝑥)=𝐴−𝐵exp−𝛽𝑥, (2.19) we find 𝑉₋(𝑥)=Φ²(𝑥)+Φ^′(𝑥)=𝐴²+𝐵²𝛽exp−2𝛽𝑥−2𝐵𝐴+2,𝑉exp−𝛽𝑥₊(𝑥)=Φ²(𝑥)−Φ^′(𝑥)=𝐴²+𝐵²𝛽exp−2𝛽𝑥−2𝐵𝐴−2exp−𝛽𝑥. (2.20) Obviously, the partner potentials are shape invariant via a mapping of the form 𝐴₁=𝐹(𝐴)=𝐴−𝛽. (2.21) For the energy relation, as 𝐴₁(𝐴)𝐴=𝐹=𝐴−𝛽,₂=𝐹𝐴₁=𝐴₁⋮𝐴−𝛽=𝐴−2𝛽,_𝑠=𝐹𝐴_𝑠−1=𝐴−𝑛𝛽, (2.22) using (2.11) and (2.12), we obtain 𝐸_𝑛=^𝑛_𝑠=1𝑅𝐴_𝑠=𝑅𝐴₁+𝑅𝐴₂+⋯+𝑅𝐴_𝑛=𝐴²−𝐴²₁+𝐴²₁−𝐴²₂+⋯+𝐴²_𝑛−1−𝐴²_𝑛=𝐴²−𝐴²_𝑛=𝐴²−𝐴−𝑛𝛽². (2.23) For the wave functions, based on (2.13) to (2.18), we have 𝑦^𝑠−𝑛exp−𝑦2𝐿_𝑛^{2𝑠−2𝑛}𝑦, (2.24) where 𝑦=2𝐵𝛽,𝐴exp−𝛽𝑥𝑠=𝛽. (2.25)

3. Approximate Analytical Solution of the Radial Part

We now see that the problem appears as the latter SUSY problem. Choosing the potential as 𝑈_𝑣=𝑉𝑒^{−𝑎(𝑟−𝑟0)}, we get 𝑑²𝑑𝑟²−𝐽(𝐽+1)𝑟²+𝐸²+2𝐸𝑉exp−𝑎𝑟−𝑟₀+𝑉²exp−2𝑎𝑟−𝑟₀−𝑚²𝐹(𝑟)=0. (3.1) We wish to emphasize that, on the one hand, exponential-type potentials find application in a wide class of physical sciences including cosmology [37–40], nuclear and particle physics [41–45], solid state [46–48], atomic and molecular physics [49–58], and chemical physics [59, 60]. On the other hand, the DKP equation itself, as an outstanding relativistic wave equation which well explains both microscale phenomena in particle physics and large-scale ones in cosmology, motivated us to do the present calculations. Naturally, depending on the system under consideration, the phenomenological fits are quite different. To be able to analytically solve the problem, let us use the well-known approximation [61] 1𝑟²≈𝐶₀+𝐶₁𝑒^−𝛼𝑥+𝐶₂𝑒^−2𝛼𝑥, (3.2) where𝑥=𝑟−𝑟₀𝑟₀,𝛼=𝑎𝑟₀,𝐶₀=1𝑟²₀31−𝛼+3𝛼²,𝐶₁=11𝑟²₀4𝛼−6𝛼²,𝐶₂=1𝑟²₀−1𝛼+3𝛼². (3.3a)(3.3b)(3.3c)We find 𝑑²𝑑𝑥²−𝑒^−𝛼𝑥𝐶₁𝐽(𝐽+1)−2𝐸𝑉𝑟²₀+𝑒^−2𝛼𝑥𝑉²−𝐶₂𝐽(𝐽+1)𝑟²₀+𝐶₀𝐽(𝐽+1)+𝐸²−𝑚²𝑟²₀×𝐹(𝑥)=0, (3.4) which is a Schrödinger-like equation corresponding to the Morse potential. Before proceeding further, it should be noted that the choice of the system under consideration definitely puts limitations on the approximation (3.2) via (3.3a), (3.3b) and (3.3c) to yield acceptable results. Here, however, as we intend to give a general background for related studies, we avoid focusing on a special system since the numerical data are quite different from one system, for example, a diatomic molecule, to another such as cosmological model.

Also, there exist many other papers which use Pekeris-type approximations for Schrödinger, Dirac, and KG equations for a lengthy list of potentials including the Hulthen, Eckart, Rosen-Morse, and Pöschl-Teller (see [62–69] and references therein).

In addition, the interested reader might find attractive discussion on the SUSY structure of DKP equation in [70]. Moreover, we wish to address the interesting papers [71–76] which first discussed investigation of half-line problems on the basis of full-line SUSY examples. Let us now return to our problem. Comparison of (3.4) with (2.20) indicates the correspondence 𝐵=±𝑟₀𝑉²−𝐶₂𝐽(𝐽+1),𝐴=±𝑟₀𝐶₀𝐽(),𝑟𝐽+1𝛽=²⁰𝐶₁𝐽(𝐽+1)−2𝐸𝑉𝐵−2𝐴. (3.5) Therefore, the eigenfunctions are 𝐹(𝑥)=𝐶_𝑛𝑦^𝑠−𝑛exp−𝑦2𝐿_𝑛^{2𝑠−2𝑛}𝑦,𝑦=2𝐵𝛽,𝐴exp−𝛽𝑥𝑠=𝛽, (3.6) and the energy eigenvalues satisfy 𝜀_𝑛=𝑟²₀𝐸²−𝑚²−𝜆₀=⋯=𝐶₀𝐽(𝐽+1)𝑟²₀−𝑟₀±𝐶₀𝐽()𝐶𝐽+1−𝑛¹𝐽(𝐽+1)−2𝐸𝑉±𝑉²−𝐶₂𝐽(𝐽+1)∓2𝐶₀𝐽(𝐽+1)². (3.7)

4. Conclusion

We obtained approximate analytical solutions of the DKP equation for an exponential term which in many cases provides more exact solutions than linear or quadratic terms. The results are applicable to many branches of physics from cosmology to particle physics.