# Quasi-Analytical Solutions of DKP Equation under the Deng-Fan Interaction

Authors: S. Hassanabadi; A. A. Rajabi; B. H. Yazarloo; S. Zarrinkamar; H. Hassanabadi
Journal: Advances in High Energy Physics
 (2012)

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## Abstract

Quasianalytical solutions of Duffin-Kemmer-Petiau equation under scalar and vector Deng-Fan potentials are reported via a novel ansatz.


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## Body

## 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 scalarUs  and vector Uvo  interactions is
(2.1)(β.p→c+mc2+Us+β0Uvo)ψ(r→)=β0Eψ(r→),
where
(2.2)ψ(r→)=(ψupperiψlower),
and the upper and lower components, respectively, are
(2.3)ψupper≡(ϕφ),ψlower≡(A1A2A3).The engaged matrices are(2.4a)β0=(θ0~0-T0),βi=(0~ρi-ρTi0),
with 0~, 0- and 0, respectively, being 2×2,2×3,3×3  zero matrices
(2.4b)θ=(0110),ρ1=(-100000),ρ2=(0-10000),ρ3=(00-1000).The equation, in (3+0)-dimensions, is written as [1–4]
(2.5)(mc2+Us)ϕ=(E-Uvo)φ+ℏc∇→.A→,∇→ϕ=(mc2+Us)A→,(mc2+Us)φ=(E-Uvo)ϕ,
where  A→=(A1,A2,A3).  In (2.4a) and (2.4b), ψ is a simultaneous eigenfunction of J2  and  J3, that is,
(2.6)J2(ψupperψlower)=(L2ψupper(L+S)2ψlower)=J(J+1)(ψupperψlower),J3(ψupperψlower)=(L3ψupper(L3+s3)ψlower)=M(ψupperψlower),
and the general solution is considered as
(2.7)ψJM(r)=(fnJ(r)YJM(Ω)gnJ(r)YJM(Ω)i∑LhnJL(r)YJL1M(Ω)),
where spherical harmonics YJM(Ω) are of order J, YJL1M(Ω) are the normalized vector spherical harmonics, and fnJ,gnJ,  and hnJL  represent the radial wavefunctions. The above equations yield the following coupled differential equations [30–38]:
(2.8)(En,J-Uv0)Fn,J(r)=(mc2+Us)Gn,J(r),(dFn,J(r)dr-J+1rFn,J(r))=-1αJ(mc2+Us)H1,n,J(r),(dFn,J(r)dr+JrFn,J(r))=1ζJ(mc2+Us)H-1,n,J(r),-αJ(dH1,n,J(r)dr+J+1rH1,n,J(r))+ζ(dH-1,n,J(r)dr-JrH-1,n,J(r))=1ℏc((mc2+Us)Fn,J(r)-(En,J-Uv0)Gn,J(r)),
which give [39–41]
(2.9)d2Fn,J(r)dr2[1+ζJ2αJ2]-dFn,J(r)dr[Us′(m+Us)(1+ζJ2αJ2)]+Fn,J(r)[-J(J+1)r2(1+ζJ2αJ2)+Us′(m+Us)(J+1r-ζJ2αJ2Jr)-1αJ2((m+Us)2-(En,J-Uv0)2)]=0,
Where αJ=(J+1)/(2J+1),fnJ(r)=F(r)/r,gnJ(r)=G(r)/r,hnJJ±1=H±1/r  and  ζJ=J/(2J+1). When Us=0, we recover the well-known formula [30–32]
(2.10)(d2dr2-J(J+1)r2+(En,J-Uv0)2-m2)Fn,J(r)=0.We consider the Deng-Fan vector and scalar potentials [42–46]
(2.11)Us=V0+V1eαr-1+V2(eαr-1)2,Uv0=u0+u1eαr-1+u2(eαr-1)2.
By a change of variable of the form
(2.12)Fn,J(r)=m+Usun,J(r),(2.9) is written as
(2.13)[d2dr2+Us′′21m+Us-34Us′2(m+Us)2-J(J+1)r2+CUs′A(m+Us)1r-1AαJ2[(m+Us)2(En,J-Uv0)2]]un,J(r)=0,
where
(2.14)A=1+ζJ2αJ2,C=J+1-ζJ2αJ2J.From (2.11), (2.13) can be written as
(2.15)[d2dr2+12(-α2V1eαr(eαr-1)2+2α2V1e2αr(eαr-1)3-2α2V2eαr(eαr-1)3+6α2V2e2αr(eαr-1)4)×1((1/m)+V0+(V1/(eαr-1))+(V2/(eαr-1)2))-341(m+V0+(V1/(eαr-1))+(V2/(eαr-1)2))2(-αV1eαr(eαr-1)2-2αV2eαr(eαr-1)3)2-J(J+1)r2+CA(m+V0+(V1/(eαr-1))+(V2/(eαr-1)2))(-αV1eαr(eαr-1)2-2αV2eαr(eαr-1)3)1r-1AαJ2[(m+V0+V1eαr-1+V2(eαr-1)2)2-(En,J-u0-u1eαr-1-u2(eαr-1)2)2]]un,J(r)=0.Here, we use the following approximations for the centrifugal term [47]:(2.16a)1r2≈α2(eαr-1)2,(2.16b)1r≈αeαr-1.Equation (2.16a) is a quite logical alternative for α<0.1  (see Figure 1), and (2.16a) and (2.16b) brings (2.15) into the form
(2.17)[{-1AαJ2{(m+V0+V1eαr-1+V2(eαr-1)2)2-(En,J-u0-u1eαr-1-u2(eαr-1)2)2}}d2dr2+(-α2V12eαr(eαr-1)2+α2V1e2αr(eαr-1)3-α2V2eαr(eαr-1)3+3α2V2e2αr(eαr-1)4)×1m+V0+(V1/(eαr-1))+(V2/(eαr-1)2)+(-34α2V12e2αr(eαr-1)4-3α2V22e2αr(eαr-1)6-3α2V1V2e2αr(eαr-1)5)×1(m+V0+(V1/(eαr-1))+(V2/(eαr-1)2))2-J(J+1)α2(1eαr-1)2+CαA(1eαr-1)(-αV1eαr(eαr-1)2-2αV2eαr(eαr-1)3)×1m+V0+(V1/(eαr-1))+(V2/(eαr-1)2)-1AαJ2{(m+V0+V1eαr-1+V2(eαr-1)2)2-(En,J-u0-u1eαr-1-u2(eαr-1)2)2}]un,J(r)=0.Figure 1
1
/
r
2and its approximation for different value of α.By introducingz=eαr and making the transformation  un,J(z)=ϕn,J(z)/(z), we obtain
[d2dz2+141z2+(-V12(m+V0)1z+V1(m+V0)1z-1-V2(m+V0)1z(z-1)+3V2(m+V0)1(z-1)2)×(1(z+z1)(z+z2))+(-3V124(m+V0)2-3V22(m+V0)21(z-1)2-3V2V1(m+V0)21z-1)(1(z+z1)(z+z2))2-J(J+1)1z2(z-1)2+1(z+z1)(z+z2){-CV1A(m+V0)1z(z-1)-2CV2A(m+V0)1z(z-1)2}-1Aα2αJ2((m+V0)2(En,J-u0)2)1z2-1Aα2αJ2(V12-u12+2V2(m+V0)+2u2(En,J-u0))×1z2(z-1)2-1Aα2αJ2(2V1(m+V0)+2u1(En,J-u0))1z2(z-1)-1Aα2αJ2(2V2V1-2u1u2)1z2(z-1)3-1Aα2αJ2(V22-u22)1z2(z-1)4]ϕn,J(z)=0,
which after decomposition of fractions gives
(2.19)[d2dz2+λz+ωz2+sz-1+q(z-1)2+p(z-1)3+N(z-1)4+gz+z1+h(z+z1)2+χz+z2+κ(z-1)2]ϕn,J(z)=0
with
z1=12(V1-2(m+V0)m+V0+(V1-2(m+V0)m+V0)2-4m+V0-V1+V2m+V0),z2=12(V1-2(m+V0)m+V0-(V1-2(m+V0)m+V0)2-4m+V0-V1+V2m+V0),λ=-V12z1z2(m+V0)+1z1z2(V2m+V0+CV1A(m+V0))+2V1(m+V0)AαJ2α2+2u1(En,J-u0)AαJ2α2-2J(J+1)-2AαJ2α2(V12-u12+2V2(m+V0)+2u2(En,J-u0))+3AαJ2α2(2V1V2-2u1u2)-4AαJ2α2(V22-u22)-2CV2Az1z2(m+V0),ω=14-1AαJ2α2((m+V0)2-(En,J-u0)2)+2V1(m+V0)AαJ2α2+2u1(En,J-u0)AαJ2α2-J(J+1)-1AαJ2α2(V12-u12+2V2(m+V0)+2u2(En,J-u0))+1AαJ2α2(2V1V2-2u1u2)-1AαJ2α2(V22-u22),S=V1(m+V0)(z1+1)(z2+1)+1(z1+1)(z2+1)(-V2m+V0-CV1A(m+V0))-3V2(z1+z2+2)(z1+1)2(z2+1)2(m+V0)+6V22(z1+z2+2)(z1+1)3(z2+1)3(m+V0)2-3V1V2(m+V0)2(z1+1)2(z2+1)2-2V1(m+V0)AαJ2α2-2u1(En,J-u0)AαJ2α2+2J(J+1)+2AαJ2α2(V12-u12+2V2(m+V0)+2u2(En,J-u0))-3AαJ2α2(2V1V2-2u1u2)+4AαJ2α2(V22-u22)+2CV2(z1z2+2z1+2z2+3)A(m+V0)(z1+1)2(z2+1)2,q=3V2(m+V0)(z1+1)(z2+1)-3V22(m+V0)2(z1+1)2(z2+1)2-J(J+1)-1AαJ2α2(V12-u12+2V2(m+V0)+2u2(En,J-u0))+2AαJ2α2(2V1V2-2u1u2)-3AαJ2α2(V22-u22)-2CV2A(m+V0)(z1+1)(z2+1),g=-V12z1(z1-z2)(m+V0)+V1(z1+1)(z1-z2)(m+V0)+1z1(z1-z2)(z1+1)(V2m+V0+CV1A(m+V0))-3V2(m+V0)(z1+1)2(z1-z2)-6V124(m+V0)2(z1-z2)3-6V22(-z2+2z1+1)(m+V0)2(z1-z2)3(z1+1)3+3V2V1(-z2+3z1+2)(m+V0)2(z1-z2)3(z1+1)2-2CV2Az1(m+V0)(z1-z2)(z1+1)2,p=-1AαJ2α2(2V1V2-2u1u2)+2AαJ2α2(V22-u22),N=-1AαJ2α2(V22-u22),h=-3V124(m+V0)2(z2-z1)2-3V22(m+V0)2(z1-z2)2(z1+1)2+3V1V2(m+V0)2(z1-z2)2(z1+1),χ=V12z2(z1-z2)(m+V0)+V1(z2+1)(z2-z1)(m+V0)+1z2(z2-z1)(z2+1)(V2m+V0+CV1A(m+V0))-3V2(m+V0)(z2+1)2(z2-z1)+6V124(m+V0)2(z1-z2)3-6V22(-z1+2z2+1)(m+V0)2(z2-z1)3(z2+1)3+3V2V1(-z1+3z2+2)(m+V0)2(z2-z1)3(z2+1)2-2CV2Az2(m+V0)(z2-z1)(z2+1)2,κ=-3V124(m+V0)2(z2-z1)2-3V22(m+V0)2(z1-z2)2(z2+1)2+3V1V2(m+V0)2(z1-z2)2(z2+1).Let us now consider an ansatz of the form(2.21)ϕn,J(z)=fn(z)exp⁡(gJ(z)),
with
(2.22)fn(z)={1,n=0,∏i=1n(z-αin),n>0,gJ(z)=γln⁡(z)+βln⁡(z-1)+ηz-1+ξln⁡(z+z1)+δln⁡(z+z2).
Substitution of the latter in (2.20) yields
ϕ′′0,J(z)={+(δ2-δ)1(z+z2)2}[(-2βγ-2γη+2γξz1+2γδz2)1z+(γ2-γ)1z2+(2βγ+2γη+2βξ1+z1+2βδ1+z2+2ηξ(1+z1)2+2ηδ(1+z2)2)1z-1+(β2-β-2γη-2ηξ1+z1-2ηδ1+z2)1(z-1)2+(2η-2βη)1(z-1)3+η21(z-1)4+(-2γξz1-2βξ1+z1-2ηξ(1+z1)2+2ξδz2-z1)1z+z1+(ξ2-ξ)1(z+z1)2+(-2δγz2-2βδ1+z2-2ηδ(1+z2)2-2ξδz2-z1)1z+z2+(δ2-δ)1(z+z2)2]ϕ0,J(z).For the sake of simplicity, here we consider only the nodless solutions. Substitution of the proposed ansatz solution and equating the corresponding powers on both sides give(2.24)-2βγ-2γη+2γξz1+2γδz2=-λ,γ2-γ=-ω,2βγ+2γη+2βξ1+z1+2βδ1+z2+2ηξ(1+z1)2+2ηδ(1+z2)2=-S,β2-β-2γη-2ηξ1+z1-2ηδ1+z2=-q,2η-2βη=-p,η2=-N-2γξz1-2βξ1+z1-2ηξ(1+z1)2+2ξδz2-z1=-g,ξ2-ξ=-h,-2δγz2-2βδ1+z2-2ηδ(1+z2)2-2ξδz2-z1=-χ,δ2-δ=-κ.
For the fixed values of  V1,u1,α,andm, in particular, the system of ten equations (2.24) determines the sets of variables E0,J,V0,V2,u0,u2,γ,β,ξ,η, and δ. Therefore, the spectrum and eigenfunctions of the system are easily obtained for a particular system. For the higher states, the mathematical process is more cumbersome and complicated but can be followed by the same token we did here, that is, by choosing f1(z)=(z-α11)for the first node,  f2(z)=(z-α12)(z-α22)  for the second node, and so forth.


## 3. Conclusion

Motivation behind our study was the high number of spin-zero relativistic systems that we frequently face as well as the attractive structure of the Deng-Fan potential. The corresponding ordinary differential equation was too complicated to be solved by common analytical techniques. We, therefore, performed some novel transformations and applied an acceptable approximation to the centrifugal term. We next proposed an interesting physical ansatz solution by which we were able to find a quasi-analytical solution. Our results are particularly useful in particle and nuclear physics and can be directly used after prerequisite fits performed.



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