Spin-One DKP Equation in the Presence of Coulomb and Harmonic Oscillator Interactions in ( 1 + 3 )-Dimension

In this work, we study Duffin-Kemmer-Petiau equation in the presence of coulomb and harmonic oscillator potentials in ()-dimension for spin-one particles and we obtain energy eigenvalues and corresponding eigenfunctions.


Introduction
The Duffin-Kemmer-Petiau Duffin, 1938;Kemmer, 1938;Petiau, 1936 equation is a firstorder relativistic wave equation for spin-zero and one bosons 1-3 .It is similar with Dirac equation in which we replace the algebra of the gamma matrices by beta matrices.More recently, there have been a renewed interest in DKP equation; particularly it has been applied to QCD large and short distances by Gribov 4 , to covariant Hamiltonian dynamics by Kanatchikov 5 and has been generalized to curve space-time by Red'kov 6 and Lunardi et al. 7 .In addition, the relativistic model of α-nucleus elastic scattering where they have been treated by the formalism of the DKP theory 8 and covariant Hamiltonian 9 in the casual approach 10, 11 and there has been an increasing interest on the DKP oscillator 12-17 .Recently, many articles have been devoted to investigate DKP theory under different types of potential; hence, we can cite the following 18-29 .Since the wave function includes all the necessary information about considering systems, the energy eigenvalues and corresponding eigenfunctions between interaction systems in relativistic quantum mechanics Advances in High Energy Physics and in nonrelativistic quantum mechanics are studied more efficiently in recent years.In this study, we have investigated DKP equation with coulomb and harmonic oscillator potentials in 1 3 -dimension.

DKP Equation
The DKP equation in free field is given by in natural units c 1 1-3 β μ are the DKP matrices which are satisfied in this algebra: where g μν diag 1, −1, −1, −1 and g μν 2 1, being the metric tensor of Minkowski spacetime.

DKP Equation in Three-Dimensional Space-Time
Furthermore, for an elastic scattering, the interaction is 31 where each term has a specific Lorentz character.Two Lorentz vectors may be written as β μ and Pβ μ by assuming rotational invariance and parity conservation.DKP matrices have three irreducible representations: one-dimension representation where is trivial, five-dimension representation that is for spin-zero particles, and ten-dimension representation that is for spin-one particles 1-3 .
The DKP equation in the presence of interaction is written as As usual, a solution of the following form removes the time component as the problem is considered in one spatial dimension, we consider one quantum number and write the wave function as

3.5
Substituting of the above relations in 3.6 , we have determined ten coupled equations as follows 3.9 Advances in High Energy Physics

3.15
Combining the above equations, we have

3.16
In obtaining the last result as follows, we combine the above equations:

3.17
Then, So we have, Thus, if we suppose that V r 0, this equation reduces to P 2 E 2 n,l − m 2 0 .

Exact Solutions of DKP Equation under Coulomb Potential
Now for deriving eigenvalues of energy and wave functions of 3.20 , we have We determine the energy eigenvalues from 4.2 as follows:

4.3
We have plotted wave function versus r in Figure 1.The energy eigenvalues for different l are reported in Table 1 to give a better view of the obtained results.Also, we have displayed the energy eigenvalues versus V 0 in Figure 2. We can now demonstrate that the spectra given in Table 1 present the pattern appearing in supersymmetry quantum mechanics 32 where the levels of energy E n,l E n 1,l−1 are degenerate.

Exact Solutions of DKP Equation under Harmonic Oscillator Potential
In this section, we study 3.20 with harmonic oscillator potential, n,l (r)   We have plotted wave functions versus r in Figure 3.The energy eigenvalues for different l are shown in Table 2, and in Figure 4 we have plotted energy eigenvalues versus α.

Conclusion
We have investigated DKP equation in the presence of coulomb and harmonic oscillator potentials in three-dimensional of space-time for spin-one particles.Thus, we have derived   energy eigenvalues and wavefunctions where we have plotted the wavefunctions versus r also the energy eigenvalues have been determined in Tables 1 and 2. In effect with increasing the quantum numbers, the values of energy increased.In order to describe the behavior of the energy versus V 0 and α, we have displayed in Figures 2 and 4 that with decreasing of the α and V 0 the values of energy tend to one point.Furthermore we have discussed the solutions of the DKP equation.Hence, our results are useful in the study of relativistic spin-one particles.