Research Article | Open Access
H. Hassanabadi, Z. Molaee, M. Ghominejad, S. Zarrinkamar, "Spin-One DKP Equation in the Presence of Coulomb and Harmonic Oscillator Interactions in (1 + 3)-Dimension", Advances in High Energy Physics, vol. 2012, Article ID 489641, 10 pages, 2012. https://doi.org/10.1155/2012/489641
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.
The Duffin-Kemmer-Petiau (Duffin, 1938; Kemmer, 1938; Petiau, 1936) equation is a first-order 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 , to covariant Hamiltonian dynamics by Kanatchikov  and has been generalized to curve space-time by Red’kov  and Lunardi et al. . In addition, the relativistic model of -nucleus elastic scattering where they have been treated by the formalism of the DKP theory  and covariant Hamiltonian  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 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 ()-dimension.
2. DKP Equation
For the spin-one case, matrices are with matrices being ones, where is 1, −1, 0 for an even permutation, an odd permutation, and repeated indices, respectively. Matrices are , , that is, and , respectively,represent unit and null matrices and s are ones .
3. DKP Equation in Three-Dimensional Space-Time
Furthermore, for an elastic scattering, the interaction is  where each term has a specific Lorentz character. Two Lorentz vectors may be written as and 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 So we choose, Substituting of the above relations in (3.6), we have determined ten coupled equations as follows Combining the above equations, we have In obtaining the last result as follows, we combine the above equations: Then, Moreover, So we have, Thus, if we suppose that , this equation reduces to .
4. 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: So We have plotted wave function versus in Figure 1. The energy eigenvalues for different are reported in Table 1 to give a better view of the obtained results. Also, we have displayed the energy eigenvalues versus in Figure 2. We can now demonstrate that the spectra given in Table 1 present the pattern appearing in supersymmetry quantum mechanics  where the levels of energy are degenerate.
5. Exact Solutions of DKP Equation under Harmonic Oscillator Potential
In this section, we study (3.20) with harmonic oscillator potential, So the energy eigenvalues can be derived from the below equation: and the wave function is
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 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 and , we have displayed in Figures 2 and 4 that with decreasing of the and 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.
The authors wish to give our sincere gratitude to the referees for their instructive comments and careful reading of the paper.
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