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

Abstract

In this work, we study Duffin-Kemmer-Petiau equation in the presence of coulomb and harmonic oscillator potentials in (1+3)-dimension for spin-one particles and we obtain energy eigenvalues and corresponding eigenfunctions.

1. Introduction

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 [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 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.

2. DKP Equation

The DKP equation in free field is given by (in natural units ℏ=𝑐=1) [1–3] 𝑖𝛽^𝜇𝜕_𝜇−𝑚₀Ψ=0 (2.1)𝛽^𝜇 are the DKP matrices which are satisfied in this algebra: 𝛽^𝜇𝛽^𝜈𝛽^𝜆+𝛽^𝜆𝛽^𝜈𝛽^𝜇=𝑔^𝜇𝜈𝛽^𝜆+𝑔^𝜆𝜈𝛽^𝜇, (2.2) where 𝑔^𝜇𝜈=diag(1,−1,−1,−1) and (𝑔^𝜇𝜈)²=1, being the metric tensor of Minkowski space-time.

For the spin-one case, 𝛽^𝜇 matrices are 𝛽⁰=00000^𝑇0𝐼00^𝑇𝐼000^𝑇,𝛽000_𝑖=00𝑒_𝑖00^𝑇00−𝑖𝑆_𝑖−𝑒_𝑖^𝑇0000^𝑇−𝑖𝑆_𝑖00 (2.3) with 𝑆 matrices being 3×3 ones, (𝑆_𝑖)_𝑗𝑘=−𝑖𝜀_𝑖𝑗𝑘 where 𝜀_𝑖𝑗𝑘 is 1, −1, 0 for an even permutation, an odd permutation, and repeated indices, respectively. 𝑒_𝑖 Matrices are 1×3, (𝑒_𝑖)_1𝑗=𝛿_𝑖𝑗, that is, 𝑒₁=(1,0,0),𝑒₂=(0,1,0),𝑒₃=(0,0,1) (2.4)𝐼 and 0, respectively,represent unit and null 3×3 matrices and 0s are 1×3 ones [30].

3. DKP Equation in Three-Dimensional Space-Time

Furthermore, for an elastic scattering, the interaction is [31] 𝑈=𝑆(𝑟)+𝑃𝑆_𝜇(𝑟)+𝛽⁰𝑉(𝑟)+𝛽⁰𝑃𝑉_𝑃(𝑟), (3.1) 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 𝑖𝛽^𝜇𝜕_𝜇−𝑚₀−𝛽⁰𝑃𝑉(𝑟)Ψ=0. (3.2) As usual, a solution of the following form removes the time component Ψ𝑥,𝑦,𝑧,𝑡=exp−𝑖𝐸_𝑛,𝑙𝑡𝜓_𝑛,𝑙𝑥,𝑦,𝑧 (3.3) as the problem is considered in one spatial dimension, we consider one quantum number and write the wave function as 𝜓^𝑇_𝑛,𝑙𝑥,𝑦,𝑧=𝜑⁽¹⁾_𝑛,𝑙,𝜑⁽²⁾_𝑛,𝑙,𝜑⁽³⁾_𝑛,𝑙,𝜑⁽⁴⁾_𝑛,𝑙,𝜑⁽⁵⁾_𝑛,𝑙,𝜑⁽⁶⁾_𝑛,𝑙,𝜑⁽⁷⁾_𝑛,𝑙,𝜑⁽⁸⁾_𝑛,𝑙,𝜑⁽⁹⁾_𝑛,𝑙,𝜑⁽¹⁰⁾_𝑛,𝑙^𝑇. (3.4) So we choose, 𝜑⁽¹⁾_𝑛,𝑙=𝑖𝜙_𝑛,𝑙,^→𝐹=𝜑⁽²⁾_𝑛,𝑙,𝜑⁽³⁾_𝑛,𝑙,𝜑⁽⁴⁾_𝑛,𝑙,𝐺^→=𝜑⁽⁵⁾_𝑛,𝑙,𝜑⁽⁶⁾_𝑛,𝑙,𝜑⁽⁷⁾_𝑛,𝑙,𝐻^→=𝜑⁽⁸⁾_𝑛,𝑙,𝜑⁽⁹⁾_𝑛,𝑙,𝜑⁽¹⁰⁾_𝑛,𝑙. (3.5) Substituting of the above relations in (3.6), we have determined ten coupled equations as follows 𝑖𝜕𝐹⁽²⁾_𝑛,𝑙−𝜕𝑥𝜕𝐹⁽¹⁾_𝑛,𝑙𝜕𝑦=𝑚₀𝐻⁽³⁾_𝑛,𝑙,𝑖𝜕𝐹⁽¹⁾_𝑛,𝑙−𝜕𝑧𝜕𝐹⁽³⁾_𝑛,𝑙𝜕𝑥=𝑚₀𝐻⁽²⁾_𝑛,𝑙,𝑖𝜕𝐹⁽³⁾_𝑛,𝑙−𝜕𝑦𝜕𝐹⁽²⁾_𝑛,𝑙𝜕𝑧=𝑚₀𝐻⁽¹⁾_𝑛,𝑙,𝜕𝐺⁽³⁾_𝑛,𝑙+𝜕𝑧𝜕𝐺⁽²⁾_𝑛,𝑙+𝜕𝑦𝜕𝐺⁽¹⁾_𝑛,𝑙𝜕𝑥=𝑚₀𝜙_𝑛,𝑙,𝐸_𝑛,𝑙𝐺⁽¹⁾_𝑛,𝑙+𝑖𝜕𝐻⁽³⁾_𝑛,𝑙−𝜕𝑦𝜕𝐻⁽²⁾_𝑛,𝑙𝜕𝑧=𝑚₀𝐹⁽¹⁾_𝑛,𝑙,𝐸_𝑛,𝑙𝐺⁽²⁾_𝑛,𝑙+𝑖𝜕𝐻⁽¹⁾_𝑛,𝑙−𝜕𝑧𝜕𝐻⁽³⁾_𝑛,𝑙𝜕𝑥=𝑚₀𝐹⁽²⁾_𝑛,𝑙,𝐸_𝑛,𝑙𝐺⁽³⁾_𝑛,𝑙+𝑖𝜕𝐻⁽²⁾_𝑛,𝑙−𝜕𝑥𝜕𝐻⁽¹⁾_𝑛,𝑙𝜕𝑦=𝑚₀𝐹⁽³⁾_𝑛,𝑙,𝐸_𝑛,𝑙−𝑉(𝑟)𝐹⁽¹⁾_𝑛,𝑙+𝜕𝜑𝜕𝑥=𝑚₀𝐺⁽¹⁾_𝑛,𝑙,𝐸_𝑛,𝑙−𝑉(𝑟)𝐹⁽²⁾_𝑛,𝑙+𝜕𝜑𝜕𝑦=𝑚₀𝐺⁽²⁾_𝑛,𝑙,𝐸_𝑛,𝑙−𝑉(𝑟)𝐹⁽³⁾_𝑛,𝑙+𝜕𝜑𝜕𝑧=𝑚₀𝐺⁽³⁾_𝑛,𝑙. (3.6)(3.7)(3.8)(3.9)(3.10)(3.11)(3.12)(3.13)(3.14)(3.15) Combining the above equations, we have 𝑖∇×^→𝐹=𝑚𝐻₀^→,∇⋅𝐺^→=𝑚₀𝜙_𝑛,𝑙,𝐸^→_𝑛,𝑙𝐺+𝑖∇×𝐻^→=𝑚₀^→𝐹,𝐸_𝑛,𝑙−𝑉(𝑟)^→𝐹+∇𝜙_𝑛,𝑙=𝑚𝐺₀^→. (3.16) In obtaining the last result as follows, we combine the above equations: 𝐸_𝑛,𝑙𝐸_𝑛,𝑙−𝑉(𝑟)−𝑚²₀^→𝐹−∇×∇×^→𝐹+∇∇⋅^→𝐹=0,∇×∇×^→𝐹=∇∇⋅^→𝐹−∇^2→𝐹. (3.17) Then, 𝐸_𝑛,𝑙𝐸_𝑛,𝑙−𝑉(𝑟)−𝑚²₀^→𝐹+∇^2→𝐹=0. (3.18) Moreover, ∇²=𝜕²𝜕𝑟²+2𝑟𝜕−𝐿𝜕𝑟𝑟²², (3.19) So we have, 𝑑²𝑑𝑟²+2𝑟𝑑𝑑𝑟+𝐸²_𝑛,𝑙−𝐸_𝑛,𝑙𝑉(𝑟)−𝑚²₀−𝑙(𝑙+1)𝑟^2→𝐹=0. (3.20) Thus, if we suppose that 𝑉(𝑟)=0, this equation reduces to 𝑃²=𝐸²_𝑛,𝑙−𝑚²₀.

4. Exact Solutions of DKP Equation under Coulomb Potential

Now for deriving eigenvalues of energy and wave functions of (3.20), we have 𝑑^2→𝐹𝑑𝑟²+2𝑟𝑑^→𝐹+1𝑑𝑟𝑟²𝐸²_𝑛,𝑙𝑟²−𝐸_𝑛,𝑙𝑉₀𝑟−𝑚²₀𝑟²−𝑙(𝑙+1)^→𝐹=0. (4.1) We determine the energy eigenvalues from (4.2) as follows: (2𝑛+1)𝑚²₀−𝐸²_𝑛,𝑙+𝐸_𝑛,𝑙𝑉₀+214+𝑙(𝑙+1)𝑚²₀−𝐸²_𝑛,𝑙=0. (4.2) So 𝐸_𝑛,𝑙=±2𝑚₀(𝑛+𝑙+1)𝑉²₀+4(𝑛+𝑙+1)²,𝜑⁽²⁾_𝑛,𝑙^→𝑟=𝑁_𝑛,𝑙𝑟^𝑙𝑒^{−𝑚20−𝐸2𝑛,𝑙𝑟}𝐿_𝑛^2𝑙+12𝑚²₀−𝐸²_𝑛,𝑙𝑟. (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 𝑉₀ 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.

Table 1 

Energy eigenvalues for coulomb interaction with (𝑚₀=1, 𝑉₀=−0.1).

Figure 1 

Wave functions for coulomb interaction versus 𝑟 for (𝑚0=1, 𝑉0=−0.1, and 𝑙=0).

Figure 2 

𝐸𝑛,𝑙 versus 𝑉0.

5. Exact Solutions of DKP Equation under Harmonic Oscillator Potential

In this section, we study (3.20) with harmonic oscillator potential, 𝑑²𝑑𝑟²+2𝑟𝑑𝑑𝑟+𝐸²_𝑛,𝑙−𝐸_𝑛,𝑙𝛼𝑟²−𝑚²₀−𝑙(𝑙+1)𝑟^2→𝐹=0. (5.1) So the energy eigenvalues can be derived from the below equation: 3𝑛+4+𝑙2𝛼𝐸_𝑛,𝑙+𝑚²₀−𝐸²_𝑛,𝑙4=0, (5.2) and the wave function is 𝜑⁽²⁾_𝑛,𝑙=𝑁_𝑛,𝑙𝑟^𝑙𝑒^{−(𝛼𝐸𝑛,𝑙/2)𝑟2}𝐿_𝑛^(𝑙+1/2)𝛼𝐸_𝑛,𝑙𝑟². (5.3)

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.

Table 2 

Energy eigenvalues for harmonic oscillator with (𝑚₀=1, 𝛼=0.1).

Figure 3 

Wave functions for harmonic oscillator potential versus 𝑟 for (𝑚0=1, 𝛼=0.1, and 𝑙=0).

Figure 4 

Energy eigenvalues versus 𝛼.

6. 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 𝑉₀ and 𝛼, we have displayed in Figures 2 and 4 that with decreasing of the 𝛼 and 𝑉₀ 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.

Acknowledgment

The authors wish to give our sincere gratitude to the referees for their instructive comments and careful reading of the paper.