The Dunkl–Duffin–Kemmer–Petiau Oscillator

Abstract

In the context of the Dunkl derivative, we present the Dunkl–Duffin–Kemmer–Petiau oscillator (DDKPO) model of spin 0 and 1. For the spin 0 case , the exact solutions are determined, the wave functions are obtained using Cartesian and polar coordinates in the general form for the different eigenvalues of the reflection operators, and the corresponding spectrum energies are deduced. For the spin 1 case, the problem is complicated by the direct method, we obtain a differential equation of order four and we limit ourselves only to the study of the non-relativistic case.

Appendix

We expose an explicit representation of $\beta ^{\mu }$ matrices ,

• For the case of spin 0, the form of the $\beta ^{\mu }$ is given by

  $$\begin{aligned} \beta ^{0}=\left( \begin{array} [c]{ll} \theta &{} \quad \mathbf {0}\\ \mathbf {0} &{} \quad \mathbf {0} \end{array} \right) ,\ \text {and }\ \ \beta ^{i}=\left( \begin{array} [c]{ll} \mathbf {0} &{} \quad \rho ^{i}\\ -\rho _{T}^{i} &{} \quad \mathbf {0} \end{array} \right) ,i=1,2,3 \end{aligned}$$

  (63)

  where

  $$\begin{aligned} \ \ \theta =\left( \begin{array} [c]{ll} 0 &{} \quad 1\\ 1 &{} \quad 0 \end{array} \right) \end{aligned}$$

  (64)

  and

  $$\begin{aligned} \rho ^{1}=\left( \begin{array} [c]{ccc} -1 &{} \quad 0 &{} \quad 0\\ 0 &{} \quad 0 &{} \quad 0 \end{array} \right) ,\rho ^{2}=\left( \begin{array} [c]{ccc} 0 &{} -1 &{} 0\\ 0 &{} 0 &{} 0 \end{array} \right) ,\rho ^{3}=\left( \begin{array} [c]{ccc} 0 &{} \quad 0 &{} \quad -1\\ 0 &{} \quad 0 &{} \quad 0 \end{array} \right) \end{aligned}$$

  (65)

  with $\rho _{T}$ denotes the transposed matrix of $\rho$ and $\mathbf {0}$ is the zero matrix.

• For the case of spin 1,  the form of the $\beta ^{\mu }$ is given as

  $$\begin{aligned} \beta ^{0}=\left( \begin{array} [c]{cccc} 0 &{} \quad \overline{0} &{} \quad \overline{0} &{} \quad \overline{0}\\ \overline{0}^{T} &{} \quad \mathbf {0} &{} \quad \mathbf {1} &{} \quad \mathbf {0}\\ \overline{0}^{T} &{} \quad \mathbf {1} &{} \quad \mathbf {0} &{} \quad \mathbf {0}\\ \overline{0}^{T} &{} \quad \mathbf {0} &{} \quad \mathbf {0} &{} \quad \mathbf {0} \end{array} \right) \text { and }\ \ \beta ^{i}=\left( \begin{array} [c]{cccc} 0 &{} \quad \overline{0} &{} \quad e_{i} &{} \quad \overline{0}\\ \overline{0}^{T} &{} \quad \mathbf {0} &{} \quad \mathbf {0} &{} -is_{i}\\ -e_{i}^{T} &{} \quad \mathbf {0} &{} \quad \mathbf {0} &{} \quad \mathbf {0}\\ \overline{0}^{T} &{} \quad -is_{i} &{} \quad \mathbf {0} &{} \quad \mathbf {0} \end{array} \right) i=1,2,3 \end{aligned}$$

  (66)

  where the matrices $s_{i}$ are the standard $(3\times 3)$ spin 1 matrices and, $\mathbf {0}$ and $\mathbf {1}$ denote respectively the zero matrix and the unity matrix. The matrices $\overline{0}$ and $e_{i}$ are given as

  $$\begin{aligned} \overline{0}=\left( \begin{array} [c]{ccc} 0&\quad 0&\quad 0 \end{array} \right) ,e_{1}=\left( \begin{array} [c]{ccc} 1&\quad 0&\quad 0 \end{array} \right) ,e_{2}=\left( \begin{array} [c]{ccc} 0&\quad 1&\quad 0 \end{array} \right) \text { and }e_{3}=\left( \begin{array} [c]{ccc} 0&\quad 0&\quad 1 \end{array} \right) . \end{aligned}$$

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