The Statistical Properties of Spin-One DKP Oscillator under an External Magnetic Field in Noncommutative Space

The first-order relativistic Duffin-Kemmer-Petiau (DKP) equation has been frequently used to study interactions of spin-one/zero particles [1] in various studies of particle physics. As typical examples, DKP equation has been also used to study deuteron-nucleus scattering [2], α-nucleus elastic scattering, meson-nuclear interaction [3, 4], and the scattering of K+ nucleus in the presence of an AharonovBohm potential [5]. On the other hand, we have motivating evidences that we have to work on the noncommutative (NC) formulation of quantum mechanics where the position or momentum operators have nonvanishing commutations. During the past years, some important problems have been investigated in the NC formalism [6–11]. Typical examples of such cases include an open string attached to D-branes in the presence of background B-field inducing noncommutativity in its both end points [12–15], the quantum Hall effect [16] and the DKP oscillator [17].The interface with solid-state physics and semiconductor theories are also studied in [17–20]. Due to the physical significance of the magnetic field, Pacheco et al. studied the thermal properties of the onedimensional Dirac oscillator problem [21, 22]. Although there are very recent papers which consider the NC DKP oscillator, no one, to our best knowledge, has discussed the corresponding thermodynamical properties. In our work, we investigate the spin-one DKP oscillator with the magnetic field in NC phase-space and search for the corresponding thermodynamical properties including partition function Z, mean internal energy U, Helmholtz free energy F, specific heat capacity CV, and entropy S.


Introduction
The first-order relativistic Duffin-Kemmer-Petiau (DKP) equation has been frequently used to study interactions of spin-one/zero particles [1] in various studies of particle physics.As typical examples, DKP equation has been also used to study deuteron-nucleus scattering [2], -nucleus elastic scattering, meson-nuclear interaction [3,4], and the scattering of K+ nucleus in the presence of an Aharonov-Bohm potential [5].
On the other hand, we have motivating evidences that we have to work on the noncommutative (NC) formulation of quantum mechanics where the position or momentum operators have nonvanishing commutations.During the past years, some important problems have been investigated in the NC formalism [6][7][8][9][10][11].Typical examples of such cases include an open string attached to -branes in the presence of background B-field inducing noncommutativity in its both end points [12][13][14][15], the quantum Hall effect [16] and the DKP oscillator [17].The interface with solid-state physics and semiconductor theories are also studied in [17][18][19][20].
Due to the physical significance of the magnetic field, Pacheco et al. studied the thermal properties of the onedimensional Dirac oscillator problem [21,22].Although there are very recent papers which consider the NC DKP oscillator, no one, to our best knowledge, has discussed the corresponding thermodynamical properties.In our work, we investigate the spin-one DKP oscillator with the magnetic field in NC phase-space and search for the corresponding thermodynamical properties including partition function , mean internal energy , Helmholtz free energy , specific heat capacity  V , and entropy .

Noncommutative Quantum Mechanics
One way to deal with the NC space is to construct a new kind of field theory, changing the standard product of the fields by the star product (Weyl-Moyal) where   =   is an antisymmetric matrix with real elements and represents the noncommutativity of the space.In (1),  and  are both infinitely differentiable functions [23] and the NC space corresponds to the commutation relations To map the ordinary quantum mechanics into its NC version, it is sufficient to imply the so-called Bopp-shift.The latter implies the transformation 2
All thermodynamical quantities of a physical system are obtained through the partition function [28].We start with the Helmholtz free energy  which is defined as Once the Helmholtz free energy is obtained, other statistical quantities are obtained in a straightforward manner.The mean energy  and the specific heat capacity at constant volume are respectively given by Entropy is related to the other quantities with the relations Accordingly, the NC DKP oscillator partition function is By using the Euler-Maclaurin integration formula [29] (see Appendix B) and after a simple calculation, we calculate the partition function as where  = 1/ 2 .We can extend DKP oscillator partition function to an -body system (without any interaction between the ingredients) via  = [ 1 ]  .

Conclusion
We obtained the statistical quantities of the charged DKP oscillator in a uniform magnetic field in the NC space in Table 1: Energy spectra  , for various quantum numbers ,  for ℏ =  = 1,  =  = 1,  = 0.5,  = 0.8, and  = 1 in noncommutative case.

|𝑛, 𝑙⟩ 𝐸
(2 + 1)-dimensions in an exact analytical manner and thereby the effect of the noncommutative parameter on the thermal properties was obtained.Tables 1 and 2 give the energies in both commutative and NC cases.In Figure 1, we have depicted the energy (  ) of the relativistic spin-one bosons as a function of .We see that the effect of the NC parameter on the energy levels is considerable.The thermodynamic quantities in the NC and commutative cases are, respectively, plotted in Figures 2 and 3.In Figure 2, the behavior of the partition function versus  is plotted for various values of the magnetic field.It shows that with increasing  the partition function has an increasing behavior.In Figure 3, where the Helmholtz free energy is plotted versus , it is seen that the energy decreases in a nearly linear behavior for increasing .

A. The Parametric NU Method
The NU method, named after Nikiforov and Uvarov, can solve a wide class of ordinary differential equations at most of second order.It has been already applied to other wave equations of quantum mechanics including Schrödinger, Dirac, Klein-Gordon, and Duffin-Kemmer-Petiau (DKP) equations.
Here, for the sake of simplicity, we use its parametric version which solves a second-order differential equation of the form [30,31], In the NU method, the energy eigenvalues satisfy [30,31]  2  − (2 + 1)  where

B. Formulation of Euler-Maclaurin
The partition function  of the DKP oscillator is We should test the convergence of the series of (B.1).For that, we apply the integral test which shows that the series and the

Figure 1 :
Figure 1: The comparison of the   for both commutative and noncommutative space.