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

Abstract

The spin-one Duffin-Kemmer-Petiau oscillator in uniform magnetic field is studied in noncommutative formalism. The corresponding energy is obtained and thereby the corresponding thermal properties are obtained for both commutative and noncommutative cases.

1. 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],  𝛼-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 [6–11]. 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 one-dimensional 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 𝐶_𝑣, and entropy 𝑆.

2. 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) 𝑓∗𝑔(𝑥)=exp𝑖𝜃^𝑖𝑗𝜕_𝑥𝑖𝜕_𝑥𝑗𝑓𝑥^𝑖𝑔𝑥^𝑗, (1) where𝜃^𝑖𝑗=𝜃𝜀^𝑖𝑗is an antisymmetric matrix with real elements and represents the noncommutativity of the space. In (1),𝑓 and𝑔are both infinitely differentiable functions [23] and the NC space corresponds to the commutation relations 𝑥^𝑖,𝑥^𝑗=𝑖𝜃^𝑖𝑗,𝑝^𝑖,𝑝^𝑗=0,𝑥^𝑖,𝑝^𝑗=𝑖ℎ⁻𝛿^𝑖𝑗. (2) 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 𝑥^𝑖⟶𝑥^𝑖−12𝜃𝜀^𝑖𝑗𝑝^𝑗. (3) As we know, the DKP equation for free particles with spin-one and spin-zero is (in natural units ℎ⁻=𝑐=1) 𝑖𝛽^𝜇𝜕_𝜇−𝑚𝜓=0, (4) where the DKP matrices satisfy the algebra 𝛽^𝜇𝛽^𝜈𝛽^𝜆+𝛽^𝜆𝛽^𝜈𝛽^𝜇=𝑔^𝜇𝜈𝛽^𝜆+𝑔^𝜆𝜈𝛽^𝜇,𝑔^𝜇𝜈=diag(1,−1,−1,−1),𝜇=0,1,2,3. (5) Within the ten-dimensional representation of spin-one DKP equation, 𝛽⁰=0_3×30_1×30_1×30_1×30^𝑇_1×30_3×3𝐼_3×30_3×30^𝑇_1×3𝐼_3×30_3×30_3×30^𝑇_1×30_3×30_3×30_3×3,𝛽^𝑖=0_3×30_1×3𝑒_𝑖0_1×30^𝑇_1×30_3×30_3×3−𝑖𝑆_𝑖−𝑒^𝑇_𝑖0_3×30_3×30_3×30^𝑇_1×3−𝑖𝑆_𝑖0_3×30_3×3, (6) with the 𝑆_𝑖 matrices being 3×3 ones, (𝑆_𝑖)_𝑗𝑘=−𝑖𝜀_𝑖𝑗𝑘, where𝜀_𝑖𝑗𝑘 is the Levi-Civita tensor,  𝑒_𝑖 matrices are1×3 ones, and(𝑒_𝑖)_1𝑗=𝛿_𝑖𝑗, that is, 𝑒₁=(1,0,0),𝑒₂=(0,1,0),𝑒₃=(0,0,1). 𝐼and0_3×3, respectively, represent unit and null3×3 matrices and0are1×3 ones [24–27]. By considering a magnetic field of the form 𝐵^→=(0,0,𝐵)=𝐵𝑘 in the symmetric gauge𝐴^→(^→𝑟)=(−𝑥₂𝐵/2,𝑥₁𝐵/2,0)=(𝐵^→×^→𝑟)/2, the momentum is transformed as 𝑝^→→(𝑝^→−𝑒𝐴^→/𝑐). The DKP equation in (2+1)-dimensions under a uniform magnetic field has the form (𝑒=1)𝛽⁰𝐸_𝑛,𝑚−𝛽¹𝑝₁+𝐵2𝑥₂−𝑖𝑚𝑤𝜂⁰𝑥₁−𝛽²𝑝₂−𝐵2𝑥₁−𝑖𝑚𝑤𝜂⁰𝑥₁−𝑚𝜓_𝑛,𝑙𝜓𝑥,𝑦,𝑡=0,_𝑛,𝑙𝑥,𝑦,𝑡=𝑒^{−𝑖𝐸𝑛,𝑙𝑡}𝜑_𝑛,𝑙𝑥,𝑦, (7) where 𝜑_𝑛,𝑙𝑥,𝑦=𝜑¹_𝑛,𝑙,𝜑²_𝑛,𝑙,𝜑³_𝑛,𝑙,𝜑⁴_𝑛,𝑙,𝜑⁵_𝑛,𝑙,𝜑⁶_𝑛,𝑙,𝜑⁷_𝑛,𝑙,𝜑⁸_𝑛,𝑙,𝜑⁹_𝑛,𝑙,𝜑¹⁰_𝑛,𝑙^𝑇. (8) Combining the above equations, we have the system of equations𝑝₁+𝐵2𝑥₂−𝑖𝑚𝑤𝑥₁𝜑⁵_𝑛,𝑚+𝑝𝑥,𝑦₂−𝐵2𝑥₁−𝑖𝑚𝑤𝑥₂𝜑⁶_𝑛,𝑚𝑥,𝑦−𝑚𝜑¹_𝑛,𝑚𝑥,𝑦=0,𝐸_𝑛,𝑚𝜑⁵_𝑛,𝑚𝑥,𝑦+𝑝₂−𝐵2𝑥₁+𝑖𝑚𝑤𝑥₂𝜑¹⁰_𝑛,𝑚𝑥,𝑦−𝑚𝜑²_𝑛,𝑚𝑥,𝑦=0,𝐸_𝑛,𝑚𝜑⁶_𝑛,𝑚𝑥,𝑦−𝑝₁+𝐵2𝑥₂+𝑖𝑚𝑤𝑥₁𝜑¹⁰_𝑛,𝑚𝑥,𝑦−𝑚𝜑³_𝑛,𝑚𝑥,𝑦=0,𝐸_𝑛,𝑚𝜑⁷_𝑛,𝑚𝑥,𝑦−𝑝₁+𝐵2𝑥₂+𝑖𝑚𝑤𝑥₁𝜑⁹_𝑛,𝑚+𝑝𝑥,𝑦₂−𝐵2𝑥₁+𝑖𝑚𝑤𝑥₂𝜑⁸_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁴_𝑛,𝑚𝑥,𝑦=0,𝐸_𝑛,𝑚𝜑²_𝑛,𝑚𝑥,𝑦−𝑝₁+𝐵2𝑥₂+𝑖𝑚𝑤𝑥₁𝜑¹_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁵_𝑛,𝑚𝑥,𝑦=0,𝐸_𝑛,𝑚𝜑³_𝑛,𝑚𝑥,𝑦−𝑝₂−𝐵2𝑥₁+𝑖𝑚𝑤𝑥₂𝜑¹_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁶_𝑛,𝑚𝑥,𝑦=0,𝐸_𝑛,𝑚𝜑⁴_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁷_𝑛,𝑚𝑥,𝑦=0,−𝑝₂−𝐵2𝑥₁−𝑖𝑚𝑤𝑥₂𝜑⁴_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁸_𝑛,𝑚𝑥,𝑦=0,−𝑝₁−𝐵2𝑥₂−𝑖𝑚𝑤𝑥₁𝜑⁴_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁹_𝑛,𝑚𝑥,𝑦=0,−𝑝₁+𝐵2𝑥₂−𝑖𝑚𝑤𝑥₁𝜑⁴_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁸_𝑛,𝑚𝑥,𝑦=0,𝑝₁+𝐵2𝑥₂−𝑖𝑚𝑤𝑥₁𝜑³_𝑛,𝑚−𝑝𝑥,𝑦₂−𝐵2𝑥₁−𝑖𝑚𝑤𝑥₂𝜑²_𝑛,𝑚𝑥,𝑦−𝑚𝜑¹⁰_𝑛,𝑚𝑥,𝑦=0. (9a)(9b)(9c)(9d)(9e)(9f)(9g)(9h)(9i)(9j)(9k)

Assuming𝜑²_𝑛,𝑚(𝑥,𝑦)=𝜑³_𝑛,𝑚(𝑥,𝑦)=0, from (9a)–(9k) we have𝐸_𝑛,𝑚𝜑⁷_𝑛,𝑚𝑥,𝑦−𝑝₁+𝐵2𝑥₂+𝑖𝑚𝑤𝑥₁𝜑⁹_𝑛,𝑚+𝑝𝑥,𝑦₂−𝐵2𝑥₁+𝑖𝑚𝑤𝑥₂𝜑⁸_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁴_𝑛,𝑚𝑥,𝑦=0,𝐸_𝑛,𝑚𝜑⁴_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁷_𝑛,𝑚𝑥,𝑦=0,−𝑝₂−𝐵2𝑥₁−𝑖𝑚𝑤𝑥₂𝜑⁴_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁸_𝑛,𝑚𝑥,𝑦=0,−𝑝₁+𝐵2𝑥₂−𝑖𝑚𝑤𝑥₁𝜑⁴_𝑛,𝑚𝑥,𝑦−𝑚𝜑⁹_𝑛,𝑚𝑥,𝑦=0. (10a)(10b)(10c)(10d)

Equations (10a)–(10d), after some algebra, give 𝐸²_𝑛,𝑙−𝑝₁+𝐵2𝑥₂+𝑖𝑚𝑤𝑥₁𝑝₁+𝐵2𝑥₂−𝑖𝑚𝑤𝑥₁−𝑝₂−𝐵2𝑥₁+𝑖𝑚𝑤𝑥₂𝑝₂−𝐵2𝑥₁−𝑖𝑚𝑤𝑥₂−𝑚²×𝜑⁴_𝑛,𝑙𝑥,𝑦=0. (11) To map the problem into the NC formulation, we consider 𝑥₁=𝑥₁−𝜃𝑃₂2,𝑥₂=𝑥₂+𝜃𝑃₁2, (12) which transform (11) into 𝐸²_𝑛,𝑙−𝑝²𝑚1+²𝑤²𝜃²4+𝐵𝜃2+𝐵²𝜃²16−𝑟²𝑚²𝑤²+𝐵²4+𝐵²𝜃4+𝑚²𝑤²𝜃+𝐵𝐿_𝑧+𝑚𝑤𝐵𝜃+2𝑚𝑤−𝑚²𝜑⁴_𝑛,𝑙𝑥,𝑦=0. (13) Proposing the well-known separation of variables 𝜑⁴_𝑛,𝑙(𝑟,𝜙)=𝑅⁴_𝑛(𝑟)𝑒^𝑖𝑙𝜙 followed by a transformation of the form 𝑅(𝑟)=𝑟^−1/2𝑢(𝑟), (13) gives 𝑑₁𝑑²𝑑𝑟²+1𝑟²−𝑙²+14𝑑₁+𝑑₂−𝑟²𝑑₃𝑢(𝑟)=0, (14) where 𝑑₁=1+𝐵𝜃2+𝐵²𝜃²+𝑚16²𝑤²𝜃²4,𝑑₂=𝐸²_𝑛,𝑙−𝑚²𝐵+𝑙𝐵+²𝜃4+𝑚²𝑤²𝜃−𝑚𝑤−2𝑚𝑤,𝑑₃=𝐵²4+𝑚²𝑤². (15)(16)(17) Proposing a change of variable of the form 𝑠=𝑟², (14) transforms into 𝑑²𝑑𝑠²+1𝑑2𝑠+1𝑑𝑠𝑠²𝑑₄+𝑑₅𝑠+𝑑₆𝑠²𝑢(𝑠)=0, (18) where 𝑑₄=1/4−𝑙²4,𝑑₅=𝑑₂4𝑑₁,𝑑₆𝑑=−³4𝑑₁. (19) To obtain the wave function, we simply use the parametric Nikiforov-Uvarov (NU) method from the Appendix. A simple comparison reveals the correspondence 𝛼₁=12,𝛼₂=𝛼₃=0,𝜉₁=−𝑑₆,𝜉₂=𝑑₅,𝜉₃=−𝑑₄,𝛼₄=14,𝛼₅=0,𝛼₆=−𝑑₆,𝛼₇=−𝑑₅,𝛼₈=116−𝑑₄,𝛼₉=−𝑑₆. (20) Using (20), the energy relation is found as 𝐸_𝑛,𝑙=±𝑚4𝑑₃𝑑₁𝑛+2𝑑₃𝑑₁−𝑘+𝑑₃𝑑₁+1−16𝑑₄𝑚², (21) or 𝐸^±_𝑛,𝑙=±𝑚𝜂+4𝜆𝑛,𝑛=1,2,…,𝑙=0,1,…, (22) with𝜆=𝑑₃𝑑₁𝑚²,2𝜂=𝑑₃𝑑₁−𝑘+𝑑₃𝑑₁1−16𝑑₄𝑚². (23a)(23b)

The partition function of the DKP oscillator at temperature𝑇is obtained from 𝑍(𝑉,𝑇)=^𝑛_𝑛=0𝑒^{−𝛽(𝐸𝑛−𝐸0)}, (24) where 𝛽=1/𝑘_𝐵𝑇 and𝑘_𝐵is the Boltzmann constant. All thermodynamical quantities of a physical system are obtained through the partition function [28]. We start with the Helmholtz free energy𝐹 which is defined as 𝐹(𝑁,𝑉,𝑇)=−𝑘_𝐵𝑇ln𝑍(𝑉,𝑇). (25) Once the Helmholtz free energy is obtained, other statistical quantities are obtained in a straightforward manner. The mean energy𝑈 and the specific heat capacity at constant volume are respectively given by𝑈=−𝜕ln𝑍𝜕𝛽=−𝑁𝜕ln𝑍₁𝜕𝛽=𝑁𝑚𝑐²𝜏²𝜕ln𝑍₁,𝜕𝜏𝐶_𝑣=𝜕⟨𝑈⟩=𝑘𝜕𝑇𝑚𝑐^𝐵2𝜕⟨𝑈⟩𝜕𝜏=−𝑘_𝐵𝛽²𝜕⟨𝑈⟩.𝜕𝛽 (26a)(26b)

Entropy is related to the other quantities with the relations 𝜕𝑆=−𝑁𝜕𝑇𝐹=𝑁𝑘_𝐵ln𝑍₁+𝑁𝑘_𝐵𝑇𝜕𝜕𝑇ln𝑍₁=𝑁𝑘_𝐵ln𝑍₁−𝑁𝑘_𝐵𝛽𝜕𝜕𝛽ln𝑍₁. (27) Accordingly, the NC DKP oscillator partition function is 𝑍₁=^𝑛_𝑛=0𝑒^{−𝛽𝑚𝑐2𝜂+4𝜆𝑛}=𝑒^{𝛽𝑚𝑐2√𝜂𝑛}_𝑛=0𝑒^{−𝛽𝑚𝑐2𝜂+4𝜆𝑛}. (28) By using the Euler-Maclaurin integration formula [29] (see Appendix B) and after a simple calculation, we calculate the partition function as 𝑍₁=12+√𝜂2𝜆1𝜏+2𝜆𝜏²+𝜆6√𝜂−𝜆³30𝜂⁵1𝜏−130𝜂²𝜆³𝜏²−𝜆³60𝜂³𝜏³, (29) where𝜏=1/𝛽𝑚𝑐². We can extend DKP oscillator partition function to an 𝑁-body system (without any interaction between the ingredients) via 𝑍=[𝑍₁]^𝑁.

3. Conclusion

We obtained the statistical quantities of the charged DKP oscillator in a uniform magnetic field in the NC space in (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 (𝐸_𝑛) of the relativistic spin-one bosons as a function of 𝑛. 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 𝑇 is plotted for various values of the magnetic field. It shows that with increasing𝑇the partition function has an increasing behavior. In Figure 3, where the Helmholtz free energy is plotted versus 𝑇, it is seen that the energy decreases in a nearly linear behavior for increasing𝑇.

Table 1 

Energy spectra 𝐸_𝑛,𝑙 for various quantum numbers 𝑛,𝑙 for ℎ⁻=𝑐=1,𝑚=𝑒=1,𝜔=0.5,𝜃=0.8, and 𝐵=1 in noncommutative case.

Table 2 

Energy spectra 𝐸_𝑛,𝑙 for various quantum numbers 𝑛,𝑙 for ℎ⁻=𝑐=1,𝑚=𝑒=1,𝜔=0.5,𝜃=0, and 𝐵=1 for commutative space.

Figure 1 

The comparison of the𝐸𝑛for both commutative and noncommutative space.

Figure 2 

The comparison of the partition functions for both commutative and noncommutative cases.

Figure 3 

The comparison of the Helmholtz free energy𝐹for both commutative and noncommutative cases.

Appendices

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], 𝑑²𝑑𝑠²+𝛼₁−𝛼₂𝑠𝑠1−𝛼₃𝑠𝑑+𝑑𝑠−𝜉₁𝑠²+𝜉₂𝑠−𝜉₃𝑠1−𝛼₃𝑠²𝜓(𝑠)=0. (A.1) In the NU method, the energy eigenvalues satisfy [30, 31] 𝛼₂𝑛−(2𝑛+1)𝛼₅+(2𝑛+1)√𝛼₉+𝛼₃√𝛼₈+𝑛(𝑛−1)𝛼₃+𝛼₇+2𝛼₃𝛼₈+2√𝛼₈𝛼₉=0, (A.2) where 𝛼₄=121−𝛼₁,𝛼₅=12𝛼₂−2𝛼₃,𝛼₆=𝛼²₅+𝜉₁,𝛼₇=2𝛼₄𝛼₅−𝜉₂,𝛼₈=𝛼²₄+𝜉₃,𝛼₉=𝛼₃𝛼₇+𝛼²₃𝛼₈+𝛼₆. (A.3)

B. Formulation of Euler-Maclaurin

The partition function 𝑍 of the DKP oscillator is 𝑍(𝑉,𝑇)=^𝑛_𝑛=0𝑒^{−𝛽(𝐸𝑛−𝐸0)}=𝑒^{𝛽√𝑏∞}_𝑛=0𝑒^{√−𝛽𝑎𝑥+𝑏}. (B.1) 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 integral converge or diverge together. So, from (B.1), we can see that the function 𝑓(𝑥), where 𝑓(𝑥)=𝑒^{√−𝛽𝑎𝑥+𝑏}, (B.2) is a decreasing positive function, and the integral ^∞₀𝑓(𝑥)2𝑑𝑥=𝑒𝑎𝛽^{√−𝛽𝑏}1+𝛽√𝑏, (B.3) is convergent. In order to evaluate this function, we use the Euler-Maclaurin formula defined as follows: ^𝑛_𝑛=0𝑓(𝑥)=12𝑓(0)+^∞₀𝑓(𝑥)𝑑𝑥−^∞_𝑝=11!𝐵2𝑝^2𝑝𝑓^(2𝑝−1)(0). (B.4) Here,𝐵_2𝑝are the Bernoulli numbers,𝐵₂=1/6,𝐵₄=−1/30,…. In our case, we have used 𝑓⁽¹⁾(𝑥)=𝑎𝛽2√𝑒𝑏+𝑎𝑥^{√−𝛽𝑏+𝑎𝑥},𝑓⁽³⁾(𝑥)=−3𝛽𝑎³8(𝑏+𝑎𝑥)^5/2−3𝛽²𝑎³8(𝑏+𝑎𝑥)²−𝛽³𝑎³8(𝑏+𝑎𝑥)^3/2𝑒^{√−𝛽𝑏+𝑎𝑥}. (B.5)(B.6) Using above equations (B.4), (B.5), and (B.6), the partition function can be then written as 1𝑧=2+2𝑎₁𝛽²1+𝛽𝑏₁−𝐵₂2𝑓⁽¹⁾−𝐵₄𝑓24⁽³⁾+⋯. (B.7)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

We wish to give our sincere gratitude to the referee for his technical comments.