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

Authors: S. Hassanabadi; M. Ghominejad
Journal: Advances in High Energy Physics
 (2014)

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## 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.


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## Body

## 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 ofK+ 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 Cv, 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)
(1)(f*g)(x)=exp⁡(iθij∂xi∂xj)f(xi)g(xj),
whereθij=θεijis 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
(2)[x^i,x^j]=iθij,[p^i,p^j]=0,[x^i,p^j]=iħδij.
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
(3)x^i⟶xi-12θεijpj.
As we know, the DKP equation for free particles with spin-one and spin-zero is (in natural units ħ=c=1)
(4)(iβμ∂μ-m)ψ=0,
where the DKP matrices satisfy the algebra
(5)βμβνβλ+βλβνβμ=gμνβλ+gλνβμ,gμν=diag⁡(1,-1,-1,-1),μ=0,1,2,3.
Within the ten-dimensional representation of spin-one DKP equation,
(6)β0=|03×30-1×30-1×30-1×30-1×3T03×3I3×303×30-1×3TI3×303×303×30-1×3T03×303×303×3|,βi=|03×30-1×3ei0-1×30-1×3T03×303×3-iSi-eiT03×303×303×30-1×3T-iSi03×303×3|,
with the Si matrices being 3×3 ones, (Si)jk=-iεijk, whereεijk is the Levi-Civita tensor,  ei matrices are1×3 ones, and(ei)1j=δij, that is, e1=(1,0,0),e2=(0,1,0),e3=(0,0,1). Iand03×3, respectively, represent unit and null3×3 matrices and0-are1×3 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)(7){β0En,m-β1(p^1+B2x^2-imwη0x^1)-β2(p^2-B2x^1-imwη0x^1)-m}ψn,l(x,y,t)=0,ψn,l(x,y,t)=e-iEn,ltφn,l(x,y),
where
(8)φn,l(x,y)=(φn,l1,φn,l2,φn,l3,φn,l4,φn,l5,φn,l6,φn,l7,φn,l8,φn,l9,φn,l10)T.
Combining the above equations, we have the system of equations(9a)(p^1+B2x^2-imwx^1)φn,m5(x,y)+(p^2-B2x^1-imwx^2)φn,m6(x,y)-mφn,m1(x,y)=0,(9b)En,mφn,m5(x,y)+(p^2-B2x^1+imwx^2)φn,m10(x,y)-mφn,m2(x,y)=0,(9c)En,mφn,m6(x,y)-(p^1+B2x^2+imwx^1)φn,m10(x,y)-mφn,m3(x,y)=0,(9d)En,mφn,m7(x,y)-(p^1+B2x^2+imwx^1)φn,m9(x,y)+(p^2-B2x^1+imwx^2)φn,m8(x,y)-mφn,m4(x,y)=0,(9e)En,mφn,m2(x,y)-(p^1+B2x^2+imwx^1)φn,m1(x,y)-mφn,m5(x,y)=0,(9f)En,mφn,m3(x,y)-(p^2-B2x^1+imwx^2)φn,m1(x,y)-mφn,m6(x,y)=0,(9g)En,mφn,m4(x,y)-mφn,m7(x,y)=0,(9h)-(p^2-B2x^1-imwx^2)φn,m4(x,y)-mφn,m8(x,y)=0,(9i)-(p^1-B2x^2-imwx^1)φn,m4(x,y)-mφn,m9(x,y)=0,(9j)-(p^1+B2x^2-imwx^1)φn,m4(x,y)-mφn,m8(x,y)=0,(9k)(p^1+B2x^2-imwx^1)φn,m3(x,y)-(p^2-B2x^1-imwx^2)φn,m2(x,y)-mφn,m10(x,y)=0.Assumingφn,m2(x,y)=φn,m3(x,y)=0, from (9a)–(9k) we have(10a)En,mφn,m7(x,y)-(p^1+B2x^2+imwx^1)φn,m9(x,y)+(p^2-B2x^1+imwx^2)φn,m8(x,y)-mφn,m4(x,y)=0,(10b)En,mφn,m4(x,y)-mφn,m7(x,y)=0,(10c)-(p^2-B2x^1-imwx^2)φn,m4(x,y)-mφn,m8(x,y)=0,(10d)-(p^1+B2x^2-imwx^1)φn,m4(x,y)-mφn,m9(x,y)=0.Equations (10a)–(10d), after some algebra, give
(11){En,l2-(p^1+B2x^2+imwx^1)(p^1+B2x^2-imwx^1)-(p^2-B2x^1+imwx^2)(p^2-B2x^1-imwx^2)-m2}×φn,l4(x,y)=0.
To map the problem into the NC formulation, we consider
(12)x^1=x1-θP22,x^2=x2+θP12,
which transform (11) into
(13){En,l2-p2(1+m2w2θ24+Bθ2+B2θ216)-r2(m2w2+B24)+(B2θ4+m2w2θ+B)Lz+mwBθ+2mw-m2(1+m2w2θ24+Bθ2+B2θ216)}φn,l4(x,y)=0.
Proposing the well-known separation of variables φn,l4(r,ϕ)=Rn4(r)eilϕ followed by a transformation of the form R(r)=r-1/2u(r), (13) gives
(14){d1d2dr2+1r2(-l2+14)d1+d2-r2d3}u(r)=0,
where
(15)d1=1+Bθ2+B2θ216+m2w2θ24,(16)d2=En,l2-m2+l(B+B2θ4+m2w2θ-mw-2mw),(17)d3=B24+m2w2.
Proposing a change of variable of the form s=r2, (14) transforms into
(18){d2ds2+12sdds+1s2{d4+d5s+d6s2}}u(s)=0,
where
(19)d4=(1/4-l2)4,d5=d24d1,d6=-d34d1.
To obtain the wave function, we simply use the parametric Nikiforov-Uvarov (NU) method from the Appendix. A simple comparison reveals the correspondence
(20)α1=12,α2=α3=0,ξ1=-d6,ξ2=d5,ξ3=-d4,α4=14,α5=0,α6=-d6,α7=-d5,α8=116-d4,α9=-d6.
Using (20), the energy relation is found as
(21)En,l=±m4d3d1n+2d3d1-k+d3d1(+1-16d4)m2,
or
(22)En,l±=±mη+4λ-n,n=1,2,…,l=0,1,…,
with(23a)λ-=d3d1m2,(23b)η=2d3d1-k+d3d1(1-16d4)m2.The partition function of the DKP oscillator at temperatureTis obtained from
(24)Z(V,T)=∑n=0ne-β(En-E0),
where β=1/kBT andkBis 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
(25)F(N,V,T)=-kBTln⁡Z(V,T).
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 by(26a)U=-∂ln⁡Z∂β=-N∂ln⁡Z1∂β=Nmc2τ2∂ln⁡Z1∂τ,(26b)Cv=∂〈U〉∂T=kBmc2∂〈U〉∂τ=-kBβ2∂〈U〉∂β.Entropy is related to the other quantities with the relations(27)S=-N∂∂TF=NkBln⁡Z1+NkBT∂∂T(ln⁡Z1)=NkBln⁡Z1-NkBβ∂∂β(ln⁡Z1).
Accordingly, the NC DKP oscillator partition function is
(28)Z1=∑n=0ne-βmc2η+4λ-n=eβmc2η∑n=0ne-βmc2η+4λ-n.
By using the Euler-Maclaurin integration formula [29] (see Appendix B) and after a simple calculation, we calculate the partition function as
(29)Z1=(12+η2λ-τ+12λ-τ2+(λ-6η-λ-330η5)1τ-130η2λ-3τ2-λ-360η3τ3),
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 (En) 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 spectraEn,l for various quantum numbers n,l for ħ=c=1,m=e=1,ω=0.5,θ=0.8, and  B=1 in noncommutative case.
|
n
,
l
〉
E
n
,
l
(
NCS
)
|
1,0
〉
2.1820
|
1,1
〉
2.2542
|
1,2
〉
2.3242
|
1,3
〉
2.3922
|
1,4
〉
2.4583
|
1,5
〉
2.5226
|
2,0
〉
2.8639
|
2,1
〉
2.9193
|
2,2
〉
2.9737
|
2,3
〉
3.0271
|
2,4
〉
3.0796
|
2,5
〉
3.1312
|
3,0
〉
3.4122
|
3,1
〉
3.4588
|
3,2
〉
3.5048
|
3,3
〉
3.5503
|
3,4
〉
3.5951
|
3,5
〉
3.6394
|
4,0
〉
3.8838
|
4,1
〉
3.9248
|
4,2
〉
3.9654
|
4,3
〉
4.0056Table 2
Energy spectraEn,l for various quantum numbers n,l for ħ=c=1,m=e=1,ω=0.5,θ=0, and  B=1 for commutative space.
|
n
,
l
〉
E
n
,
l
(
CS
)
|
1,0
〉
2.0597
|
1,1
〉
2.1579
|
1,2
〉
2.2519
|
1,3
〉
2.3402
|
1,4
〉
2.4288
|
1,5
〉
2.5127
|
2,0
〉
2.6591
|
2,1
〉
2.7359
|
2,2
〉
2.8106
|
2,3
〉
3.8833
|
2,4
〉
3.9543
|
2,5
〉
3.0235
|
3,0
〉
3.1463
|
3,1
〉
3.2114
|
3,2
〉
3.2753
|
3,3
〉
3.3379
|
3,4
〉
3.3994
|
3,5
〉
3.4598
|
4,0
〉
3.5676
|
4,1
〉
3.6252
|
4,2
〉
3.6818
|
4,3
〉
3.7377Figure 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.



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*Source: 185169-2014-05-22.xml*