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Authors: Bing-Qian Wang; Zheng-Wen Long; Chao-Yun Long; Shu-Rui Wu
Journal: Advances in High Energy Physics
 (2017)

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## Abstract

Using the momentum space representation, we study the (2 + 1)-dimensional Duffin-Kemmer-Petiau oscillator for spin 0 particle under a magnetic field in the presence of a minimal length in the noncommutative space. The explicit form of energy eigenvalues is found, and the wave functions and the corresponding probability density are reported in terms of the Jacobi polynomials. Additionally, we also discuss the special cases and depict the corresponding numerical results.


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## Body

## 1. Introduction

The interest in the study of the minimal length uncertainty relation combination with the noncommutative space commutation relations in nonrelativistic wave equation and relativistic wave equation has drawn much attention [1, 2] in recent years. Motivated by string theory, loop quantum gravity, and quantum geometry [3–15], the modification of the ordinary uncertainty relation has become an appealing case of research. In the so-called minimal length formulation, Kempf and Hinrichsen [16–18] have shown that the minimal length can be introduced as an additional uncertainty in position measurement, so that the usual canonical commutation relation between position and momentum operator is substituted by x,p=iħ1+βp2, where β is a small positive parameter called the deformation parameter. Recently, various topics have been studied in connection with the minimal length uncertainty relation [19–28]. On the other hand, the issue of noncommutative (NC) quantum mechanics has also been extensively discussed. The noncommutativity of space-time coordinates was first introduced by Snyder [29] aiming to improve the problem of infinite self-energy in quantum field theory, and the noncommutative geometry has been put forward because of the discovery in string theory and matrix model of M-theory [30]. Recently, various aspects of both NC classical [31] and quantum [32] mechanics have been extensively studied devoted to exploring the role of NC parameter in the physical observables [33–40]. Recently, the effect of the quantum gravity on the quantum mechanics by modifying the basic commutators among the canonical variables has been an attractive topic; therefore the combination of the minimal length uncertainty relation and the noncommutative space commutation relations is a colorful problem. In the present work, we are interested to study the two sectors in the framework of the relativistic DKP equation and analyze the effects of them on the energy spectrum and the corresponding wave functions.The organization of this work is as follows. In Section2, we consider the DKP oscillator for spin 0 particle in the presence of a minimal length in the NC space. In Section 3, we study the problem under a magnetic field. In Section 4, we discuss some special cases of the solutions to check the validity of our results. Finally, the work is summarized in last section.


## 2. The(2+1)-DKP Oscillator for Spin 0 Particle in the Presence of the Minimal Length in NC Space

In NC space, the canonical variables satisfy the following commutation relations:(1)xiNC,xjNC=iθij,PiNC,PjNC=0,xiNC,PjNC=iħδij,i,j=1,2,where θij=ϵijθ is the antisymmetric NC parameter, representing the noncommutativity of the space, and xiNC and PiNC are the coordinate and momentum operators in the NC space. By replacing the normal product with star product, the DKP equation in commutation space will change into the DKP equation in NC space as(2)HP,x∗ψx=Eψx.Usually, one way to deal with the problem of NC space is via the star product or Moyal-Weyl product on the commutative space functions:(3)f∗gx=exp⁡i2θij∂ix∂jyfxgyy→x,where fx and gy are arbitrary infinitely differentiable functions.Then the Moyal-Weyl product can be replaced by a Bopp shift of the form(4)xiNC=x^i-12ħθijP^j,PiNC=P^i,and therefore in the two-dimensional NC space, (4) can be expressed as(5)xNC=x^-θ2ħP^y,yNC=y^+θ2ħP^x,PxNC=P^x,PyNC=P^y,where x^, y^, P^x, and P^y are the position and momentum operators in the usual quantum mechanics, respectively, which satisfy the canonical Heisenberg commutation relations.Thus, the relativistic DKP equation for a free boson of massm is given by [41, 42](6)cβ→·P→+mc2ψ′=iħβ0∂∂tψ′,where β→=(β1,β2,β3) and β0 is the DKP matrices which meet the following algebra relation: (7)βμβνβλ+βλβνβμ=gμνβλ+gλνβμ,where gμν=diag(1,-1,-1,-1) is the metric tensor in Minkowski space. For spin 0 particle, βi are 5 × 5 matrices expressed as(8)β0=I00-0-Tб,βi=0^ki-kiTб,i=1,2,3,with б, 0-, and 0^ being 3 × 3, 2 × 3 and 2 × 2 zero matrices, respectively. Other matrices in (8) are given as follows:(9)I0=0110,k1=-100000,k2=0-10000,k3=00-1000.The DKP oscillator is introduced by using substitution of the momentum operator P→ with (P→→P→-imωη0r→), where the additional term is linear in r, w is the oscillator frequency, and η0=2β02-I with (η0)2=I.It is easy to get the 2D DKP oscillator from the above equation:(10)cβ→·P→-imωη0r→+mc2ψ=iħβ0∂∂tψ.Given this situation above, considering the NC formalism, and via the Bopp shift (4)-(5), the (2 + 1)-dimensional DKP oscillator equation in NC space becomes(11)cβ1PxNC-imwη0xNC+cβ2PyNC-imwyNC+mc2ψ=β0E.For a boson of spin 0, the spinor ψ is a vector with five components [43, 44], which reads(12)ψ=ψ1,ψ2,ψ3,ψ4,ψ5T.Substituting ψ into (11) we have(13)-mc2ψ1+Eψ2+cPxNC+imwxNCψ3+cPyNC+imwyNCψ4=0,Eψ1-mc2ψ2=0,cPxNC-imwxNCψ1+mc2ψ3=0,cPyNC-imwyNCψ1+mc2ψ4=0,mc2ψ5=0.Combination of (13) gives (14)P^x+imwx^-θ2ħP^yP^x-imwx^-θ2ħP^y+P^y+imwy^+θ2ħP^xP^y-imwy^+θ2ħP^x+m2c2-E2c2ψ1=0.In addition, in the minimal length formalism, the Heisenberg algebra is given by(15)x^i,P^j=iħδij1+αP2,where α is the minimal length positive parameter. Moreover, in the momentum space, the position vector and momentum vector can be expressed as(16)x^i=iħ1+αP2∂∂pi,P^i=Pi,and substituting (15) and (16) into (14) we have(17)1+m2w2θ24ħ2Px2+Py2+m2w2x2+y2-2mwħ1+αP2-m2w2θħ1+αP2Lz+m2c2-E2c2ψ1=0.Now, in order to solve (17), an auxiliary wave function is defined as ψ1P,ϑ=eilϑϕP, and for the sake of simplicity, we bring the problem into the polar coordinates, recalling that(18)Px=Pcos⁡ϑ,Py=Psin⁡ϑ,∂∂Px=cos⁡ϑ∂∂P-sin⁡ϑP∂∂ϑ,∂∂Py=sin⁡ϑ∂∂P+cos⁡ϑP∂∂ϑ;then (17) becomes(19)1+αP22∂2∂P2+1P1+αP22+2αP1+αP2∂∂P-l2P21+αP22+-1+m2w2θ2/4ħ2+2mwαħ+m2w2θαlm2w2ħ2P2+E2-m2c4/c2+2mwħ+m2w2θlm2w2ħ2ϕP=0;with the help of a variable transformation (20)q=1αtan-1⁡αP,which will map the variable P∈0,∞ to q∈0,π/2α, we simplify (19) to(21)∂2∂q2+υμ+μυα∂∂q-αl2υ2μ2+αk′μ2υ2+αε′ϕq=0,where (22)μ=sin⁡αq,υ=cos⁡αq,k′=-l2+-1+m2w2θ2/4ħ2+2mwαħ+m2w2θαlα2m2w2ħ2,ε′=E2-m2c4/c2+2mwħ+m2w2θlαm2ω2ħ2-2l2.Next, for convenience, another auxiliary function is introduced by ϕq=υλFq, with λ being a constant to be determined. Thus we have (23)∂2∂q2+υμ+1-2λμυα∂∂q-αl2υ2μ2+αk′+λλ-1-λμ2υ2+αε′-2λFq=0.Here, in order to simplify above mathematical expression, we select k′+λλ-1-λ=0; then it leads to the following expression of λ:(24)λ=1+1-k′,λ′=1-1-k′.Since the second solution leads to a nonphysically acceptable wave function, then (23) turns into(25)∂2∂q2+υμ+1-2λμυα∂∂q-αl2υ2μ2+αε′-2λFq=0.Then with the help of another auxiliary function Fq=μlζq, thus (25) reads (26)∂2∂q2+2l+1υμ+1-2λμυα∂∂q+αε′-2λ-2λlζq=0.Now we make a variable transformation by demanding z=2μ2-1, where the variable interval is z∈(-1,1); then one can obtain (27)1-z2∂2∂z2+l+1-λ-l+1+λz∂∂z+14ε′-2λ-2λlζz=0.It is important to point out that the wave function we used here will be regular at z=±1 on the condition that ζz is a polynomial, which is obtained by imposing the following constraint 1/4ε′-2λ-2λl=nn+l+λ, with n being a nonnegative integer. Then (27) turns into (28)1-z2∂2∂z2+l+1-λ-l+1+λz∂∂z+nn+l+λζz=0,whose solution can be written in terms of Jacobi polynomials as ζz=Pna,bz, where a=λ-1,b=l. In this case, the energy eigenvalue of the system can be expressed as(29)ε′′=2l2+4n2+4nl+4n+2l+11+1-k′with ε′′=E2+2mωħc2-m2c4/αm2ω2ħ2c2.Therefore the energy eigenvalues can be derived from (29) as (30)Enl=±mc24n2+4nl+2l2αω2ħ2c2-2ħωmc2-w2θlc2+1+2αω2ħ2c22n+l+11+1+l2+1+m2w2θ2/4ħ2-2mwαħ-m2w2θαlα2m2w2ħ21/2n,l=0,1,2,…;furthermore, the wave function may be expressed by(31)ψ1P=Neilϑ11+αP2λ/2αP21+αP2l/2Pna,b2αP21+αP2-1,n,l=0,1,2,….Then the wave function of the system is(32)ψ=N1Enlmc2-cos⁡ϑ+iθ/2ħmwsin⁡ϑP+ħmw1+αP2cos⁡ϑ∂/∂P-sin⁡ϑ/P∂/∂ϑmc-sin⁡ϑ-iθ/2ħmwcos⁡ϑP+mωħ1+αP2sin⁡ϑ∂/∂P+cos⁡ϑ/P∂/∂ϑmc0ψ1P,ϑ.Now the Jacobi polynomial [45] is employed to obtain the other components wave: (33)dPna,btdt=12n+a+b+1Pn-1a+1,b+1t;we finally have(34)ψ2P,ϑ=Enlmc2ψ1P,ϑ=NEnlmc2eilϑ11+αP2λ/2αP21+αP2l/2Pna,bαP2-11+αP2,ψ3P,ϑ=-cos⁡ϑ+iθ/2ħmwsin⁡ϑP+ħmw1+αP2cos⁡ϑ∂/∂P-sin⁡ϑ/P∂/∂ϑmcψ1P,ϑ=-Neilϑ11+αP2λ/2αP21+αP2l/21mccos⁡ϑ+iθ2ħmwsin⁡ϑP+ħmwcos⁡ϑ-λαP+lP-iħmwl1+αP2sin⁡ϑPpnλ-1,l+ħmwcos⁡ϑ2αP1+αP2n+l+λpn-1λ,l+1,ψ4P,ϑ=-sin⁡ϑ-iθ/2ħmwcos⁡ϑP+mwħ1+αP2sin⁡ϑ∂/∂P+cos⁡ϑ/P∂/∂ϑmcψ1P,ϑ=-Neilϑ11+αP2λ/2αP21+αP2l/21mcsin⁡ϑ-iθ2ħmwcos⁡ϑP+mwħsin⁡ϑ-λαP+lP+iħmwl1+αP2cos⁡ϑPpnλ-1,l+ħmwsin⁡ϑ2αP1+αP2n+l+λpn-1λ,l+1,ψ5P,ϑ=0.After ending this part, we determine the normalization constant N by demanding(35)∫-∞+∞d2p1+αP2ψ-Pβ0ψP=1,besides, according to the property of the Jacobi polynomial,(36)∫-1+1dt1-ta1+tbPna,bt2=2a+b+1Γa+n+1Γb+n+1n!a+b+1+2nΓa+b+n+1;we obtain(37)N=α1/2mc2n!2n+λ+lΓn+λ+lEnlΓn+λΓn+l+11/2;then one can obtain the corresponding probability density of every component given by(38)P′i=∫0π∫0∞ψ-iP,ϑβ0ψiP,ϑdpdϑ,i=1,2,3,4,5.


## 3. The Problem under a Magnetic Field

Now, in the presence of an external magnetic field, that is,A→=-ByNC/2BxNC/20, (11) is transformed into(39)β0E~-cβ1PxNC+eByNC2c-imωη0xNC-cβ2PyNC-eBxNC2c-imwη0yNC-mc2ψ~=0;here the spinor ψ~ is also a vector with five components which reads(40)ψ~=ψ~1,ψ~2,ψ~3,ψ~4,ψ~5T.Substituting ψ~ into (39) one can obtain(41)-mc2ψ~1+E~ψ~2+cPxNC+eByNC2c+imwxNCψ~3+cPyNC-eBxNC2c+imwyNCψ~4=0,Eψ~1-mc2ψ~2=0,cPxNC+eByNC2c-imwxNCψ~1+mc2ψ~3=0,cPyNC-eBxNC2c-imwyNCψ~1+mc2ψ~4=0,mc2ψ~5=0.Simplifying (41) gives(42)1+eBθ4ħcP^x+eBy^2c+imwx^-θ2ħP^y1+eBθ4ħcP^x+eBy^2c-imwx^-θ2ħP^y+1+eBθ4ħcP^y-eBx^2c+imwy^+θ2ħP^x1+eBθ4ħcP^y-eBx^2c-imwy^+θ2ħP^x+m2c4-E~2c2ψ~1=0,and considering (16) and (42) one can obtain(43)1+eBθ4ħc2+m2w2θ24ħ2-2ħαmw1+eBθ4ħc+eBθmw4ħc-αħleBc1+eBθ4ħc+m2w2θħ-2ħαeBmwcP2+m2w2+eB2c2x2+y2+m2c4-E~2c2-2ħmw1+eBθ4ħc+eBθmw4ħc-ħleBc1+eBθ4ħc+m2w2θħ-2ħαeBmwcψ~1P,ϑ=0.Now, by a series of analogical algebraic operations and for the sake of simplification, we just give the following results:The energy eigenvalues of the system are(44)E~n~l=±mc2αħ2m2c24n~2+4n~l+2l2m2w2+eB2c2-2ħm2c2mw1+eBθ4ħc+eBθmw4ħc-ħlm2c2eBc1+eBθ4ħc+m2w2θħ-2ħαeBmwc+1+2ħ2m2c22n~+l+1α+α2-α2K~m2w2+eB2c21/2,n~,l=0,1,2,…,where (45)K~=-l2-1+eBθ/4ħc2+m2w2θ2/4ħ2-2ħαmw1+eBθ/4ħc+eBθmw/4ħc-αħleB/c1+eBθ/4ħc+m2w2θ/ħ-2ħαeBmw/cα2ħ2m2w2+eB/2c2.The wave function can be expressed as(46)ψ~1P,ϑ=N~eilϑ11+αP2λ′/2αP21+αP2l/2Pn~a′,b′2αP21+αP2-1,n~,l=0,1,2,3,4,…;thus the wave function of the system is(47)ψ~=N~1En~lmc2-1+eBθ/4ħccos⁡ϑ+imwθ/2ħsin⁡ϑP+1+αP2iħeB/2c∂/∂Py+ħmw∂/∂Pxmc1+eBθ/4ħcsin⁡ϑ-imwθ/2ħcos⁡ϑP+1+αP2-iħeB/2c∂/∂Px+ħmw∂/∂Pymc0ψ~1P,ϑ;then the other components’ wave functions of the system are(48)ψ~2P,ϑ=En~lmc2ψ~1P,ϑ=N~En~lmc2eilϑ11+αP2λ′/2αP21+αP2l/2Pn~a′,b′2αP21+αP2-1,ψ~3P,ϑ=-1+eBθ/4ħccos⁡ϑ+imwθ/2ħsin⁡ϑP+1+αP2iħeB/2c∂/∂Py+ħmw∂/∂Pxmcψ~1P,ϑ=-N~eilϑmc11+αP2λ′/2αP21+αP2l/21+eBθ4ħccos⁡ϑ+imwθ2ħsin⁡ϑP+iħeB2csin⁡ϑ+ħmwcos⁡ϑ-λ′αP+lP-ħeBl2c1+αP2cos⁡ϑP-ilħmw1+αP2sin⁡ϑPPn~λ′-1,l+iħeB2csin⁡ϑ+ħmwcos⁡ϑ2αP1+αP2n+l+λ′Pn~-1λ′,l-1,ψ~4P,ϑ=-1+eBθ/4ħcsin⁡ϑ-imwθ/2ħcos⁡ϑP+1+αP2-iħeB/2c∂/∂Px+ħmw∂/∂Pymcψ~1P,ϑ=-N~eilϑmc11+αP2λ′/2αP21+αP2l/21+eBθ4ħcsin⁡ϑ-imwθ2ħcos⁡ϑP+-iħeB2ccos⁡ϑ+ħmwsin⁡ϑ-λαP+lP+iħeB2csin⁡ϑP+ħmwcos⁡ϑPil1+αP2Pn~λ′-1,l+-iħeB2ccos⁡ϑ+ħmwsin⁡ϑ2αP1+αP2n+l+λ′Pn~-1λ′,l-1,ψ~5P,ϑ=0.The normalization constant N is(49)N~=α1/2Mc2n~!2n~+λ′+lΓn~+λ′+lEn~lΓn~+λ′Γn~+l+11/2;finally, the corresponding probability density of every component can be expressed as(50)P′′i=∫0π∫0∞ψ~-iP,ϑβ0ψ~iP,ϑdpdϑ,i=1,2,3,4,5.


## 4. Special Cases and Discussions

In this section, the natural unit(ħ=c=m=w=1) is employed. Now, let us check the special cases. First, when the minimal length parameter α→0, the energy relation (44) reduces to(51)E~n~l=±mc21-2ħm2c2mw1+eBθ4ħc+eBθmw4ħc-ħlm2c2eBc1+eBθ4ħc+m2w2θħ+K′~1/2,n~,l=0,1,2,…,where(52)K′~=2ħ2m2c22n~+l+1m2w2+eB2c2l2+1+eBθ/4ħc2+m2w2θ2/4ħ2ħ2m2w2+eB/2c21/2.The energy has plotted the energy eigenvalues versus B in Figure 1. We see that the energy E increases monotonically with the magnetic field parameter B and the tendency of the spectrum can be observed for large numbers. It also shows that, for one principal quantum number, the energy E increases with the increase of the azimuthal quantum number. The energy relation in the special case of θ=0, that is, for vanishing NC parameter, gives(53)E~n~l=±mc2αħ2m2c24n~2+4n~l+2l2m2w2+eB2c2-2ħmwm2c2-ħleB-2ħαeBmwm2c3+1+2ħ2m2c22n~+l+1m2w2+eB2c2α+α1-K′′~1/2,where(54)K′′~=1-2ħαmw-2ħ2α2eBmwl/c-l2α2ħ2m2w2+eB/2c2α2ħ2m2w2+eB/2c2.From the result shown in Figure 2, we see that the energy eigenvalues also increase monotonically with the magnetic field variable B, and the profile shows that the energy eigenvalues first has a slow-growth and then rapidly increases. Finally, when both the noncommutative and minimal length parameters are absent, that is, α=ϑ=0, the energy spectrum degrades into(55)E~n~l=±mc21-ħleBm2c3-2ħwmc2+ħ4n~+2l+2w2+eB/2mc21/2mc21/2;obviously, it is strictly consistent with [46]. In Figure 3, we have depicted the energy values versus B. It is also observed that the energy E increases monotonically with the magnetic field parameter B, and for one principal quantum number, the energy E increases with the increase of the azimuthal quantum number.Figure 1
The energy eigenvalues versusB (θ=0.00003, m=1).Figure 2
The energy eigenvalues versusB (α=0.05, m=1).Figure 3
The energy eigenvalues versusB (m=1).


## 5. Conclusions

This paper is devoted to study of the (2 + 1)-dimensional Duffin-Kemmer-Petiau oscillator for spin 0 particle under a magnetic field in the presence of a minimal length in the NC space. We first analyze the DKP oscillator in the presence of a minimal length in NC space; by employing the momentum space representation, the energy spectrum is obtained as well as the wave functions and the corresponding probability densities of the system are reported in terms of the Jacobi polynomials. Subsequently, we generalize this quantum model into the framework of a magnetic field and report the corresponding results, respectively. Finally, this quantum model for special cases is discussed and the numerical results are depicted, respectively. It shows that the energy eigenvalues increase monotonically with the magnetic field variableB for the minimal length parameter and the NC parameter, respectively, and for one principal quantum number, the energy E increases with the increase of the azimuthal quantum number.



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*Source: 2843020-2017-07-18.xml*