In this paper, we investigated the two-dimensional Klein–Gordon oscillator in non-commutative quantum mechanics (NCQM). We also studied the case of a spin-0 particle moving in a background magnetic field with the Cornell potential in commutative space, non-commutative space, and non-commutative space by using a quasi-exact methodology. The Hamiltonian was modified by the non-commutative parameter θ. We observed that the terms related to the deformation parameter can be taken as perturbation terms in QM. It was demonstrated that the non-commutative Hamiltonian was derived from the Moyal–Weyl multiplication and the Bopp shift method. We numerically calculated the energy spectrum in both commutative and non-commutative spaces. The behavior of all energies (the first, second, third, and fourth states) for the magnetic field was shown graphically. Furthermore, we derive the non-relativistic limit of the energy eigenvalues, which were comparable to the energy eigenvalues in the presence of the magnetic field in commutative space, known as the Zeeman effect.

String theory and quantum gravity suggest the non-commutative (NC) formulation of quantum mechanics, in which common commutation relations are modified.1–3 Non-commutative position operators are referred to as non-commutative space (NCS). The case in which both space–space and momentum–momentum are non-commuting variables is distinct from the case where only the space–space is a non-commuting variable, which is called the non-commutative phase space (NCPS). Therefore, in NCPS, both coordinate and momentum operators are non-commutative.4,5 The NC formulation of quantum mechanics has significant implications; therefore, the important concepts and methods of quantum mechanics, including the quantum Hall effect, have been reviewed in this context.6,7 In addition, the case of M-theory has been scrutinized in Ref. 8, and quantum gravity has been studied in Ref. 9. An immediate consequence of the NC formulation of quantum mechanics is the mathematical modification of wave equations.

So far, the non-relativistic Schrödinger equation and relativistic Dirac, Klein–Gordon (KG), and Duffin–Kemmer–Petiau (DKP) equations have been investigated in this context by a quasi-exact methodology.10,11 Furthermore, many papers have been dedicated to studying several features of quantum mechanics in commutative space (CS), NCS, and NCPS.12–31 In this paper, we consider the phenomenological Cornell potential, which has a solid physical interpretation.

The Cornell potential is the sum of the two asymptotic limits and is supported by theoretical calculations using Wilson loops32 and renormalization cancellation techniques33 and has yielded successful results in the annals of particle physics. The interaction potential contains both the coulomb and linear confining terms, and has been among the most successful interactions in high-energy physics.34–36 The particle–particle, particle–antiparticle, or antiparticle–antiparticle potential in a plasma environment is generally defined as U(r, T) = zizjV(rij), where zi = +1 or zi = −1 when the particle corresponds to a particle or antiparticle, respectively. The Cornell potential contains both the coulomb and linear confining terms, and has been among the most successful interaction of high energy physics.32–36 

Additionally, we derive the Moyal–Weyl multiplication and Bopp’s shift for Klein–Gordon oscillators (KGO) in the NCS. The KG equation (KGE) provides us a relativistic basis to study spin-zero bosons. Therefore, the solution of the equation under various phenomenological interactions becomes the first, and perhaps the most important, step-in further studies. There are many interesting parallel works, based on a variety of methodologies, on the D-dimensional space for a wide range of physical potentials, including the Rosen–Morse potential, Mie-type potential, etc.37–65 

The organization of this paper is as follows:

Section II introduces the NC quantum mechanics (NCQM). Section III, investigates the (1 + 2) dimensions of the KGO in a CS in the presence of a uniform magnetic field. In Secs. IV–VI, we solve the KGO equation with the Cornell potential in the presence of a magnetic field in NCS and in the NCPS, respectively. Finally, we present our conclusions in Sec. VII.

In N-dimensional CS, the coordinates (xi) and momentum (pi) are satisfied by the well-known commutation relations is given by
xi,xj=0,pi,pj=0,xi,pj=iδij.
(1)
However, at very small scales, on the order of the string scale, the space–momentum operators do not commute. In fact, the same may occur for the space–space components, leading to a different scenario. Several problems in quantum mechanics have been studied in the context of NCS or NCPS.28,29 In quantum mechanics, several problems have been considered in NCS or NCPS.37–39 NCS maps onto CS by replacing the coordinates xi and the momenta pi, respectively, by the operators x̂i and p̂i, which obey the algebra
x̂i,x̂j=iθij,x̂i,p̂j=iδij,p̂i,p̂j=0,
(2)
where θ is a real positive parameter, θij = θɛij, and ɛij is an anti-symmetric tensor. In two dimensions, the matrix θij can be written as
θij=00000θ0θ0,
(3)
where θi = 1/2 (εijkθjk) denotes the constant non-commutativity parameter. Thus, we can change the xi by their Bopp shifts as
x̂ixi12θεijpj,
(4)
where x and p are the quantum phase–space variables. The Moyal product is defined as
f*gx=expiθijxixjfxigxj=f(x)g(x)+i2θijifjgxi=xj+(θ2),
(5)
where f (x) and g (x) are two arbitrary functions. In this case, the Moyal–Weyl (MW) product can be replaced by a Bopp shift of Eq. (4), where the *-product can be changed into a commutative product. In this case, the operators xi and pi obey the commutation relations37,
x̂i,x̂j=iθij,p̂i,p̂j=iθ̄ij,x̂i,p̂j=iδij,
(6)
where θ̄ij is also an anti-symmetric constant tensor and θ̄ij=θ̄εij. Thus, to map the NCPS onto the CS, we must change x̂ and p̂ as
x̂iλxi12λθεijpj,p̂iλpi+12λθ̄εijxj
(7)
Here, the constant λ is a scaling factor. The parameters λ, θ̄, and θ are the NC parameters of the phase space related via37,38
θ̄ijθij=θijθ̄ij=θθ̄I=4λ2λ21I,
(8)
where I is the unit matrix. Finally, for future use, we evaluate the Moyal product of a potential as follows:
Vx*ψ(x)=expi2θjkj(1)k(2)V(x1)ψ(x2)x1=x2=x={V(x1)+n=1(i/2)nn!j1jnV(x1)k1kn}ψ(x2)x1=x2=x.
(9)
We use the following Fourier transform of F (x) to the solution of Eq. (9):
j1jnV(x)p̃k1p̃knψ(x)=ind2keikx(kp̃)nV(k)ψ(x),
(10)
where p̃i=θijpj, and summing over n in Eq. (10), it is transformed as60 
Vx*ψ(x)=d2keikx12p̃V(k)ψ(x)=Vx12p̃ψ(x).
(11)
The KGO in two dimensional in CS can be defined by the following equation:
2c22+mc2+S(r)2ψn,lr=εn,l(CS)V(r)2ψn,lr,
(12)
where P=i, where m, ε, and c are the mass of the particle, the energy, and the velocity of light, respectively. S(r) and V(r) are the scalar and vector parts of the potential, respectively. We consider the Hamiltonian of the KGO via minimal coupling р → pimωr, which given by
c2P+imωrPimωrψn,lr={(εn,l(C))2+V2m2c4S22mc2S2εn,l(C)V}ψn,lr.
(13)
where ω = eB/m is the oscillator frequency of the moving particle in the constant magnetic field, p is the momentum PPeArc and Ar=B×r2, A is the vector potential. We investigate a uniform magnetic field as B=(0,0,B)=Bk̂. Hence, from Eq. (13), we can write the KGO, in a constant magnetic field, in the form,
c2Pe2cB×r+imωrPe2cB×rimωrψn,lr={(εn,l(C))2+V2m2c4S22mc2S2εn,l(C)V}ψn,lr.
(14)
We represent the scalar and vector Cornell interaction by Ref. 37,
Vr=V0r+V1r,Sr=S0r+S1r.
(15)
Substitution of the latter in Eq. (14) gives
p2+m2ω2r2+e2B2r24c22mωeBLzc(εn,l(C))2+m2+S02r2+S12r2+2S0S1V02r2V12r22V0V1+2V0εn,l(C)r+2V1En,l(C)r+2S0mr+2S1mrψn,lr=0
(16)
Equation (16) in cylindrical coordinates is written as
d2dr2+1rddr+1r2d2dφ2m2ω2r2e2B2r24c2+2mω+eBl(C)c+(εn,l(C))2m2S02r2S12r22S0S1+V02r2+V12r2+2V0V12V0εn,l(C)r2V1εn,l(C)r2S0mr2S1mrψn,lr=0
(17)
Now, for the solution of Eq. (17), applying an ansatz to the radial wave function ψn,l(r)=eil(C)φRn,l(C)(r), After some generalization and collection, we have
{d2dr2+1rddrl2r2}Rn,l(C)(r)+m2ω2r2+2mωe2B2r24c2+eBl(C)c+(εn,l(C))2m2S02r2S12r22S0S1+V02r2+V12r2+2V0V12V0εn,l(C)r2V1εn,l(C)r2S0mr2S1mrRn,l(C)(r)=0
(18)
To remove the first-order derivative, we apply the transformation Rn,l(C)(r)=r1/2Un,l(C)r and bring Eq. (18) into the form
d2Un,l(C)rdr2+1r214l2+V12S12+r2V02S02m2ω2e2B24c2r2V0εn,l(C)+2S0m1r2V1εn,l(C)+2S1m+2mω+eBl(C)c+(εn,l(C))2m22S0S1+2V0V1Un,l(C)r=0
(19)
This leads to the following solution in parameters of ζi:10,38
ζ1(C)=1/4(l(C))2+V12S12,
(20a)
ζ2(C)=2V1εn,l(C)2S1m,
(20b)
ζ3(C)=2mωeBl(C)/c(εn,l(C))2+m2+2S0S12V0V1,
(20c)
ζ4(C)=2V0εn,l(C)+2S0m,
(20d)
ζ5(C)=V02+S02+m2ω2+e2B2/4c2,
(20e)
we calculate
d2Un,l(C)rdr2+ζ3(C)+ζ1(C)r2+ζ2(C)rζ4(C)rζ5(C)r2Un,l(C)r=0.
(21)
Now, we use an ansatz of the following form for the radial wave function:56,57
Un,lr=n=0anrn+μeβr+1/2qr2,
(22)
we have
n=0ann+μn+μ1rn+μ2+n=12an1βn+μ1rn+μ2+n=2[2qan2n+μ2+qan2+β2an2]rn+μ2+n=32βqan3rn+μ2+n=4q2an4rn+μ2ζ3n=2an2rn+μ2+ζ1n=2anrn+μ2+ζ2n=2an1rn+μ2ζ4n=2an3rn+μ2ζ5n=2an4rn+μ2=0
(23)
Herein,
a0μμ1+ζ1=0,
(24a)
a1=ζ2+2βμμμ+1+ζ1,
(24b)
a2=2a1βμ+1+2a0qμ+a0q+a0β2ζ3a0+ζ2a1μ+1μ+2+ζ1,
(24c)
a3=2a2βμ+2+2a1qμ+1+a1q+a1β2+2βqa0ζ3a1+ζ2a2ζ4a0μ+2μ+3+ζ1,
(24d)
an=q2ζ5an4+2βqζ4an3+2qn+μ2+q+β2ζ3an2+2βn+μ1+ζ2an1μ+n1μ+n+ζ1.
(24e)
We know that there are some boundaries, and the series must be bounded for n = nr. The latter propels to the relations
μ(C)μ(C)1+ζ1(C)=0,
(25a)
(q(C))2ζ5(C)=0,
(25b)
2β(C)q(C)ζ4(C)=0,
(25c)
2q(C)n+μ(C)2+q(C)+βC2ζ3(C)=0,
(25d)
2β(C)n+μ(C)1+ζ2(C)=0.
(25e)
Corresponding to
μ(C)=1±4(l(C))2V12+S122,
(26a)
q(C)=±V02+S02+m2ω2+e2B2/4c2,S0>V0
(26b)
β(C)=±V0εn,l(C)+S0mV02+S02+m2ω2+e2B2/4c2,
(26c)
where the coefficients β(C), q(C), and μ(C) are NCS parameters. The acceptable physical limit is the negative sign for β(C) and q(C) and the positive sign for μ(C). Therefore, the energy relation, from Eqs. (26a), (26b), (26c), and (20), is
V02+S02+m2ω2+e2B2/4c2(2(l(C))2V12+S12+2n+2)+V0εn,l(C)+S0m2V02+S02+m2ω2+e2B2/4c2+2mω+eBl(C)/c+(εn,l(C))2m22S0S1+2V0V1=0.
(27)
It would be interesting to investigate the obtained results of Eq. (21) for the special case of A2 = A4 = 0, which corresponds to the equation of harmonic oscillator. From Eqs. (25e) and (25b), we can find
(V0εn,l(C)+S0m)n12+(l(C))2V12+S12V02+S02+m2ω2+e2B2/4c2=V1εn,l(C)+S1m.
(28)
The above relation is a constraint relation.
By using the Moyal product, we can map the NCS onto the CS. In other words, we have to consider coordinates change as follows:
rr+12θ×p,xx12θpyyy+12θpx.
(29)
Now, we consider the NC-KGO with a Cornell interaction in the presence of a magnetic field. To do this, we take the KGO equation in NCS as follows:
PeB×r2c+imωrPeB×r2cimωr[V2S22mS2εn,l(C)V]*ψn,lr={(εn,l(C))2m2}ψn,lr.
(30)
We can map the NCS onto the CS via Bopp shifts; thus, we can rewrite Eq. (30),
PeB×r2ce(B×(θ×P))4c+imωr+imω(θ×P)2PeB×r2ce(B×(θ×P))4cimωrimω(θ×P)2(V2S22mS2εn,l(C)V)r+12θ×pψn,lr={(εn,l(C))2m2}ψn,lr.
(31)
Under these Bopp shifts, a radial form potential in NCS takes the form60 
Vr=Vr12p=Vxi12θijpjxi12θijpj=Vr+12θ×p.V(r)+O(θ2)=VrLθ2rVr+O(θ2)VrLθ2rVr.θ=(θi),
(32)
where r=xixi and L=r×p is the angular momentum operator. The other higher order terms, besides being higher powers in θ which in its own turn is very small, are also higher powers in momenta. In this paper, we find Cornell potential in NCS,
Vr*ψ(r)=V0r+V1rLθV02r+LθV12r3+Oθ2,
(33a)
Sr*ψ(r)=S0r+S1rLθS02r+LθS12r3+Oθ2.
(33b)
By the same token as the previous section, we have
1+eBθ2c+eBθ4c2+mωθ22P2+r2m2ω2+eB2c2+S02V02+r2V0εn,l(C)+2S0m+1r2V1εn,l(C)+2S1mθLz(mS0+εn,lV0)+θLz(S1m+V1εn,l(C))1r3θLz(V12S12)1r4eBc+m2ω2θ+e2B2θ4c2+θLz(V02S02)Lz2mωmωeBθ2c(εn,l(C))2+m2+2S0S12V0V1+1r2(S12V12)ψn,lr=0
(34)
By considering a vanishing coefficient for Lz, we have (eBc+m2ω2θ+e2B2θ4c2+θLz(V02S02))=0. Alternatively, one can express this equation explicitly in the following form:
θ=eBcm2ω2c2+e2B2/4+Lz(V02S02)c2,
(35)
and therefore, the critical value of magnetic field is obtained as
B=±2ceθ(+2θ2m2ω2θ2Lz(V02S02)).
(36)
Note that, taking the two terms of (34) into account as perturbation terms [i.e., Vθ = −1/r3 Lzθ (V1εnl + S1m) + 1/r4 Lzθ (V12 − S12)], first we solve the equation in the absence of perturbation terms, and we compute the energy eigenvalues for KGO using the ansatz method. Thus, Eq. (34) becomes
δd2dr2+1rddr+1r2d2dφ2r2m2ω2+eB2c2+S02V02r2V0εn,l(NC)+2S0m1r2V1εn,l(NC)+2S1mθLz(mS0+εn,lV0)+eBc+m2ω2θ+e2B2θ4c2+θLz(V02S02)l(NC)+2mω+mωeBθ2c+(εn,l(NC))2m22S0S1+2V0V11r2S12V12ψn,lr=0
(37)
We choose the ansatz to the solution Eq. (37), ψn,lr=eil(NC)φRn,l(NC)(r) and Rn,l(NC)(r)=r1/2Un,l(NC)r, and choose the following change of variable, taking S0 > V0:
δ=1+eBθ2c+eBθ4c2+mωθ22,
(38a)
ζ1(NC)=1/4(l(NC))2+V12/δS12/δ,
(38b)
ζ2(NC)=2V1εn,l(NC)/δ2S1m/δ+θLz(mS0+εn,lV0)/δ,
(38c)
ζ3(NC)=1δeBc+m2ω2θ+e2B2θ4c2+θLz(V02S02)l(NC)2mωmωeBθ2c(εn,l(NC))2+m2+2S0S12V0V1,
(38d)
ζ4(NC)=2V0εn,l(NC)+2S0m/δ,
(38e)
ζ5(NC)=(V02+S02+m2ω2+e2B2/4c2)/δ.
(38f)
By similar transformations, we obtain
d2Un,lNCrdr2+ζ3(NC)+ζ1(NC)r2+ζ2(NC)rζ4(NC)rζ5(NC)r2Un,lNCr=0.
(39)
As the series solution of the latter must be bound for a typical n = nr, we obtain
μ(NC)=1±4l2+ζ1(NC)/δ2,
(40a)
q(NC)=±ζ5(NC)/δ,
(40b)
β(NC)=±ζ4(NC)/δ2ζ5(NC)/δ,
(40c)
and
2ζ5(NC)δ1+l2+ζ1(NC)/δ2+n2ζ5(NC)δ+(ζ4(NC))24ξ5(NC)δζ3(NC)δ=0,
(41)
where β(NC), q(NC), and μ(NC) are the parameters of NCS. By placing the appropriate parameters, we obtain
(V02+S02+m2ω2+e2B2/4c2)1+eBθ2c+eBθ4c2+mωθ224(l(NC))2V12S121+eBθ2c+eBθ4c2+mωθ22+n2+V0εn,l(NC)+S0m2(V02+S02+m2ω2+e2B2/4c2)1+eBθ2c+eBθ4c2+mωθ2211+eBθ2c+eBθ4c2+mωθ22×{eBc+m2ω2θ+e2B2θ4c2+θLz(V02S02)l(NC)2mωmωeBθ2c(εn,l(NC))2+m2+2S0S12V0V1}=0.
(42)
If θ = 0, similar to [Eq. (27)] that in CS, the zero coefficients r, 1/r in Eq. (37) reduce, which corresponds to that of harmonic oscillator with a centrifugal potential barrier.
From Eq. (38c), we can find the constraint relation as
12ζ5(NC)ζ4(NC)n12+(l(NC))2δ+ζ1(NC)=V1εn,l(NC)+S1mθLz(mS0+εn,lV0),
(43)
where n and is the radial quantum number angular quantum number, respectively. In Fig. 1, where we have plotted the energies in NC and commutative cases for the sets (n = 1, l = 0), (n = 2, l = 0) and (n = 3, l = 0), respectively, we see the B value for which the energies in the two cases coincide. Above such a value, which in the above-mentioned cases, respectively, is 0.0238, −0.029, and −0.0502, the NC energy has a higher value, and below that situation, it is vice versa. In Fig. 1, we plotted the behavior of the energy spectra in the NCS and CS cases for the series (n = 1, l = 0), (n = 2, l = 1), (n = 3, l = 1), and (n = 4, l = 3), respectively, and we observe the role of the magnetic field B. We observe that, which in the aforementioned cases are 3.630, 4.218, 4.619, and 5.472, the NC energy values have a higher value, and another situation is the opposite, respectively.
FIG. 1.

The energies for non-commutative and commutative cases vs the magnetic field B for the values S0 = 2.9, S1 = 1.9, V0 = −2, V1 = 1, and θ = 0.09.

FIG. 1.

The energies for non-commutative and commutative cases vs the magnetic field B for the values S0 = 2.9, S1 = 1.9, V0 = −2, V1 = 1, and θ = 0.09.

Close modal

Also, in Table I, we have derived the values of the magnetic fields where the εn,lNC=εn,lC, for arbitrary of θ.

TABLE I.

The coinciding value of the magnetic field non-commutative space for (=c=e=ω=V1=1,V0=2,θ=0.09,S0=3,S1=1.9).

n,lɛn,ln,lɛn,l
1,0 3.647 320 703, −2.476 588 996 3,2 4.850 496 171, −3.679 764 463 
2,0 4.120 328 803, −2.949 597 096 4,0 4.913 963 105, −3.743 231 397 
2,1 4.211 681 319, −3.040 949 612 4,1 4.988 882 057, −3.818 150 350 
3,0 4.537 119 719, −3.366 388 011 4,2 5.201 829 021, −4.031 097 314 
3,1 4.619 044 002, −3.448 312 295 4,3 5.437 247 426, −4.266 515 719 
n,lɛn,ln,lɛn,l
1,0 3.647 320 703, −2.476 588 996 3,2 4.850 496 171, −3.679 764 463 
2,0 4.120 328 803, −2.949 597 096 4,0 4.913 963 105, −3.743 231 397 
2,1 4.211 681 319, −3.040 949 612 4,1 4.988 882 057, −3.818 150 350 
3,0 4.537 119 719, −3.366 388 011 4,2 5.201 829 021, −4.031 097 314 
3,1 4.619 044 002, −3.448 312 295 4,3 5.437 247 426, −4.266 515 719 

In Tables II and III, we have provided the values of the magnetic field where the energies in the NC and commutative cases coincide, εn,lNC=εn,lC. It can be shown that we recover the case of CS as a special case of NCS when the non-commutativity parameter goes to zero. This is because the NC parameter is a measure of the deviation from the commutative case, and when it becomes zero, the system becomes commutative (θ = 0).42–49 

TABLE II.

The coinciding value of the magnetic field in commutative and non-commutative space for =c=e=ω=m=1,θ=0.09.

n,lɛn,lBn,lɛn,lB
1,0 3.630 0.0135 3,2 4.879 0.3950 
2,0 4.082 0.0136 4,0 4.843 0.0524 
2,1 4.218 0.6360 4,1 4.974 0.1890 
3,0 4.482 0.0013 4,2 5.223 0.2690 
3,1 4.619 0.3950 4,3 5.472 0.4070 
n,lɛn,lBn,lɛn,lB
1,0 3.630 0.0135 3,2 4.879 0.3950 
2,0 4.082 0.0136 4,0 4.843 0.0524 
2,1 4.218 0.6360 4,1 4.974 0.1890 
3,0 4.482 0.0013 4,2 5.223 0.2690 
3,1 4.619 0.3950 4,3 5.472 0.4070 
TABLE III.

The perturbation energy eigenvalues in non-commutative space.

n,lɛn,ln,lɛn,l
1,0 3,2 −4.989 556 555 × 10−16 
2,0 4,0 
2,1 −2.251 428 841 × 10−15 4,1 −2.995 032 490 × 10−15 
3,0 4,2 −5.700 171 891 × 10−16 
3,1 −2.735 512 069 × 10−15 4,3 −5.169 944 951 × 10−16 
n,lɛn,ln,lɛn,l
1,0 3,2 −4.989 556 555 × 10−16 
2,0 4,0 
2,1 −2.251 428 841 × 10−15 4,1 −2.995 032 490 × 10−15 
3,0 4,2 −5.700 171 891 × 10−16 
3,1 −2.735 512 069 × 10−15 4,3 −5.169 944 951 × 10−16 
In this section, we solve the non-commutative KGO equation for the Cornell potential, via the perturbation theory. One can observe that Eq. (37) has not been solved yet exactly in the presence of the perturbation term, i.e., [−1/r3 Lzθ (V1εnl + S1m) + 1/r4 Lzθ (V12S12)]. In this section, we solve Eq. (37) with the perturbation term. Let us write NC-KGO equation with the Cornell interaction
d2Un,lrdr2+1r214l2S12V12ξ+r2m2ω2e2B24c2+S02V02r(2V0εn,l+2S0m)1r(2V1εn,l+2S1mlθV0εn,llθS0m)1r3(lθS1m+lθV1εn,l)+θl(V12S12)1r4+eBc+m2ω2θ+e2B2θ4c2+θ(V02S02)l+2mω+mωeBθ2c+εn,l2m22S0S1+2V0V/ξUn,lr=0
(44)
In this case, we consider the perturbation
H1=0ψ*H1ψ(2πr)dr,
(45)
where
ψn,l(r)=eilφn=0anrn+μeβr+12qr2r12.
(46)
Therefore,
H1=lθ2π(S1m+V1εn,l)0an2(rn+μ)2e2pr+qr21r3dr+(V12S12)0an2(rn+μ)2e2βr+qr21r4dr.
(47)
We see that we recover the solution of the CS, and the perturbation energy is zero for θ = 0. In the first-order perturbation theory, the expectation value of 1/r3 and 1/r4 with respect to the accurate solution can be written as66 
nlm1rknlm=0Rnl2(r)1rkdrδmm=2kakn!2(n+v+1)Γ(n+2v+2)×0x2v+2ke2x[Ln2v+1(x)]2dxδmm=f(k)k=3,4,5,6
(47a)
By using the formula between the confluent hypergeometric function and the related Laguerre polynomials, such as Lnv(x)=Γ(n+v+1)Γ(n+1)Γ(v+1)F(n;v+1;x) and
0xv1ex[F(n;γ;x)]2dx=n!Γ(v)γ(γ+1)(γ+n1)1+n(γv1)(γv)12γ+n(n1)(γv2)(γv1)(γv)(γv+1)1222γ(γ+1)++n(n1)1(γvn)(γv1)(γv+n1)1222n2γ(γ+1)(γ+n1).
(47b)
So,
nlm1r4nlm=0Rnl2(r)1r4drδmm=16!a4n!2(n+v+1)Γ(n+2v+2)×0x2v11ex[Ln2v+1(x)]2dxδmm=8!a4n!(n+v+1)Γ(n+2v+2)Γ(n+2v+2)Γ(n+1)Γ(2v+2)×0x2v2ex[F(n;2v+2;x)]2dxδmm=4a4(2v1)v(2v+1)(n+v+1)×1+3n(v+1)+3n(n1)(v+1)(2v+3)δmm=f(4).
(47c)
In this section, we analyze the KGE in NCPS, to map NCPS into CS, using the following expressions:
xxλ12λθpy,pxpxλ+12λθ̄y,yyλ+12λθpx,pypyλ12λθ̄x,
(48)
where the scale factor λ is an arbitrary constant parameter. Thus, we rewrite (14) in the form
pλθ̄×r2λeB2c×rλ+θ×p2λ+imωrλ+θ×p2λpλθ̄×r2λeB2c×rλ+θ×p2λimωrλ+θ×p2λV2+S2+2mS+2εn,l(NC)Vψn,lr={(εn,l(NC))2m2}ψn,lr.
(49)
By calculations, we can write in the form
λ2+m2ω2θ24λ2+e2B2θ216c2λ2+eBθ2cp2+r2V0εn,l(NC)+2S0m+1r2V1εn,l(NC)+2S1m+1r2S12V12+r2θ̄24λ2+e2λ2B24c2+m2ω2λ2+eBθ̄2c+S02V02θ̄+eλ2Bc+m2ω2θ+eθθ̄B4cλ2+e2B2θ4c2Lz+2mωλ2+mωeBθcmωθθ̄2λ2(εn,l(NC))2+m2+2S0S12V0V1ψn,lr=0.
(50)
If the coefficient of the angular momentum is zero, we can obtain the NC parameters as θ̄+eλ2B/c+m2ω2θ+eθθ̄B/4cλ2+e2B2θ/4c2=0, and then they become
θ=θ̄c+eλ2Bm2ω2c+eθ̄B/4λ2+e2B2/c4
(51)
and the magnetic field
B=±c2θλ2e4λ4θθ̄+16λ88λ4θθ̄+θ2θ̄216θ2m2ω2λ4.
(52)
Thus, Eq. (50) becomes
ηd2dr2+1rddr+1r2d2dφ2r2V0εn,l(NCP)+2S0m1r2V1εn,l(NCP)+2S1m1r2S12V12r2θ̄24λ2+e2λ2B24c2+m2ω2λ2eBθ̄2c+S02V02+θ̄+eλ2Bc+m2ω2θ+eθθ̄B4cλ2+e2B2θ4c2l(NCP)2mωλ2mωeBθc+mωθθ̄2λ2+(εn,l(NCP))2m22S0S1+2V0V1ψn,lr=0,
(53)
and using the following change of variables ψn,lr=eil(NCP)φRn,l(NCP)(r) and Rn,l(NCP)(r)=r1/2Un,l(NCP)r, so
η=λ2+m2ω2θ24λ2+e2B2θ216c2λ2+eBθ2c,
(54a)
ζ1(NCP)=1/4(l(NCP))2+V12/ηS12/η,
(54b)
ζ2(NCP)=(2V1εn,l(NCP)+2S1m)/η,
(54c)
ζ3(NCP)=1ηθ̄+eλ2Bc+m2ω2θ+eθθ̄B4cλ2+e2B2θ4c2l(NCP)2mωλ2mωeBθc+mωθθ̄2λ2(εn,l(NCP))2+m2+2S0S12V0V1,
(54d)
ζ4(NCP)=1η(2V0εn,l(NCP)+2S0m),
(54e)
ζ5(NCP)=1ηθ̄24λ2+e2λ2B24c2+m2ω2λ2+eBθ̄2c+S02V02.
(54f)
By similar transformations, we obtain
d2Un,lNCprdr2+ζ3(NCP)+ζ1(NCP)r2+ζ2(NCP)rζ4(NCP)rζ5(NCP)r2Un,lNCpr=0.
(55)
We see that Eq. (55) corresponding to Eq. (21). On the other hand, the series must be bounded for n = nr. The latter leads to the equations
μ(NCP)=1/21±4(l(NCP))2+S12V12λ2+m2ω2θ24λ2+e2B2θ216c2λ2+eBθ2c,
(56a)
q(NCP)=±θ̄24λ2+e2λ2B24c2+m2ω2λ2+eBθ̄2c+S02V02λ2+m2ω2θ24λ2+e2B2θ216c2λ2+eBθ2c,
(56b)
β(NCP)=±V0εn,l(NCP)+S0mθ̄24λ2+e2λ2B24c2+m2ω2λ2+eBθ̄2c+S02V02λ2+m2ω2θ24λ2+e2B2θ216c2λ2+eBθ2c,
(56c)
where the coefficients β(NCP), q(NCP)μ(NCP) are parameters of NCPS and the eigenvalue relation can be written as
2ζ5(NCP)η1±4(l(NCP))2+ζ1(NCP)/η2+n2ζ5(NCP)η+(ζ4(NCP))24ηζ5(NCP)ζ3(NCP)η=0.
(57)
By placing the appropriate parameters, we have
2θ̄24λ2+e2λ2B24c2+m2ω2λ2+eBθ̄2c+S02V02(n2)λ2+m2ω2θ24λ2+e2B2θ216c2λ2+eBθ2c+(l(NCP))2λ2+m2ω2θ24λ2+e2B2θ216c2λ2+eBθ2c+S12V12+V0εn,l(NCP)+S0m2θ̄24λ2+e2λ2B24c2+m2ω2λ2+eBθ̄2c+S02V02θ̄+eλ2Bc+m2ω2θ+eθθ̄B4cλ2+e2B2θ4c2l(NCP)2mωλ2+mωeBθcmωθθ̄2λ2(εn,l(NCP))2+m2+2S0S12V0V1=0.
(58)
If θ=θ̄=0,λ=1 (because λ2=1/2±1/21+θθ̄) similar to Eq. (27) of CS, we have a centrifugal potential barrier. From Eq. (54c), we can find the constraint relation as
12ζ5(NCP)ζ4(NCP)n12+(l(NCP))2δ+ζ1(NCP)=V1εn,l(NCP)+S1m.
(59)
In Fig. 2, where we have, respectively, plotted the energies in NC and commutative cases for the sets (n = 1, l = 0), (n = 2, l = 0), and (n = 3, l = 0), we see the B value for which the energies in the two cases coincide. Above such a value, which in the above-mentioned cases, respectively, is 0.0026, 0.0114, and 0.6470, the NC energy has a higher value below which the situation is vice versa. However, the non-commutativity parameter, if it is non-zero, should be very tiny (plank scale) compared to the length scales of the system, one can always treat the NC effects as some perturbations of the commutative counterpart, and hence, up to first order in θ, we can use the usual wave functions and probabilities.
FIG. 2.

The energies for commutative and non-commutative phase space vs the magnetic field B for θ=0.09,θ̄=0.01,λ=1.000112469.

FIG. 2.

The energies for commutative and non-commutative phase space vs the magnetic field B for θ=0.09,θ̄=0.01,λ=1.000112469.

Close modal

Also, in Table IV, we have derived the values of the magnetic field for where εn,lNCP=εn,lC.

TABLE IV.

The coinciding value of the magnetic field in non-commutative phase space for (=c=e=ω=V1=1,V0=2,θ=0.09,θ̄=0.01,λ=1.000112469,S0=3,S1=1.9).

n,lɛn,lBn,lɛn,lB
1,0 4.413 0.0238 3,0 5.626 −0.0502 
2,0 5.059 −0.0290 3,1 5.833 0.1720 
2,1 5.280 0.3410 3,2 6.219 0.2670 
n,lɛn,lBn,lɛn,lB
1,0 4.413 0.0238 3,0 5.626 −0.0502 
2,0 5.059 −0.0290 3,1 5.833 0.1720 
2,1 5.280 0.3410 3,2 6.219 0.2670 

In this current Letter, the Klein–Gordon oscillator was studied in the background of an external magnetic field with the Cornell potential in a CS, NCS, and NCPS. Also, we have obtained the energy eigenvalues and the critical values of the magnetic field. We have also calculated the wave functions of the system explicitly by using the quasi-exact methodology. In addition, we have found the non-relativistic limit of the energy eigenvalues, which are similar to the energy eigenvalues under the action of a magnetic field in commutative space, that it is so called the Zeeman effect. The non-commutativity proposes a technique for introducing an issue in degenerate perturbation theory to a non-degenerate issue. Our results show explicitly the dependence of the energy eigenvalues on the NC parameter. Moreover, we have confirmed that, in the limit of θ → 0, we recover the results of the commutative space.

The authors take great pleasure in thanking the referee for his/her several suggestions and comments.

The authors have no conflicts to disclose.

M. Qolizadeh: Investigation (equal); Writing – original draft (equal). S. M. Motevalli: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Supervision (equal); Writing – review & editing (equal). S. S. Hosseini: Conceptualization (equal); Data curation (equal); Investigation (equal); Software (lead); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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