Exact Solutions of the (2+1)-Dimensional Dirac Oscillator under a Magnetic Field in the Presence of a Minimal Length in the Non-commutative Phase Space

1 Introduction

The Dirac relativistic oscillator is an important potential for both theory and application. For the first time, it was studied by Ito et al. [1]. They considered a Dirac equation in which the momentum p→ is replaced by p→ − imβωr→, with r→ being the position vector, m the mass of the particle, and ω the frequency of the oscillator. The interest in the problem was revived by Moshinsky and Szczepaniak [2], who gave it the name of Dirac oscillator (DO) because, in the non-relativistic limit, it becomes a harmonic oscillator with a very strong spin–orbit coupling term. Physically, it can be shown that the DO interaction is a physical system that can be interpreted as the interaction of the anomalous magnetic moment with a linear electric field [3, 4]. The electromagnetic potential associated with the DO has been found by Benitez et al. [5]. The DO has attracted a lot of interest because not only it provides one of the examples of the Dirac equation exact solvability but also for its numerous physical applications [6–10]. Recently, Franco-Villafane et al. [11] exposed the proposal of the first experimental microwave realisation of the one-dimensional DO. The experiment relies on a relation of the DO to a corresponding tight-binding system. The experimental results obtained, concerning the spectrum of the one-dimensional DO with and without the mass term, are in good agreement with those obtained in the theory. In addition, Quimbay and Strange [12, 13] showed that the DO can describe a naturally occurring physical system. Specifically, the case of a two-dimensional DO can be used to describe the dynamics of the charge carriers in graphene, and hence its electronic properties. Also, the exact mapping of the DO in the presence of a magnetic field with a quantum optics leads to regarding the DO as a theory of an open quantum systems coupled to a thermal bath [6].

The unification between the general theory of relativity and the quantum mechanics is one of the most important problems in theoretical physics. This unification predicts the existence of a minimal measurable length on the order of the Planck length. All approaches of quantum gravity show the idea that near the Planck scale, the standard Heisenberg uncertainty principle should be reformulated. The minimal length uncertainty relation appears in the context of the string theory, as a consequence of the fact that the string cannot probe distances smaller than the string scale ℏβ where β is a small positive parameter called the deformation parameter. This minimal length can be introduced as an additional uncertainty in position measurement, so that the usual canonical commutation relation between position and momentum operators becomes[x^, p^] = iℏ(1 + βp2). This commutation relation leads to the standard Heisenberg uncertainty relation Δx^Δp^ ≥ iℏ(1 + β(Δp)2), which clearly implies the existence of a non-zero minimal length Δxmin = ℏβ. This modification of the uncertainty relation is usually termed the generalised uncertainty principle (GUP) or the minimal length uncertainty principle [14–17]. Investigating the influence of the minimal length assumption on the energy spectrum of quantum systems has become an interesting issue primarily for two reasons. First, this may help to set some upper bounds on the value of the minimal length. In this context, we can cite some studies of the hydrogen atom and a two-dimensional Dirac equation in an external magnetic field. Moreover, the classical limit has also provided some interesting insights into some cosmological problems. Second, it has been argued that quantum mechanics with a minimal length may also be useful to describe non-point-like particles, such as quasi-particles and various collective excitations in solids, or composite particles (see [18] and references therein).

Nowadays, the reconsideration of the relativistic quantum mechanics in the presence of a minimal measurable length has been studied extensively. In this context, many papers were published where a different quantum system in space with the Heisenberg algebra was studied. They are the Abelian Higgs model [19], the thermostatics with minimal length [20], the one-dimensional hydrogen atom [21], the casimir effect in minimal length theories [22], the effect of minimal lengths on electron magnetism [23], the DO in one and three dimensions [24–28], the non-commutative (NC) (2+1)-dimensional DO and quantum phase transition [10], the solutions of a two-dimensional Dirac equation in the presence of an external magnetic field [29], the NC phase space Schrödinger equation [30], and the Schrödinger equation with harmonic potential in the presence of a magnetic field [31].

The study of NC spaces and their implications in physics is an extremely active area of research. It has been argued in various instances that non-commutativity should be considered as a fundamental feature of space time at the Planck scale. On the other side, the study of quantum systems in an NC space has been the subject of much interest in past years, assuming that non-commutativity may be, in fact, a result of quantum gravity effects. In these studies, some attention has been paid to the models of NC quantum mechanics (NCQM). The interest in this approach lies on the fact that NCQM is a fruitful theoretical laboratory where we can get some insight on the consequences of non-commutativity in field theory by using standard calculation techniques of quantum mechanics. Various NC field theory models have been discussed as well as many extensions of quantum mechanics. Of particular interest is the so-called phase space non-commutativity, which has been investigated in the context of quantum cosmology, black holes physics, and the singularity problem. This specific formulation is necessary to implement the Bose–Einstein statistics in the context of NCQM (see [32–36]).

The purpose of this work is to investigate the formulation of a two-dimensional DO in the presence of a magnetic field by solving fundamental equations in the framework of relativistic quantum mechanics with minimal length in the NC phase space. To do this, we first mapped the problem in question into a commutative space by using an appropriate transformations. Then, we solved it in the presence of a minimal length. We would like to mention here that the origin of relativistic Landau problem and the DO is entirely different. In the former case, the magnetic field is introduced via minimal coupling, whereas in the latter case, the interaction is introduced via non-minimal coupling and can be viewed as anomalous magnetic interaction [37, 38].

The article is organised as follows. In Section 2, we solve the DO in the presence of magnetic field in NC phase space. Then, in Section 3, we study this problem in the framework of relativistic quantum mechanics with minimal length. Finally, in Section 4, we present the conclusion.

2 The Solutions in Non-commutative Phase Space

To begin with, we note that the NC phase space is characterised by the fact that their coordinate operators satisfy the following equation [32–36]:

(1)[xν(NC), xμ(NC)] = iθ˜μν,  [pμ(NC), pν(NC)] = iθ¯μν,[xμ(NC), pν(NC)] = iℏδμν, (1)

where θ˜μν and iθ¯μν are antisymmetric tensors of space dimension. In order to obtain a theory that includes the aspects of being unitary and causal, we choose θ˜0ν = 0 and θ̅_0ν=0, which implies that the time remains as a parameter and the non-commutativity affects only the physical space. By replacing the normal product with star product, the Dirac equation in commuting space will change into the Dirac equation in NC space:

(2)H^D(p, x)⋆ψD(x) = EψD(x), (2)

where the ⋆–product Moyal between two functions is defined by

(3)(f ⋆ g)(x) = exp[i2θ˜ab∂xa∂xb]f(x)g(y)|x = y. (3)

Since the system in which we study is two dimensional, we limit our analysis to the xy plane, where the NC algebra is written by

(4)[xi(NC), xj(NC)] = iθ˜ϵij, [pi(NC), pj(NC)] = iθ¯ϵij,[xi(NC), pj(NC)] = iℏδij, (i, j = 1, 2). (4)

with ϵ_ij is two-dimensional Levi-civita tensor.

Instead of solving the NC Dirac equation by using the star product procedure, we use Bopp’s shift method, that is, we replace the star product by the usual product by making a Bopp’s shift

(5)xi(NC) = xi − 12ℏθ˜ϵijpj, pi(NC) = pi + 12ℏθ¯ϵijxj. (5)

Hence, in the two-dimensional NC phase space, (5) becomes

(6)x(NC) = x − θ˜2ℏpy, y(NC) = y + θ˜2ℏpx, px(NC) = px + θ¯2ℏy,py(NC) = py − θ¯2ℏx. (6)

In this case, the two-dimensional DO equation, in commutative space, which is written by

{cαx(px − im0ωβ˜x) + cαy(py − im0ωβ˜y) + β˜m0c2}ψD = EψD,

is modified and transformed into

(7){cαx(px(NC) − im0ωβ˜x(NC)) + cαy(py(NC) − im0ωβ˜y(NC))+ β˜m0c2}ψD = ENCψD. (7)

Using the following representation of Dirac matrices,

(8)αx = σx = (0110), αy = σy = (0−ii0), β˜ = (100−1), (8)

and with ψ_D=(ψ₁, ψ₂)^T, (7) becomes

(9)(m0c2cp−cp+−m0c2)(ψ1ψ2) = ENC(ψ1ψ2), (9)

or

(10)m0c2ψ1 + cp−ψ2 = ENCψ1, (10)

(11)cp+ψ1 − m0c2ψ2 = ENCψ2, (11)

where

(12)p− = px(NC) − ipy(NC) + im0ω(x(NC) − iy(NC)) = ϱ1(px − ipy)+ im0ωϱ2(x − iy), (12)

(13)p+ = px(NC) + ipy(NC) − im0ω(x(NC) + iy(NC)) = ϱ1(px + ipy)− im0ωϱ2(x + iy), (13)

and where

(14)ϱ1 = 1+m0ω2ℏθ˜, ϱ2 = 1 + θ¯2m0ωℏ. (14)

From (10) and (11), we have

(15){c2p−p+ − (ENC2 − m02c4)}ψ1 = 0. (15)

Now, in order to solve the last equation, and for the sake of simplicity, we bring the problem into the momentum space.

Recalling that

(16)x = iℏ∂∂px, y = iℏ∂∂px, p^x = px, p^y = py, (16)

and passing onto polar coordinates with the following definition [29]:

(17)px = pcosθ,  py = psinθ, p2 = px2 + py2, (17)

(18)x^ = iℏ∂∂px = iℏ(cosθ∂∂p − sinθp∂∂θ), (18)

(19)y^ = iℏ∂∂py = iℏ(sinθ∂∂p + cosθp∂∂θ), (19)

Equations (12) and (13) transform into

(20)p− = e−iθ{ϱ1p − λ(∂∂p − ip∂∂θ)}, (20)

(21)p+ = eiθ{ϱ1p + λ(∂∂p + ip∂∂θ)}, (21)

where

(22)λ = (1 + θ¯2m0ωℏ)m0ℏω. (22)

With the aid of these expressions, the p₋p₊ term, appears in (15), can be written by

(23)p−p+ = ϱ12p2 − 2ϱ1λ − λ2∂2∂p2 − λ2p2∂2∂θ2 − λ2p∂∂p + 2iλϱ1∂∂θ. (23)

So, (15) becomes

(24){ϱ12p2 − λ2(∂2∂p2 + 1p∂∂p + 1p2∂2∂θ2) + 2iλϱ1∂∂θ − 2λϱ1 − ζ}ψ1 = 0, (24)

with

(25)ζ = ENC2 − m02c4c2. (25)

With the help of the following relation [39]

(26)ψ1(p, θ) = f(p)eimθ, (26)

Equation (24) is modified and transforms into

(27)(d2f(p)dp2 + 1pdf(p)dp − m2p2f(p)) + (κ2 − k2p2)f(p) = 0, (27)

with

(28)κ2 = 2λϱ1(m+1) + ζλ2, k2 = ϱ12λ2. (28)

Putting that

(29)f(p) = pme−k22p2F(p), (29)

then, the differential equation

(30)F″ + (2m+1p − 2kp)F′ − [2k(m+1) − κ2]F = 0, (30)

is obtained for F(p) which by using, instead of p, the variable xt=kp², is transformed into the Kummer equation:

(31)td2Fdt2 + {m+1 − t}dFdt − 12{m+1 − κ24k}F = 0, (31)

whose solution is the confluent series ₁F₁(a; m + 1; t), with

(32)a = 12(m+1) − κ24k. (32)

The confluent series becomes a polynomial if and only if a=−n,(n=0, 1, 2,). Thus, we have [40]

(33)ψ1(p, θ) = Cn,mpme−k2p21F1(−n;|m| + 1; kp2)eimθ, (33)

(34)(ENC)n = ±m0c21 + 4(1 + m0ω2ℏθ˜)(1 + θ¯2m0ωℏ)n. (34)

The total associated wave function is

(35)ψn,m(p, θ) = (1cp+ENC + m0c2)ψ1. (35)

Now, in the presence of an external magnetic field, (7) is transformed into

(36){cαx[(px(NC) + eBy(NC)2c) − im0ωβ˜x(NC)]+ cαy[(py(NC) − eBx(NC)2c) − im0ωβ˜y(NC)] + β˜m0c2}ψD = ϵψD, (36)

where ϵ is the eigenvalue of the system. Here, the potential vectors is chosen as

(37)A→ = (−By(NC)2, Bx(NC)2, 0), (37)

and (36) can be cast into a detail form as follows:

(38)(m0c2cp˜−cp˜+−m0c2)(ψ˜1ψ˜2) = ϵ(ψ˜1ψ˜2). (38)

with

(39)p˜− = (px(NC) + eBy(NC)2c) + im0ωx(NC) − i(py(NC) − eBx(NC)2c) + m0ωy(NC), (39)

(40)p˜+ = (px(NC) + eBy(NC)2c) − im0ωx(NC) + i(py(NC) − eBx(NC)2c) + m0ωy(NC) (40)

or

(41)p˜− = px(NC) − ipy(NC) + im0ω˜(x(NC) − iy(NC)), (41)

(42)p˜+ = px(NC) + ipy(NC) − im0ω˜(x(NC)+iy(NC)), (42)

where

(43)ω˜ = ω − ωc2,  ωc = |e|Bm0c. (43)

is a cyclotron frequency.

Thus, the (2+1)-dimensional DO in a magnetic field is mapped onto the one with reduced angular frequency ω˜ in the absence of magnetic field. Hence, the only role of a magnetic field consists in reducing the angular frequency, and the entire dynamics remains unchanged.

Using the mapping defined by (6), the systems of equations become

(44)p˜− = ϱ1(px − ipy) + im0ω˜ϱ2(x − iy), (44)

(45)p˜+ = ϱ1(px + ipy) − im0ω˜ϱ2(x + iy). (45)

By the same way used above, we obtain

(46)ψ˜1(p, θ) = C˜n,mpme−k2p2 1F1(−n; |m|+1; kp2)eimθ, (46)

(47)ϵn = ±m0c21 + 4(1 + m0ω˜2ℏθ˜)(1 + θ¯2m0ω˜ℏ)n. (47)

The last equation concerning the eigenvalue is in a good agreement with the one obtained in the literature [6].

The corresponding total eigenfunction is given by

(48)ψn,m(p, θ) = (1cp˜+ϵ + m0c2)ψ˜1. (48)

In this section, we have studied the solutions of the two-dimensional DO with or without an external magnetic field by using the same way described in [29]: the authors work within a momentum space representation of the Heisenberg algebra, and by an appropriate transformation, the problem is identified as a Kummer differential equation where the solutions are well known. The solutions that we have found are in wellagreement with those obtained in the literature. This agreement allows us to extend this method by introducing the concept of the minimal length.

3 The Problem with a Minimal Length

3.1 The Solutions without a Magnetic Field

In the minimal length formalism, the Heisenberg algebra is given by

(49)[x^i, p^i] = iℏδij(1 + βp2), (49)

where β>0 is the minimal length parameter. A representation of x^i and p^i, which satisfies (49), may be taken as

(50)x^i = iℏ(1 + βp2)ddpi,  p^i = pi (50)

or

(51)x^ = iℏ(1 + βp2)ddpx, y^ = iℏ(1 + βp2)ddpy, (51)

(52)p^x = px, p^y = py. (52)

In this case, (15) is modified and becomes

(53){c2P−P+ − (ε2 − m02c4)}ψ1 = 0, (53)

with

(54)P− = ϱ1(px − ipy)−λ(1 + βp2)(∂∂px − i∂∂py), (54)

(55)P+ = ϱ1(px − ipy) + λ(1 + βp2)(∂∂px + i∂∂py). (55)

In the polar coordinates, (54) and (55) can be written as

(56)P− = e−iθ{ϱ1p − λ(1 + βp2)(∂∂p − ip∂∂θ)}, (56)

(57)P+ = eiθ{ϱ2p+λ(1 + βp2)(∂∂p + ip∂∂θ)}. (57)

When we evaluate the P₋P₊ term, we get

(58)P−P+ = ϱ12p2 + 2(1 + βp2){λϱ1(i∂∂θ − 1) − βλ2(p∂∂p + i∂∂θ)}− λ2(1 + βp2)2(∂2∂p2 + 1p∂∂p + 1p2∂2∂θ2), (58)

and then we have

(59)[ϱ12p2 + 2(1 + βp2){λϱ1(i∂∂θ − 1) − βλ2(p∂∂p + i∂∂θ)}− λ2(1 + βp2)2(∂2∂p2 + 1p∂∂p + 1p2∂2∂θ2) − ξ2]ψ1 = 0, (59)

with

(60)ξ2 = ε2 − m02c4c2. (60)

Putting that

(61)ψ1 = h(p)eimθ, (61)

Equation (60) reads

(62)[ϱ12p2 − 2(1 + βp2){λϱ1(m + 1) + βλ2(pddp − m)}−λ2(1 + βp2)2(d2dp2 + 1pddp − m2p2) − ξ2]h(p) = 0. (62)

This equation can be written in another form as follows:

(63){−a(p)d2dp2 + b(p)ddp + c(p) − ξ2}h(p) = 0, (63)

where

(64)a(p) = λ2(1 + βp2)2, (64)

(65)b(p) = −2βλ2(1 + βp2)p − λ2(1 + βp2)2p, (65)

(66)c(p) = ϱ12p2 − 2λϱ1(m+1)(1 + βp2) + 2βλ2m(1 + βp2)+ λ2(1 + βp2)2m2p2. (66)

The solutions of (63) can be found by using the following transformations [41]:

(67)h(p) = ρ(p)φ(p), q = ∫1a(p)dp, (67)

with

(68)ρ(p) = e∫χ(p)dp. (68)

Using these transformations, we obtain a form similar to the Schrödinger differential equation, so

(69)(−d2dq2 + V(q))φ(p) = ξφ(p), (69)

where

(70)χ(p) = 2b + a′4a = −12p, (70)

and

(71)V(p) = ϱ12p2 − 2λϱ1(m+1)(1 + βp2) + 2βλ2m(1 + βp2)+βλ2(1 + βp2)+λ2(1 + βp2)2p2(m2 − 14). (71)

Using that

(72)p = 1βtan(qλβ), (72)

we get

(73)V(p) = −1β + βλ2{ζ1(ζ1 − 1)sin2(qλβ) + ζ2(ζ2 − 1)cos2(qλβ)}, (73)

with

(74)ζ1(ζ1 − 1) = m2 − 14, (74)

(75)ζ2(ζ2 − 1) = (m − ϱ1βλ + 12)(m − ϱ1βλ + 32). (75)

Thus, we have

(76)(−d2dq2 + 12U0{ζ1(ζ1 − 1)sin2(αq) + ζ2(ζ2 − 1)cos2(αq)})φ(p) = ξ¯2φ(p) (76)

with ξ¯2 = ξ2 + (α12/β) and U₀=u² with u = λβ.

The last equation is the well-known Schrödinger equation in a Poschl–Teller potential where [39]

(77)U = 12U0{ζ1(ζ1 − 1)sin2(uq) + ζ2(ζ2 − 1)cos2(uq)} (77)

with ζ₁>1 and ζ₂>1. Thus, following (74) and (75), we have

(78)ζ1 = m ± 12, (78)

(79)ζ2 = 12±(m+1 − ϱ1βλ). (79)

Introducing the new variable

(80)z = sin2(uq), (80)

the Schrödinger equation is transformed into

(81)z(1 − z)φ″ + (12 − z)φ′ + 14{ξ¯2u2 − ζ1(ζ1 − 1)z − ζ2(ζ2 − 1)1 − z}φ = 0. (81)

Now, putting

(82)φ = zζ12(1 − z)ζ22Ψ(z), (82)

we arrive at

(83)z(1 − z)Ψ″ + [(ζ1 + 12)−z(ζ1 + ζ2 + 1)]Ψ′+ 14{ξ¯2u2 − (ζ1 + ζ2)2}Ψ = 0. (83)

The general solutions of this equation are [40, 42]

(84)Ψ = C1 2F1(a; b; c; z) + C2 z1 − c 2F1(a + 1 − c; b + 1 − c; 2 − c; z), (84)

where

(85)a = 12(ζ1 + ζ2 + ξ¯u2),  b = 12(ζ1 + ζ2 − ξ¯u2),  c = ζ1 + 12. (85)

With the condition a=−n, we obtain

(86)ξ¯2 = u2(ζ1 + ζ2 + 2n)2. (86)

In order to obtain the energy spectrum, it should be noted that in the limit β→ 0, the energy of spectrum should be covert to no-GUP result. Thus, we choose

(87)ζ1 = m + 12, (87)

(88)ζ2 = 12 − (m+1 − ϱ1βλ). (88)

Following this, we obtain

(89)ϵn = ±m02c4 + 4c2(1 + m0ω2ℏθ˜)(1 + θ¯2m0ωℏ)n + 4c2β(1 + θ¯2m0ωℏ)2n2, (89)

where

(90)β < β0, β0 = 1m + 32(1 + m0ω2ℏθ˜)(1 + θ¯2m0ωℏ)ℏωm0, with m > 0. (90)

So, the non-zero minimal length is

(91)Δxmin = ℏβ < (Δx min )0 = 1m + 32(1 + m0ω2ℏθ˜)(1 + θ¯2m0ωℏ)lmin, (91)

with lmin = ℏm0ω is the characteristic length of the DO, and (Δx_min)₀ is the admissible length above which the physics becomes experimentally inaccessible. We can see that the influence of the NC parameters on (Δx_min)₀ is very clear. Now, expanding to first order in terms of the variable β we have [24]

(92)ϵn ≃ ±m0c21 + 4m02c2(1 + m0ω2ℏθ˜)(1 + θ¯2m0ωℏ)n×(1 + 2β(1 + θ¯2m0ωℏ)m02c2n21 + 4m02c2(1 + m0ω2ℏθ˜)(1 + θ¯2m0ωℏ)n). (92)

The first term is the energy spectrum of the usual two-dimensional DO, and the second term represents the correction due to the presence of the minimal length. As mentioned in [23], we note the dependence on n², which is a feature of hard confinement. For a large values of n, we have

(93)ϵn = ℏω¯n, (93)

which means the energy continuum for large n for the DO without the minimal length disappears in the presence of the minimal length, and consequently the behaviour of the DO can be described by a non-relativistic harmonic oscillator with a frequency of ω¯ = 2cβℏ(1 + θ¯2m0ωℏ).

According to (85) and (87), we can see that the parameter c=m+1 is an integer; thus, either the two solutions of (84) coincide or one of the solutions will blow up. Now, when c is an integer greater than 1, which is our case, the second solution diverges. Thus, the component ψ₁ will has the following form:

(94)(ψ1)n,m(p, θ, z) = (C1)n,mp−12eimθzζ12(1−z)ζ22 2F1(−n; b, |m|+1; z). (94)

Finally, the total associated eigenfunction is determined by

(95)ψn,m(p, θ, z) = (1cP+ϵ+m0c2)ψ1. (95)

3.2 The Solutions in the Presence of a Magnetic Field

Now, in the presence of a uniform magnetic field, (7) is transformed into

(96){cαx[(px(NC) + eBy(NC)2c) − im0ωβ˜x(NC)]  + cαy[(py(NC) − eBx(NC)2c) − im0ωβ˜y(NC)] + β˜m0c2}ψD = ϵ¯ψD. (96)

In this case, (9) takes the following form:

(97)(m0c2cP˜−cP˜+−m0c2)(ψ1ψ2) = ϵ¯(ψ1ψ2), (97)

with

(98)P˜− = (px(NC) + eBy(NC)2c) + im0ωx(NC) − i (py(NC) − eBx(NC)2c)+ m0ωy(NC), (98)

(99)P˜+ = (px(NC) + eBy(NC)2c) − im0ωx(NC) + i(py(NC) − eBx(NC)2c)+ m0ωy(NC), (99)

or

(100)P˜− = ϱ1(px − ipy) + im0ω˜ϱ2(x − iy), (100)

(101)P˜+ = ϱ1(px + ipy) + im0ω˜ϱ2(x + iy), (101)

where

(102)ω˜ = ω − ωc2,  ωc = |e|Bm0c, (102)

and where ω_c is a cyclotron frequency. According to the above case, the eigen solutions are given by

(103)(ψ1)n,m(p, θ, z) = (C˜1)n,mp−12eimθzζ12(1 − z)ζ22 2F1(−n; b, |m| + 1; z), (103)

ϵ¯n = ±m02c4 + 4c2(1 + m0ω˜2ℏθ˜)(1 + θ¯2m0ω˜ℏ)n + 4c2β(1 + θ¯2m0ω˜ℏ)2n2,

where the total wavefunction is given by

(104)ψn,m(p, θ, z) = (1cP˜+ϵ¯ + m0c2)ψ1. (104)