Relativistic quantum motion of an electron in spinning cosmic string spacetime in the presence of uniform magnetic field and Aharonov-Bohm potential
RRelativistic quantum motion of an electron in spinning cosmic string spacetime in the presence ofuniform magnetic field and Aharonov-Bohm potential
M´arcio M. Cunha ∗ and Edilberto O. Silva † Departamento de F´ısica, Universidade Federal do Maranh˜ao, 65085-580, S˜ao Lu´ıs, Maranh˜ao, Brazil (Dated: June 12, 2020)In this manuscript, we study the relativistic quantum mechanics of an electron in external fields in the spinningcosmic string spacetime. We obtain the Dirac equation, write the first and second-order equations from it, andthen we solve these equations for bound states. We show that there are bound states solutions for the first-order equation Dirac. For the second-order equation, we show that its wave functions are given in terms of theKummer functions and we determine the energies of the particle. We examine the behavior of the energies as afunction of the physical parameters of the model, such as rotation, curvature, magnetic field, Aharonov-Bohmflux, and quantum numbers. Our study reveals that both curvature and rotation influence more intensely whenthese parameters have values smaller than 0.3. We also find that, depending on the values of these parameters,there are energy nonpermissible levels.
PACS numbers: 03.65.Ge, 03.65.Pm, 04.62.+v, 71.15.Rf
I. INTRODUCTION
Symmetry is key-ingredient in the description of naturalphenomena. The notion of symmetry is an essential featurein several areas of physics. In this context, the well-knownNoether’s theorem [1] establishes a connection between sym-metry and conservation laws of relevant physical quantities.In quantum mechanics, we often use symmetry to obtain cru-cial results concerning angular momenta operators [2]. Like-wise, symmetry is important in the topic of quantum informa-tion [3]. Symmetry is also important in the framework of rel-ativity [4], and for this reason, it is indispensable in researchareas such as particle physics [5] and cosmology [6].A pertinent question in the research areas cited above refersto think about what happens when some symmetry is brokenin a given physical system. When a system suffers a phasetransition, for example, it can lose some type of symmetry.Another example of symmetry-breaking occurs in condensedmatter systems: by employing the Volterra process [7], it ispossible to break some symmetry of the system due to thecreation of a topological defect, like disclinations and dislo-cations [8, 9], for instance.Topological defects can emerge in a large number of phys-ical systems covering themes such as liquid crystals [10],graphene physics [11], magnetism [12] and cosmology [13].Recent studies also have reported the importance of topologi-cal defects in Life Sciences [14, 15]. On the point of view ofcosmology, defects in the spacetime topology can be viewedas a possible consequence of the evolution of the early uni-verse, which has suffered phase transitions due to the temper-ature decreasing and the process of expansion [16, 17].In this contribution, we are particularly involved in study-ing the topological defect known as a cosmic string. A cosmicstring is a linear defect, similar to a flux tube in type-II super-conductors [16]. The spacetime around a cosmic string has ∗ [email protected] † [email protected] a conical symmetry, identically to the case of a disclination[18]. The concept of a cosmic string it was introduced in theliterature by Kibble [19]. Since then, this topic has been in-vestigated in diverse forms. An intriguing facet in this subjectrefers to the quantum mechanical description of a particle ina region of the spacetime containing a cosmic string. It canbe done both in the scenario of relativistic and nonrelativis-tic quantum mechanics. For instance, the hydrogen atom ina spacetime of a cosmic string it was analyzed in Ref. [20].In Ref. [21], it was considered the problem of a relativisticelectron in the presence of both Coulomb and scalar poten-tials in the cosmic string spacetime. Results about vacuumpolarization in a cosmic string spacetime were reported in Ref.[22]. Again, the cosmic string spacetime it was considered asa background to examine relativistic oscillators [23], quantumphases [24], and fermionic currents [25].A relevant issue in this context consists of taking into con-sideration the influence of electromagnetic fields in the quan-tum particle motion. Landau levels [26] and the Aharonov-Bohm effect [27, 28], for instance, are essential ingredients inthe investigation of quantum systems even in a flat spacetime.It can be explained because Landau levels are a quantum ana-log of classical cyclotron motion, while the Aharonov-Bohmeffect reveals the significance of the vector potential in thequantum world. Then, studying the contribution of magneticfields to the quantum mechanical description of a system inspacetime having a topological defect is a natural develop-ment. Examples of studies dealing with Landau levels and theAharonov-Bohm effect in the presence of topological defectscan be accessed in Refs. [29] and [30], respectively. In partic-ular, the inclusion of electromagnetic interactions in the caseof a cosmic string background also has been considered. Forinstance, in Ref. [31], it was analyzed the quantum dynam-ics of a charged particle in the presence of a magnetic fieldand scalar potential. In Ref. [32], various configurations ofconfined magnetic fields are examined and the existence ofinduced vacuum fermionic currents is investigated.On the other hand, we can be interested in analyzing thebehavior of a quantum system when noninertial effects turnon. These effects play a fundamental role in the description of a r X i v : . [ h e p - t h ] J un systems governed by classical mechanics. Noninertial effectsalso can take place on quantum systems, providing novel the-oretical predictions and feasible experimental developments.For instance, the emergence of quantum phases in rotatingsystems, in analogy to the Aharonov-Bohm effect [33, 34]were investigated. In addition, a relation between the Halleffect and inertial forces it was established [35]. Besides,if a given system it is put to rotate, it has consequences indiverse physical properties like spin transport [36, 37], elec-tronic structure [38], and even can present magnetization duerotation, like in the Barnett effect [39]. While a magneticfield produces a spin-field coupling, resulting in the anoma-lous Zeeman effect [40], rotation produces an analog effect,due the spin-rotation coupling [41]. Thus, rotation can con-tribute similarly to a magnetic field in the dynamics of a quan-tum system. More, noninertial effects are an interesting issuein the situation in which spacetime contains topological de-fects. In this case, the noninertial effects and the presenceof a topological defect can be included in the quantum me-chanical description by employing the same tools: we canuse a metric tensor to a spinning spacetime with a topologi-cal defect [42]. The spacetime of a spinning cosmic string hasbeen considered as background for several problems involv-ing quantum systems. For instance, the Schr¨odinger equationin that spacetime it was solved in Ref. [43]. Bound states forneutral particles in a rotating frame of a cosmic string wereanalyzed in Ref. [44]. Likewise, rotating effects on a Landau-Aharonov-Casher System in the spacetime of a cosmic stringwere investigated in Ref. [45].As we already have mentioned, in some cases rotationpresents similarities within electromagnetic fields. This way,it is also an attractive question examining how the electromag-netic interactions affect the particle quantum motion of a ro-tating system in the presence of a topological defect. A recentexample of studying dealing with both topological and nonin-ertial effects in the presence of an Aharonov-Bohm potentialcan be accessed in Ref. [46]. In Ref. [47], it was addressed theproblem of a spinless relativistic particle subjected to a uni-form magnetic field in the spinning cosmic string spacetime.The Dirac oscillator in the spacetime of a cosmic string con-sidering noninertial effects and the presence of the Aharonov-Casher effect it was analyzed in Ref. [48]. In Ref. [49], it wasanalyzed the problem of a charged half-spin particle depictedby the Dirac equation in the presence of a uniform magneticfield in the rotating cosmic string spacetime. A meaningfulaspect in this context consists of analyzing how different con-figurations of magnetic fields affect the quantum particle mo-tion.In this paper, we study the relativistic quantum mechanicsof an electron in the presence of both a uniform magnetic fieldand Aharonov-Bohm potential in the spinning cosmic stringspacetime. In other words, we solve the Dirac equation inthis scenario and investigate how the rotation, curvature andexternal magnetic fields affect the wave functions and energiesof the electron.The manuscript is organized as follows. In Section II, wepresent some algebraic elements necessary to construct thefield equations in curved spacetime and write the Dirac equa- tion describing the quantum motion of the electron in the pres-ence of external magnetic fields in the spinning cosmic stringbackground. In Section III, we deal with first-order solutionsand study the existence of isolated solutions for the particularcase of a particle at rest. In Section IV, we take our attentionto the case when the energy of the particle is different fromits rest energy. We map the Dirac equation problem in curvedspace with minimal coupling into a Sturm-Liouville problemfor the upper component of the Dirac spinor and, using an ap-propriate ansatz, we derive the radial equation. We solve theradial equation and find the wave functions and energies of theparticle. We make a detailed discussion of the results and alsocomparisons with other studies in the literature. In Section V,we present our conclusions. In our work, we use natural units, (cid:126) = c = G = 1 . II. DIRAC EQUATION IN THE SPINNING COSMICSTRING SPACETIME
In this section, we briefly present the main tools needed toconstruct the Dirac equation in the conical spacetime in thepresence of noninertial effects. The first step consists in takea look at the metric tensor characterizing this geometry. Next,we will choose an appropriate tetrad basis and implement thefields configuration involved through the performing of a min-imal substitution. The spacetime induced by a rotating cosmicstring is described by the metric ds = ( dt + adϕ ) − dr − α r dϕ − dz , (1)where −∞ < z < ∞ , r (cid:62) and (cid:54) ϕ (cid:54) π . The parameter α is related to the linear mass density µ of the cosmic stringthrough the relation α = 1 − µ and it runs in the interval (0 , . The quantity a = 4 J is the rotation parameter, with J representing the angular momentum of the spinning cosmicstring. The relativistic quantum dynamics of a spin- / par-ticle interacting with external magnetic fields in the rotatingcosmic string spacetime is governed by the Dirac equation [ iγ µ ( x ) ( ∂ µ + Γ µ ( x ) + ieA µ ( x )) − M ] Ψ ( x ) = 0 , (2)where M is the mass of the particle and γ µ ( x ) are the Diracmatrices in the rotating cosmic string spacetime, which aredefined in terms of the tetrad fields e µa and Dirac matrices inthe flat space γ a in the following way: γ µ ( x ) = e µa ( x ) γ a , (3)where γ a = (cid:0) γ , γ i (cid:1) , with γ = (cid:18) − (cid:19) , γ i = (cid:18) σ i − σ i (cid:19) , (4)are the standard Dirac matrices and σ i = ( σ x , σ y , σ z ) are theusual Pauli matrices. The matrices (3) satisfy the followingrelation: { γ µ ( x ) , γ ν ( x ) } = 2 g µν ( x ) . (5)Also, in Eq. (2), Γ µ ( x ) is the spin affine connection given by Γ µ ( x ) = 14 γ a γ b e νa ( x ) (cid:2) ∂ µ e bν ( x ) − Γ σµν e bσ ( x ) (cid:3) , (6)where Γ σµν are the Christoffel symbols of the second kind and e µa ( x ) is the tetrad field. The tetrad basis satisfies the relations e aµ ( x ) e bν ( x ) η ab = g µν ( x ) , (7) e aµ ( x ) e bν ( x ) = δ ba , (8) e µa ( x ) e aν ( x ) = δ νµ . (9)In Eq. (6), the Greek letters are used for tensor indices whilethe Latin letters are denoting Minkowski indices. We use thetetrad basis and its inverse defined as [50] e aµ ( x ) = a
00 cos ϕ − rα sin ϕ
00 sin ϕ rα cos ϕ
00 0 0 1 , (10) e µa ( x ) = a sin ϕrα − a cos ϕrα
00 cos ϕ sin ϕ − sin ϕrα cos ϕrα
00 0 0 1 . (11)For this choice, it can be shown that the non-vanishing affineconnection is given by Γ µ = (0 , , Γ ϕ , , with Γ ϕ = i − α ) Σ z , (12)with Γ ϕ = i − α ) Σ z , (13)where Σ z = (cid:18) σ z σ z (cid:19) , σ z = (cid:18) − (cid:19) . (14)By using the tetrad basis (11), the matrices (3) can be writtenexplicitly as γ t = γ − aγ ϕ , (15) γ r = (cid:18) σ r − σ r (cid:19) , γ ϕ = (cid:18) σ ϕ − σ ϕ (cid:19) , (16)with σ r = (cid:18) e − iϕ e + iϕ (cid:19) , σ ϕ = 1 rα (cid:18) − ie − iϕ ie + iϕ (cid:19) (17)being the Pauli matrices in the curved spacetime.Since we are first interested in studying the solutions of theDirac equation in its present form (Eq. (2)), we need to writethe corresponding system of first order coupled differentialequations. For this to be accomplished, let’s assume the time-dependence of the wave functions together with the decompo-sition of the fermion field in the form Ψ ( r, ϕ ) = e − iEt (cid:18) ψ ( r, ϕ ) ψ ( r, ϕ ) (cid:19) , (18) with ψ ( r, ϕ ) = (cid:18) ψ a ( r, ϕ ) ψ b ( r, ϕ ) (cid:19) = (cid:18) e imϕ f + ( r ) ie i ( m +1) ϕ f − ( r ) (cid:19) , (19) ψ ( r, ϕ ) = (cid:18) ψ c ( r, ϕ ) ψ d ( r, ϕ ) (cid:19) = (cid:18) e imϕ g + ( r ) ie i ( m +1) ϕ g − ( r ) (cid:19) . (20)The system we will analyze takes into account the particleis immersed in a region where there is a uniform magneticfield and also the potential due to a thin long solenoid alongthe z-axis. Having this field configuration in mind, we studythe physical implications due to noninertial effects and theAharonov-Bohm potential on the relativistic Landau quanti-zation. We also take into account the translational invarianceof the system along the z -direction, which allows us to elim-inate the third direction ( p z = z = 0 ) and, consequently, wecan consider only the planar motion [51–54]. In this case, theparticle experiences a superposition of potential vectors writ-ten in the Coulomb gauge as A = (0 , − αrA ϕ , , (21)with A ϕ = A ϕ, + A ϕ, , (22) A ϕ, = Br , A ϕ, = φαr , (23)where B is the magnetic field magnitude, φ = Φ / Φ , Φ isthe magnetic flux and Φ = 2 π/e is the quantum of magneticflux along the solenoid. This configuration also provides ansuperposition of magnetic fields in the z-direction B = B z, + B z, , (24)with B ,z = B, B z, = φ δ ( r ) αr , (25)Note that the particle only interacts with the magnetic fielddue to the potential vector A ϕ, . Here, we are focused onstudying the electron motion only in the r (cid:54) = 0 region, so thatwe can neglect the point interaction B z, and, consequently,consider only regular wave functions.Using the results above, the Dirac equation (2) can be writ-ten as ( E − M ) ψ + σ r i∂ r ψ + σ ϕ (cid:16) i∂ ϕ + eA ϕ − aE − s − α ) (cid:17) ψ = 0 , (26) ( E + M ) ψ + σ i i∂ r ψ + σ ϕ (cid:16) i∂ ϕ + eA ϕ − aE − s − α ) (cid:17) ψ = 0 . (27)At this point, we are ready to solve the equations (26) and (27)by considering two distinct circumstances:(i) Take our attention to isolated solutions of the first orderDirac equation by imposing the condition E = ± M ;(ii) By imposing the condition E (cid:54) = ± M , we looking forsolutions of the second order Dirac equation.We will show in the next two sections that there are boundstate solutions for both cases and discuss their main physicalproperties. To distinguish each case in (i), in the next sectionwe use the superscripts ( ± ) to label the quantities correspond-ing to E = ± M . III. SOLUTION OF THE EQUATION OF MOTION TO E = ± M To study the existence of isolated solutions of the Diracequation (2), we must set E = ± M in Eqs. (26) and (27). Inliterature, such solutions are known to be excluded from theSturm-Liouville problem. The search for isolated solutions ofthe Dirac equation has been performed in different physicalcontexts [55–59]. The bound state solution must satisfy thenormalization condition (cid:90) ∞ (cid:0) | ψ ( r ) | + | ψ ( r ) | (cid:1) rdr = 1 . (28)By making E = + M in Eqs. (26) and (27) and using Eqs.(19) and (20), we get dg (+)+ ( r ) dr − L (+) m rα g (+)+ ( r ) + eBr g (+)+ ( r ) = 0 , (29) dg (+) − ( r ) dr + L (+) m +1 rα g (+) − ( r ) − eBr g (+) − ( r ) = 0 , (30) df (+)+ ( r ) dr − L (+) m rα f (+)+ ( r ) + eBr f (+)+ ( r ) = − M g (+) − ( r ) , (31) df (+) − ( r ) dr + L (+) m +1 rα f (+) − ( r ) − eBr f (+) − ( r ) = 2 M g (+)+ ( r ) , (32)with L (+) m = m − φ + aM + s − α ) , (33) L (+) m +1 = m + 1 − φ + aM + s − α ) . (34)The solution of the coupled linear differential equations sys-tem (29)-(32) is given by f (+)+ ( r ) = e − eBr r L (+) mα (cid:34) a + a M (cid:18) − eB (cid:19) Ω a Γ (+) a (cid:35) , (35) f (+) − ( r ) = e eBr r − L (+) m +1 α (cid:34) b − b M (cid:18) eB (cid:19) − Ω b Γ (+) b (cid:35) , (36) g (+)+ ( r ) = b e − Ber r L (+) mα , (37) g (+) − ( r ) = a e Ber r − L (+) m +1 α , (38) with Ω a = 12 α (cid:16) L (+) m + L (+) m +1 − α (cid:17) , (39) Ω b = 12 α (cid:16) L (+) m + L (+) m +1 + α (cid:17) , (40)where Γ (+) a = Γ (cid:18) − Ω a , − eBr (cid:19) , (41) Γ (+) b = Γ (cid:18) Ω b , eBr (cid:19) . (42)are upper incomplete Gamma functions [60], and a , a , b and b are constants. Analyzing the solutions (35) and (37),we note that e − eBr dominates over r L (+) mα for any value of L (+) m /α , in such way both solutions converge when r → and r → ∞ . This will not occur for the function e eBr in the so-lutions (36) and (38). Moreover, since the incomplete Gammafunctions Γ (+) a and Γ (+) b always diverge, then the function f (+)+ ( r ) will only converges as r → if a = 0 while thefunction f (+) − ( r ) will always diverge when r → ∞ and, there-fore, will not be a square-integratable function. Thus, the onlysolution allowed for the equations system (29)-(32) results f (+)+ ( r ) = a e − eBr r L (+) mα , with L (+) m α (cid:62) , (43)with f (+) − ( r ) = g (+)+ ( r ) = g (+) − ( r ) = 0 . Solution (43) satis-fies equation (28) and constitutes a bound state solution for thecase E = M , i.e., an isolated solution to the Dirac equation(2) in the metric spacetime (1).Proceeding in an analogous way, now we make E = − M in Eqs. (26) and (27). We find the system of equations df ( − )+ ( r ) dr − L ( − ) m rα f ( − )+ ( r ) + eBr f ( − )+ ( r ) = 0 , (44) df ( − ) − ( r ) dr + L ( − ) m +1 rα f ( − ) − ( r ) − eBr f ( − ) − ( r ) = 0 , (45) dg ( − )+ ( r ) dr − L ( − ) m rα g ( − )+ ( r ) + eBr g ( − )+ ( r ) = 2 M f ( − ) − ( r ) , (46) dg ( − ) − ( r ) dr + L ( − ) m +1 rα g ( − ) − ( r ) − eBr g ( − ) − ( r ) = − M f ( − )+ ( r ) . (47)with L ( − ) m = m − φ − aM + s − α ) , (48) L ( − ) m +1 = m + 1 − φ − aM + s − α ) . (49)The solution of the coupled linear ordinary differential equa-tions system (44)-(47) is given by f ( − )+ ( r ) = c e − eBr r L ( − ) mα , (50) f ( − ) − ( r ) = d e Ber r − L ( − ) m +1 α , (51) g ( − )+ ( r ) = e − Ber r L ( − ) mα (cid:34) − d M (cid:18) − eB (cid:19) Λ c Γ ( − ) c + d (cid:35) , (52) g ( − ) − ( r ) = e eBr r − L ( − ) m +1 α (cid:34) c M (cid:18) eB (cid:19) − Λ d Γ ( − ) d + c (cid:35) , (53)with Λ c = 12 α (cid:16) L ( − ) m + L ( − ) m +1 − α (cid:17) , (54) Λ d = 12 α (cid:16) L ( − ) m + L ( − ) m +1 + α (cid:17) , (55)where Γ ( − ) c = Γ (cid:18) − Λ c , − Ber (cid:19) , (56) Γ ( − ) d = Γ (cid:18) Λ d , eBr (cid:19) . (57)By making the same analysis of the solutions as we have madefor the case E = M , i.e., analyzing the behavior of the func-tions for r → ± ∞ , we find that the only solution that admitsbound state is (52). Thus, the solution for the case E = − M satisfying the normalization condition (28) is given by g ( − )+ ( r ) = d e − Ber r L ( − ) mα , with L ( − ) m α (cid:62) , (58)with f ( − )+ ( r ) = f ( − ) − ( r ) = g ( − ) − ( r ) = 0 . Note that the solu-tions (43) and (58) are affected by rotation through Eqs. (33)and (48), respectively. IV. SOLUTION OF THE EQUATION OF MOTION TO E (cid:54) = ± M In this section, we solve the second order equation to ψ thatwe find from the Eqs. (26) and (27). The solution of thisequation is different from that one calculated in the previoussection and allow us to obtain an expression for the particleenergies. By isolating ψ in Eq. (27) and replacing in Eq.(26), we are able to write the second order differential equa-tion for ψ as (cid:0) E − M (cid:1) ψ + ∂ r ψ + 1 r ∂ r ψ + 1 αr σ z e ( ∂ r A ϕ ) ψ + 1 α r (cid:18) ∂ ϕ − ieA ϕ + i − α σ z + iaE (cid:19) ψ = 0 . (59) n = = = = - - - E n (>) ( a ) n = = = = - - - E n (>) ( b ) FIG. 1. Sketch of the energy levels E ( > ) n (Eq. (64)) as a function ofthe magnetic field B for different values of n . In panel (a), s = +1 and in panel (b), s = − . The positive energies are represented bysolid lines and the negative by dashed lines. We assume e = 1 and M = 1 . Using the decomposition of the fermion field (19) (ignoringthe subscript ( + )) together with Eqs. (21), (22) and (23), weobtain the radial equation for f ( r ) (cid:18) d dr + 1 r ddr − L α r − e B r k (cid:19) f ( r ) = 0 , (60)where k = E − M + eBα L + seB, (61) L = m − φ + s (1 − α )2 + aE. (62)Note that there are three other equivalent equations and thereis no need to solve them here because their respective energieswould also be equivalent. Equation (60) is the confluent hy-pergeometric equation and its solution is well known. Thus, itcan be shown that the solution to ψ a is ψ a ( r, ϕ ) = c nm (cid:18) eB (cid:19) ( | L | α ) e imϕ r | L | α e − eBr × F (cid:18) (cid:18) | L | α (cid:19) + k eB , | L | α , eBr (cid:19) , (63)where F ( a, b, z ) denotes the confluent hypergeometricfunction of the first kind or Kummer’s function M ( a, b, z ) and c nm the normalization constant. It can be shown that the hy-pergeometric function F ( a, b, z ) has a divergent behaviorfor large values of z . Because of this, bound state solutionsfor Eq. (63) are only possible if we impose that this func-tion becomes a polynomial of degree n . For this to be ac-complished, we require that / | L | / α + k / Be = − n ,where n ∈ Z ∗ , with Z ∗ denoting the set of the nonnegative in-tegers. However, as we can see in Eq. (62), the absolute value FIG. 2. Sketch of the energy (Eq. (65)) as a function of n and m for a = 0 . , α = 0 . , B = 1 , e = 1 , M = 1 , s = 1 and φ = 1 . of the effective angular moment L is defined in terms of theenergy E . In this way, to obtain the energy eigenvalues fromthe above condition, we must consider | L | > and | L | < ,respectively, and then solve them for E . By making this, weget E ( > ) n = ± (cid:112) eB (2 n − s + 1) + M , (64) E ( < ) nm = − aeBα ± α (cid:112) a e B + αQ, (65)with the following requirement: a e B + α Q (cid:62) , (66)where Q = αeB (cid:18) n − α (cid:16) m − φ + s (cid:17) + 1 (cid:19) + αM . In Eqs. (64) and (65), the superscripts ( >, < ) refer to theenergies calculated for | L | > and | L | < , respectively.For a given choice of the element of spin s , the energy E ( > ) n depends only on the quantum number n and the magnetic field FIG. 3. Sketch of the energy (Eq. (65)) as a function of α and a for B = 4 , M = 1 , e = 1 , n = 1 , φ = 2 , s = 1 and m = 1 . B . For a given value of n , the energy increases when themagnetic field is increased. In Fig. 1, we show the profile of E ( > ) n for the first four states for s = 1 . The energy levels for s = − (Fig. 1(b)) are slightly larger than the profile for thecase s = 1 (Fig. 1(a)).The energies (64) and (65) denote the relativistic Landaulevels in the present context. These energies can be directlycompared with those obtained for the relativistic oscillator(Dirac oscillator) addressed in Ref. [50]. Although that sce-nario is different from the one we are exploring here, there aresimilarities between the profiles of the energy levels in bothmodels. For example, for s = 1 , the energy (48) of the Ref.[50] depends only on the frequency of the oscillator and thequantum number n . In our case, by defining the cyclotronfrequency ω c = eB/M , Eq. (64) results ˜ E ( > ) nm = ± (cid:112) nM ω c + M , (67)which makes such a similarity clear. Since the energies (65)are the only ones that depend on all the physical parametersinvolved in the current problem, we study them in more de-tail. For a given set of fixed parameters, for example, a = 0 . , α = 0 . , B = 1 . , e = 1 . , M = 1 , s = 1 and φ = 1 ,we have the profile of the energy levels as a function of n and m (Fig. 2). We can clearly see that | E ( < ) nm | increases with n and m . The green solid bars denote the discrete energy valuesfor a given m and n . On the other hand, when we investigatethe behavior of (65) as a function of α and a for specific val-ues of the other parameters, we see that the negative spectrumchanges more rapidly when compared with the positive one(Fig. 3). In the positive spectrum, both rotation and curvaturelead to a linear change, except in the region with α < . andarbitrary a . In the negative spectrum, we see that the curva-ture effects are more predominant in the region where alphahas values smaller than 0.2. In this region, any variation in therotation parameter implies in an abrupt change in the energyspectrum. Modifications in the energies with α < . is anexpected manifestation in our analyses. Its physical implica- FIG. 4. Sketch of the energy (Eq. (65)) as a function of B and φ for a = 1 , α = 0 . , e = 1 , m = 1 , M = 1 , n = 1 and s = 1 . tion is inherent in the metric (1) and is an immediate conse-quence of the topological cone, which becomes more singularfor smaller α values. To complete our analysis, we investigatethe profile of the energy (65) as a function of magnetic field B and the magnetic flux through the solenoid, φ . Similarlyto Fig. 3, by fixing the other parameters, we see that the en-ergy of the anti-particle varies more rapidly when comparedto the energy of the particle (Fig. 4). Clearly, we observethat the energy of the particle varies very slowly throughoutthe region of flux and magnetic field . As a final commentary,we clarify that the cases discussed in Figs. 2, 3 and 4 can beinvestigated for other fixed parameter values. In this way, itcan be shown that there are forbidden energies, depending onthe values of the parameters considered. In general, this oc-curs when both the α parameter and the rotation parameter a are smaller than . and the other parameters assuming highervalues than those we use here. V. CONCLUSIONS
In the present manuscript, we have addressed the problemof the relativistic quantum motion of an electron in the spin-ning cosmic string background considering the presence of auniform magnetic field and the Aharonov-Bohm potential. Wehave shown that this combination of potentials allows boundstates configurations in the scenario of first-order solutions aswell as in the case of second-order solutions of the Dirac equa-tion. It is worth noting the role played by the two differentterms in the vector potential. As already known in the lit-erature, we have shown that the uniform field is responsiblefor a behavior analog to a harmonic oscillator, which leadsto the relativistic Landau quantization while the Aharonov-Bohm flux contributes to the angular momentum of the parti- cle. In the case of first order solutions, which were obtainedby solving Eqs. (43) and (58) for E = + M and E = − M ,respectively, the oscillator-like behavior provided by the uni-form magnetic field guarantees the convergent first-order so-lutions and, consequently, the existence of bound states. Theisolated solutions obtained (Eqs. (43) and (58)) are particularsolutions of the Dirac equation (2).We have also studied the more general problem by solvingthe second-order equation implied by equations (26) and (27)for the upper component of the Dirac spinor for E (cid:54) = ± M .Using appropriate solutions (Eq. (18)) we have derived theradial equation and shown that its solution is given in termsof the Kummer functions from which we have extracted theexpression for the energy levels of the particle (Eqs. (64)and (65)). For the field configuration considered, we havefound that the effective angular momentum of the electron de-pends on its energy and the Aharonov-Bohm flux tube whilethe potential vector that generates the uniform field leads to acharged oscillator. This implies that such field superpositionprovides distinct effects on the motion of the particle. Ad-ditionally, in some cases, the rotation produces a combinedeffect with both the uniform magnetic field and the curvature(see Eq. (65)). We have shown that the energy levels of theparticle and antiparticle depend on the values of the physicalparameters involved. In the case of energy (65), its validity isconditioned to Eq. (66). Depending on the choice we makefor the parameters, we can obtain forbidden energies. Thesketches in Figs. 2, 3, and 4 illustrate the profiles of the par-ticle and antiparticle energies and show that they belong tothe same spectrum. The effects of curvature and rotation aremore evident when α < . , being the antiparticle energy themost affected. As a final comment, we would like to empha-size that the model studied in this article generalizes othersfound in the literature, such as those of Refs. [46, 49] for thecase including a superposition of external magnetic fields andthe investigation of isolated solutions of the Dirac equation.Furthermore, we present a detailed discussion on the energylevels of the particle which, in general, is not found in theliterature. ACKNOWLEDGMENTS
We would like to thanks E.R.B. Mello (Universidade Fed-eral da Para´ıba, PB, Brazil) for his remarks and comments.This work was partially supported by the Brazilian agen-cies CAPES, CNPq and FAPEMA. EOS acknowledges CNPqGrants 427214/2016-5 and 307203/2019-0, and FAPEMAGrants 01852/14 and 01202/16. This study was financed inpart by the Coordenac¸ ˜ao de Aperfeic¸oamento de Pessoal deN´ıvel Superior - Brasil (CAPES) - Finance Code 001. MMCacknowledges CAPES Grant 88887.358036/2019-00. [1] Cornelius Lanczos,
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