A bound quantum particle in a Riemann–Cartan space with topological defects and planar potential

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A bound quantum particle in a Riemann–Cartan space with topological defects and planar potential

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  See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/275830142 A Bound Quantum Particle in a Riemann-Cartan space with Topological Defects andPlanar Potential  Article   in  Physics Letters A · January 2007 CITATIONS 0 READS 12 4 authors , including: Some of the authors of this publication are also working on these related projects: Possible Gamma-Ray Burst radio detections by the Square Kilometre Array. New perspectives   ViewprojectCANTATA - Cosmology and Astrophysics Network for Theoretical Advances and Training Actions -COST Action CA15117.   View projectSean Ali 28   PUBLICATIONS   176   CITATIONS   SEE PROFILE Salvatore CapozzielloUniversity of Naples Federico II 539   PUBLICATIONS   11,788   CITATIONS   SEE PROFILE Carlo CafaroMax-Planck-Institut für die Physik des Lichts 61   PUBLICATIONS   396   CITATIONS   SEE PROFILE All content following this page was uploaded by Sean Ali on 04 May 2015. The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the srcinal documentand are linked to publications on ResearchGate, letting you access and read them immediately.  Physics Letters A 366 (2007) 315–323www.elsevier.com/locate/pla A bound quantum particle in a Riemann–Cartan space with topologicaldefects and planar potential S.A. Alix a , C. Cafaro a , S. Capozziello b , ∗ , Ch. Corda c a  Department of Physics, University at Albany-SUNY, 1400 Washington Avenue, Albany, NY 12222, USA b  Dipartimento di Scienze Fisiche, Università di Napoli “Federico II” and INFN Sez. di Napoli, Complesso Universitario Monte S. Angelo, Ed.N, Via Cinthia, I-80126 Napoli, Italy c  INFN Sez. di Pisa and Università di Pisa, Via F. Buonarroti 2, I-56127 Pisa, Italy Received 9 February 2007; accepted 18 February 2007Available online 21 February 2007Communicated by V.M. Agranovich Abstract Starting from a continuum theory of defects, that is the analogous to three-dimensional Einstein–Cartan–Sciama–Kibble gravity, we consider acharged particle with spin  12  propagating in a uniform magnetic field coincident with a wedge dispiration of finite extent. We assume the particleis bound in the vicinity of the dispiration by long range attractive (harmonic) and short range (inverse square) repulsive potentials. Moreover,we consider the effects of spin-torsion and spin-magnetic field interactions. Exact expressions for the energy eigenfunctions and eigenvalues aredetermined. The limit, in which the defect region becomes singular, is considered and comparison with the electromagnetic Aharonov–Bohmeffect is made. © 2007 Elsevier B.V. All rights reserved. PACS:  02.40.Ky; 61.72.Lk; 03.65.Ge Keywords:  Topological defects; Riemann–Cartan geometry; Screw dislocation; Bounded quantum mechanical particle 1. Introduction The investigation of quantum mechanical systems with non-trivial boundary conditions is an active field of research. A fertileground for such systems is provided by particles moving in a background space with nonvanishing (positive or negative) curvatureand/or non-trivial topology. By non-trivial topologywe meanmultiply connectedspaces. Indeed, the prototyperepresenting the caseof non-trivial topology is the well-known electromagnetic Aharonov–Bohm (AB) effect [1]. In the context of the electromagnetic ABeffect,theeffectoftopologymanifestsasthephasefactorinthewavefunctionofanelectronmovingaroundamagneticfluxline.The gravitational analogue of this effect has been investigated in [2–4]. We are going to investigate a space with topological defects characterized by vanishing Riemann–Christoffel curvature and torsion [5] everywhere except on the defects. Such defects arise ingauge theories with spontaneous symmetry breaking and may have played some significant role in the formation of large scalecosmological structure. Some examples in cosmology are domain walls [6], cosmic strings [6,7] and monopoles [8]. Analogues of such defects in condensed matter physics include vortices in superfluids and superconductors [9], domain walls in magnetic materials, dislocations in solids and disclinations in liquid crystals or two-dimensional graphite [10].Quantum effects on particles moving in a crystalline media with topological defects have been a subject of investigation sincethe early 1950s [11]. In the geometric approach of Katanaev and Volovich [12], the theory of defects in solids is translated into * Corresponding author.  E-mail address:  capozziello@na.infn.it (S. Capozziello).0375-9601/$ – see front matter  © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2007.02.057  316  S.A. Alix et al / Physics Letters A 366 (2007) 315–323 the language of three-dimensional Einstein–Cartan–Sciama–Kibble gravity [13,14] but it could be extended considering moregeneral forms of torsion [15]. This formalism corresponds to viewing the continuous limit of a crystalline solid in which the defect configuration is characterized by a non-trivial metric describing a static three-space. In this way, elastic deformations introduced inthe medium by defects are incorporated into the metric manifold. The boundary conditions required by the presence of such defectsare accounted for by the associated non-Euclidean metric. In the continuum limit, that is, for distances much larger than the Bravaislattice spacing, the theory describes a non-Riemannian manifold where curvature and torsion are associated with disclinations anddislocations in the medium, respectively.In the case of solid crystals, a topological defect can be thought as consisting of a core region characterized by the absence of regular lattice order and an ordered (undistorted) far-field region [16]. Although in the continuum approximation in which we work  the core region is usually shrunk to a singularity, we choose to relax this condition by smearing the singularity over a finite region.By virtue of having a finite defect region, we consider the effects of spin-torsion and spin-magnetic field interactions on the system.In this work, we study a charged spin- 12  particle moving in the space of a defect comprised of a combined screw dislocation anda wedge disclination. Such a combined defect is called a wedge dispiration, a defect possessing nonvanishing local curvature andtorsion. A homogeneous, finite magnetic field, concentric with the wedge dispiration line, is included. As a working hypothesis, weassume the particle is bound in the vicinity of the wedge dispiration by a planar potential given by a long range attractive (harmonic)and short range (inverse square) repulsive terms.The Letter is organized as follows: In Section 2, we introduce the topological defect. In Section 3, we consider a simple finite defect distribution describing a non-singular dispiration. In Section 4, the corresponding time-independent Schrödinger equation is taken into account. In Section 5, exact expressions for the energy eigenfunctions of the particle moving both inside and outside thedefect core of the medium are obtained. Boundary conditions on the surface of the defect region are implemented in order to getthe wavefunction defined over the whole space. The energy spectrum is determined and we consider the limit in which the defectregion is shrunk to a null size. Conclusions are drawn in Section 6. 2. The magnetic wedge dispiration In the framework of traditional elasticity theory, a Cartesian reference frame  x i is attached to the undistorted medium of anelastic solid with Euclidean metric  δ ab . The deformation of the medium is described locally in terms of a continuous displacementfunction  u(x) . In this way, after a deformation has occurred, the point  x i will have coordinates  x i → y i (x) = x i + u i (x) . The initialmetric  δ ab  is transformed into [17,18](1) g ij   :=  ∂x a ∂y i ∂x b ∂y j   δ ab , i,j   = 1 , 2 , 3 . We consider a topological defect in three dimensions whose geometry is characterized by the spatial line element [12](2) dl 2 = g ij   dx i dx j  = dρ 2 + κ 2 ρ 2 dϕ 2 + (dz + τ dϕ) 2 where (ρ,ϕ,z) arecylindricalcoordinates,with ρ  0, 0  ϕ  2 π , −∞  z  ∞ , κ,τ   ∈ R .Theparameter κ  isrelatedtotheFrank vector   Ω  [19] of the disclination (describing curvature, i.e. the angular deficit in the manifold) while  τ   is related to the Burgers’vector   b  [17,19] of the dislocation (describing torsion). The three-dimensional geometry of the medium is therefore characterizedby nonvanishing torsion and curvature. When  κ  = 0 and  τ   = 0 we have a wedge disclination; when  κ  = 0 and  τ   = 0 we have ascrew dislocation. The metric tensor  g ij   and its inverse  g ij  = (g ij  ) − 1 are given by [20](3) g ij   =  1 0 00  κ 2 ρ 2 + τ  2 τ  0  τ   1  , g ij  =  1 0 00  1 κ 2 ρ 2  −  τ κ 2 ρ 2 0  −  τ κ 2 ρ 2  1 +  τ  2 κ 2 ρ 2  . The line element (2) describes an infinitely long linear wedge dispiration oriented along the  z -axis. We introduce the dual (1-form)basis vectors  ϑ i = e ij   dx j  which describe the background (2), with coframe components [18] (4) ϑ 1 ≡ ϑ ρ = dρ, ϑ 2 ≡ ϑ ϕ = κρdϕ  and  ϑ 3 ≡ ϑ z = dz + τ dϕ. The metric  g ij  and triad components  e ia  are related via  e ia e j b δ ab = g ij  , where the triads satisfy  e aj  e j b  = δ ab . In a Riemanniangeometry without torsion, the Ricci  R ij   and Riemann curvature tensor  R lijk  are given by [5](5) R ij   = g ab R aibj   = ∂ k Γ  kij   − ∂ j  Γ  kik + Γ  kij  Γ  nkn − Γ  mik Γ  kjm and(6) R lijk  = ∂ i Γ  ljk − ∂ j  Γ  lik + Γ  lim Γ  mjk − Γ  ljm Γ  mik  ,  S.A. Alix et al / Physics Letters A 366 (2007) 315–323  317 respectively. The Christoffel symbols  Γ  kij   appearing in (5) or (6) are defined by [5] (7) Γ  kij   :=  12 (∂ k g ij   + ∂ i g jk − ∂ j  g ki ) where  ∂ k  := ∂/∂x k . The nonvanishing components of the Christoffel symbols in the space with metric (3) are [18] (8) Γ  zϕρ  = Γ  zρϕ  =− τ ρ, Γ  ρϕϕ  =− κ 2 ρ, Γ  ϕρϕ  = Γ  ϕϕρ  =  1 ρ. By contrast, a non-Riemannian geometry is one that is characterized by nonvanishing curvature and torsion. The torsion tensor  T  iab is defined by(9) T  kij   := ∂ i e kj   − ∂ j  e ki  + Γ  kiν e νj   − Γ  kjν e νi . The Ricci scalar is given by [16,21](10) R 1212 = R 11 = R 22 = 2 π  1 − κκ  δ 2 (ρ), where  κ  = 1 + φ/ 2 π  and  δ 2 (ρ)  is a two-dimensional Dirac  δ -function. This  δ -function revels the conic singularity in the curvaturegiven in (10). The dispiration characterized by (2) can be thought as arising from a “cut and paste” process, known as the Volterra process [20]. From the perspective of the Volterra process, the disclination is generated by removing  (κ <  1 )  or inserting  (κ >  1 ) a wedge of material with deficit angle  φ = 2 π(κ − 1 ) . For 0 <κ < 1 the disclination carries positive curvature while for 1 <κ < ∞ it carries negative curvature. The only nonvanishing component of torsion 2-form  T  k = T  kij   dx i ∧ dx j  is given by [16,21](11) T  z = 2 πτδ 2 (ρ)dρ ∧ dϕ. It is clear that the space described by metric (3) features two conical singularities at the srcin as seen in (10) and (11). The Burger vector can be viewed as a flux of torsion and the Frank vector as a flux of curvature. The Burger vector is calculated by integratingaround a closed path  C  encircling the dislocation [17,18](12) b 3 =   C ϑ 3 =   S  dρ ∧ dϕT  3 ρϕ  = 2 πτ, implying that 2 πτ   is the flux intensity of the torsion source passing through a closed loop  C  in the  ˆ e z -direction. Thus, the parameter τ   is the modulus of the Burger vector. In a similar manner, the Frank vector is determined by [10,18]   Ω  :=  ijk Ω jk ˆ e i , with(13) Ω 12 =    S  dρ ∧ dϕR 12 ρϕ  = 2 π  1 − κκ  , where   ijk  is the Levi-Civita symbol  ( 123 = 1 ) ,  Ω 12 =− Ω 21 and  S   is a surface perpendicular to the defect line, implying the fluxof curvature is the surface density of the Frank vector field. 3. Finite defect distribution To avoid the singular nature of  (10) and (11), we choose a simple exactly solvable model of finite torsion and curvature. In the former case, we choose a torsion field with a homogeneous flux distribution within the dislocation region. In particular, we choosea torsion field specified by(14) T  z = T  z (ρ,ϕ) = τ   Θ(R c − ρ)dρ ∧ dϕ, where  R c  denotes the radius of the defect core,  τ    =  s T  b z πR 2 c and  s T   =± 1 denotes the handedness of the screw, where  (s T   =− 1 ) indicates a left-handed screw in which a clockwise rotation of   ϕ  = 2 πκ   ( κ   is defined in Eq. (17)) relative to the  ( + ) x -axisinduces a shift in the  ( − ) z -direction; similarly,  (s T   =+ 1 )  describes a right-handed screw in which a counter-clockwise rotation of  ϕ = 2 πκ   relative to the  ( + ) x -axis induces a shift in the  ( + ) z -direction. In (14),  Θ  denotes the Heaviside theta function(15) Θ(R c − ρ) =  1 for  ρ <R c , 0 for  ρ >R c . In the case of non-singular curvature, we consider a disclination characterized by the deficit angle(16) ¯ φ =  12 φR 2 c Θ(R c − ρ).  318  S.A. Alix et al / Physics Letters A 366 (2007) 315–323 Under  φ → ¯ φ  transformation, the angular deficit  κ  = 1 + φ/ 2 π  transforms into(17) κ  = 1 +  12 φR 2 c Θ(R c − ρ) =  κ in = 1 +  12 φR 2 c  for  ρ <R c ,κ out = 1 for  ρ >R c . By the change of variables  ρ → ¯ ρ , where(18) ρ → ¯ ρ =  ρ κ  κ   =  ρ < =  ρ κ in κ in for  ρ <R c ,ρ > = ρ  for  ρ >R c , the metric describing the non-singular wedge dispiration is given by [16](19) dl 2 = g ij   dx i dx j  = d   ¯ ρ 2 + κ  2 ¯ ρ 2 dϕ 2 + (dz + τ   dϕ) 2 . Observe that distributions (14) and (17) are chosen such that the total torsion and curvature flux within the dispiration region areequivalent to initial values 2 πτ   and 2 π( 1 − κκ  ) , respectively. Furthermore, we assume the particle is moving in the electromagneticvector potential   A( ¯ ρ)  [22](20)  A( ¯ ρ) =  B 0 2 πκ  ¯ ρ   ¯ ρ 2 R 2 c Θ(R c − ρ) + Θ(ρ − R c )  ˆ e ϕ , where  B 0  is the total magnetic flux. Vector potential (20) gives rise to a uniform magnetic field   B  within the dispirated region. 4. The Schrödinger equation We study the non-relativistic quantum dynamics of a charged spin- 12  particle propagating in a space with finite dispiration defect(19) in presence of a uniform magnetic field   B = ∇×  A  and planar potential(21) V( ¯ ρ) =  12 Mω 2 ¯ ρ 2 Θ(R c − ρ) +  f  ¯ h 2 2 M   ¯ ρ 2 , where  f   is a real, dimensionless constant characterizing the medium being probed. Observe that the harmonic potential does notextend beyond the medium  (R m )  being considered, where  R m  R c  and  R m  is the linear dimension of the medium. Furthermore,following [23], the spin-torsion interaction term − 18  σ   ·  T   is considered, where   σ   =  g e µ ¯ h  ˆ e S  ,  ˆ e S   denotes the spin direction,  µ =  q ¯ h 2 Mc is the Bohr magneton and  g e  is the gyromagnetic ratio of the electron. Finally, the time-independent Schrödinger equation becomes,(22)   12 M  +  σ   ·   B −  18  T   + V( ¯ ρ)  ψ( ¯ ρ,ϕ,z) = Eψ( ¯ ρ,ϕ,z). The wavefunction satisfying (22) has periodicity  ψ( ¯ ρ,ϕ,z)  =  ψ( ¯ ρ,ϕ  +  2 πκ  ,z)  rather than the usual situation  ψ(ρ,ϕ,z)  = ψ(ρ,ϕ + 2 π,z)  in flat space. The Laplace–Beltrami operator    is defined by(23)  :=  1 √  g   ¯ hi C ∂ i  −  qcA i  g ij  √  g   ¯ hi C ∂ j   −  qcA j   =−¯ h 2 ∇  2 +  q 2 c 2  A 2 −  q ¯ hci C ( ∇·  A +  A · ∇  ), where  i C  is the imaginary unit,  c  is the speed of light,  q  is the charge of the electron and  g  = det | g ij  |  is the determinant of themetric. The Laplacian in the space (2) is given by(24) ∇  2 =  1 ¯ ρ∂∂  ¯ ρ  ¯ ρ∂∂  ¯ ρ  +  1 κ  2 ¯ ρ 2   ∂∂ϕ − T  z  ∂∂z  2 +  ∂ 2 ∂z 2  . The divergence appearing in (23) is computed according to(25) ∇·  A =  1 ¯ ρ∂∂  ¯ ρ( ¯ ρA ¯ ρ ) +  1 κ  ¯ ρ   ∂∂ϕ − T  z  ∂∂z  A ϕ  +  ∂A z ∂z and is vanishing since  ∇·  A =  1 κ   ¯ ρ (  ∂∂ϕ  − T  z ∂∂z )  B 0 2 πκ   ¯ ρ  = 0. With the aid of  (24) and (25), operator (23) can be written as  =−¯ h 2 ∇  2 +  q 2 c 2   B 0 2 πκ  ¯ ρ  2   ¯ ρ 2 R 2 Θ(R c − ρ) + Θ(ρ − R c )  2 (26) −  qB 0 2 πcκ  ¯ ρ ¯ hi C   ¯ ρ 2 R 2 Θ(R c − ρ) + Θ(ρ − R c )   ∂∂ϕ −  s T  b z πR 2 Θ(R c − ρ)∂∂z  .
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