Exact Solutions of the Dirac Hamiltonian on the Sphere under Hyperbolic Magnetic Fields

Abstract

Two-dimensional massless Dirac Hamiltonian under the influence of hyperbolic magnetic fields is mentioned in curved space. Using a spherical surface parameterization, the Dirac operator on the sphere is presented and the system is given as two supersymmetric partner Hamiltonians which coincides with the position dependent mass Hamiltonians. We introduce two ansatzes for the component of the vector potential to acquire effective solvable models, which are Rosen-Morse II potential and the model given Midya and Roy, whose bound states are Jacobi 𝑋₁ type polynomials, and we adapt our work to these special models under some parameter restrictions. The energy spectrum and the eigenvectors are found for Rosen-Morse II potential. On the other hand, complete solutions are given for the second system. The vector and the effective potentials with their eigenvalues are sketched for each system.

1. Introduction

Since the isolation of graphene investigated in 2004 by Novoselov et al. [1], this new material has spurred an intense interest in its applications in the fields of the condensed matter physics and the quantum field theory. The charge carriers in flat graphene are modeled in continuum by the Dirac-Weyl operator for massless fermions [2, 3]. Apart from some spectacular properties of graphene such as the half-integer quantum Hall effect [1, 2, 4–6], observation of Klein tunneling for the two-dimensional massless Dirac electrons [7–9], algebraic approaches to impurities [10], and graphene wormholes [11], interesting flexural modes of the graphene have also attracted much attention [12–17]. According to the Landau-Peierls theorem, the flat graphene does not exist at all [18, 19]. The curvature effects on the quality factor of the graphene nanoresonators can be found in [20]. Moreover, it is seen that graphene nanoribbons can be the solution of improving energy storage in ultra-high capacitors and bending nanodevices [21–23]. In this manner, instead of a flat graphene, curved graphene has such mechanical and electrical properties that lead to the possibility of a bending component of some nanodevices. In the relativistic territory, the Dirac equation in curved spacetime has been studied as its generalization to the Robertson-Walker spacetime in a Cartesian tetrad gauge [24], the gravitational effects in Hydrogen atom [25], bound states of the Dirac equation in gravitational fields [26], exact solutions in (1+1) and (2+1) dimensions [27, 28], some conditions on the modified Dirac equation which admits a symmetry operator [29], and the Hawking-Unruh phenomenon on graphene [30]. On the other hand, the system on the two-sphere or on the pseudosphere is related to some interesting generalizations of the planar Landau level problems [31, 32]. The supersymmetry is known to be responsible for the mathematical structure of the Landau levels. The Pauli Hamiltonian for a nonrelativistic spin-1/2 particle possesses 𝑁=2 supersymmetry [33, 34]. The Dirac equation and its analysis and discussion within supersymmetric quantum mechanics can be found in [35–39]. Moreover, extending a study of Bochner [40], exceptional orthogonal polynomials were introduced first in [41]. Later, it is shown that there is a relation between exceptional orthogonal polynomials and the Darboux transformation [42, 43].

In this study, the Dirac equation in curved space is written for the curved graphene models to show that an exactly solvable bound state model for the Dirac equation in a two-dimensional curved space can be reduced into a Schrödinger-like operator and the bound states can be discussed using the concepts of quantum mechanics. Rosen-Morse II potential (hyperbolic Rosen-Morse) model is studied as a first example and the next one is related with the soliton models [44]. This potential family is empirically useful to investigate the polyatomic vibrational states of NH₃ molecule [45]. Also, trigonometric Rosen-Morse potential is studied in the confinement phenomena of quarks in hadrons and this potential is responsible for the quark-gluon dynamics through the QCD calculations [46, 47]. Hyperbolic potentials are popular in nonrelativistic quantum mechanics with their soliton models [48]. Considering these instanton structures, hyperbolic interactions and their connection with the QCD and Morse-Rosen potentials can be found in an interesting work [49]. Also, actual relativistic vibrational states for the molecules are studied in [50]. Thus, our study may be a motivation in the relativistic domain for the relevant QCD models.

2. Dirac Operator in Two-Dimensional Curved Space

On a two-dimensional surface, the Dirac operator is given by [51] ℋ_𝐷=−𝑖𝛾^𝑎𝐸^𝜇_𝑎𝜕_𝜇+12𝐸^𝜇_𝑎𝜕_𝜇ln√1𝑔+2𝜕_𝜇𝐸^𝜇_𝑎. (1) The symbol 𝛾^𝜇 corresponds to the generalized Dirac matrices satisfying the relation 𝛾^𝑎𝐸^𝜇_𝑎=𝛾^𝜇 (2) and the anticommutation relation is 𝛾^𝜇,𝛾^𝜈=2𝑔^𝜇𝜈. (3) Here, 𝐸^𝜇_𝑎 are the tetrad fields, 𝑔 is the determinant of the metric tensor, and 𝑔_𝜇𝜈 is the metric tensor for the curved spacetime; the Latin indices refer to local inertial frame and the Greek indices refer to curved spacetime. Generally, 𝑔_𝜇𝜈 is given by 𝑔_𝜇𝜈(𝐱)=𝜂_𝑎𝑏𝑒^𝑎_𝜇(𝐱)𝑒^𝑏_𝜈(𝐱), (4) where 𝐱=(𝑡,𝑥), 𝜂_𝑎𝑏=diag(+1,−1), and 𝑒^𝑎_𝜇 and 𝐸^𝜇_𝑎 are zweibeins and their inverse satisfies 𝛿^𝑎𝑏𝐸^𝜇_𝑎𝐸^𝜈_𝑏=𝑔^𝜇𝜈,𝑔_𝜇𝜈𝐸^𝜇_𝑎𝐸^𝜈_𝑏=𝛿_𝑎𝑏. (5) The conformal metric tensor is given by [51] 𝑔_𝜇𝜈=𝑒^2𝜎,1001 (6) where 𝜎(𝑥) is a spatial function. Using (5) and (6), we obtain 𝐸^𝜈_𝑎=𝑒^−𝜎𝛿^𝜈_𝑎. As it is explained in [51], 𝑃_𝜇=−𝑖𝜕_𝜇 is the flat momentum operator replaced by 𝑃_𝜇→𝑃_𝜇−𝐴_𝜇, where 𝐴_𝜇 is the vector potential which is spatial dependent. In two dimensions, Dirac matrices reduce to Pauli matrices; then, the Dirac operator becomes ℋ_𝐷=−2𝑖𝑒^𝜎0𝜕_𝑧−𝑖𝐴_𝑧+12𝜕_𝑧𝜎𝜕_𝑧∗−𝑖𝐴_𝑧∗+12𝜕_𝑧∗,𝜎0 (7) where ∗ denotes the complex conjugate and the transformation 𝑧=𝑥¹+𝑖𝑥² is used in the Hamiltonian above for the sake of simplicity. In this manner, we use 𝜕_𝑧=(1/2)(𝜕₁−𝑖𝜕₂), 𝐴_𝑧=(1/2)(𝐴₁−𝑖𝐴₂). The operator (7) is also studied in [52]. Now we take ℋ_𝐷 which can be mapped to the operator ℋ_𝐷 using a transformation operator 𝑈 as ℋ_𝐷=𝑈ℋ_𝐷𝑈⁻¹𝑒,𝑈=^𝜎/200𝑒^𝜎/2. (8) Therefore, we have ℋ_𝐷=𝛼0𝜕₁−𝑖𝜕₂−𝑖𝐴₁−𝐴₂𝜕₁+𝑖𝜕₂−𝑖𝐴₁+𝐴₂0, (9) where 𝛼=−𝑖𝑒^𝜎. Let us bring the attention to the isothermal coordinates and parameterization [53]. Because the topological equivalent of a layer is known as a sphere, then we may deal with a sphere with radii 𝑅 and the surface parameterization of the sphere is given by 𝐱(𝑢,𝑣)=𝑅(cos𝑢cos𝑣,sin𝑢cos𝑣,sin𝑣), (10) and the metric can be written as (𝑑𝑠)²=𝑅²cos²𝑣𝑑𝑢²+𝑑𝑣²=𝑅²cos²𝑣𝑑𝑢+𝑖𝑑𝑣cos𝑣𝑑𝑢−𝑖𝑑𝑣.cos𝑣 (11) We may rewrite the metric using the transformation 𝑤=log(tan𝑣+sec𝑣). Then, the metric of the patch is written as 𝑑𝑠²=𝜆²𝑑𝑢²+𝑑𝑤², (12) where 𝜆=𝑅cos𝑣. If we use the transformation cosh𝑤=sec𝑣, we get 𝑒^𝜎=𝑅sech𝑤. It can be pointed out that the eigenvectors of the operator ℋ_𝐷 are Ψ=𝑒^{𝑖𝑘𝑢𝜓1}_𝑖𝜓2, where we take 𝑥¹=𝑤 and 𝑥²=𝑢, and 𝑘 is the wave number. We also note that the vector potential and its components are selected as 𝐴_𝜇=[0,𝐴_𝑢(𝑤),0]. Let us now look at the eigenvalue equation ℋ_𝐷Ψ(𝑥)=𝐸Ψ(𝑥).

3. Dirac Hamiltonians

The Dirac Hamiltonian in (9) can be rewritten as 𝛼𝜕_𝑤−𝑖𝜕_𝑢−𝐴_𝑢(𝑤)𝑖𝑒^𝑖𝑘𝑢𝜓₂(𝑤)=𝐸𝑒^𝑖𝑘𝑢𝜓₁(𝑤),𝛼𝜕_𝑤+𝑖𝜕_𝑢−𝐴_𝑢(𝑤)𝑒^𝑖𝑘𝑢𝜓₁(𝑤)=𝐸𝑖𝑒^𝑖𝑘𝑢𝜓₂(𝑤). (13) The equations given above lead to ℋ^𝑗_𝐷𝜓_𝑗(𝑤)=𝐸²𝜓_𝑗(𝑤),𝐸=𝐸𝑅, (14) where 𝑗=1,2. Then, Hamiltonians, ℋ^𝑗_𝐷, shown in a joint expression can be given as ℋ^𝑗_𝐷=−cosh²𝑤𝑑²𝑑𝑤²𝑑−sinh𝑤cosℎ𝑤+𝑑𝑤𝑘−𝐴_𝑢(𝑤)²+(−1)^𝑗𝐴^′(𝑤)cosh²𝑤+(−1)^𝑗𝐴_𝑢(𝑤)−𝑘cosh𝑤sinh𝑤. (15) We may transform (14) into a Sturm-Liouville type equation using a similarity transformation. We obtain 𝔥_𝑗=𝜌⁻¹ℋ^𝑗_𝐷√𝜌,𝜌=cosh𝑤⟹𝔥_𝑗=−cosh²𝑤𝑑²𝑑𝑤²𝑑−2sinh𝑤cosℎ𝑤𝑑𝑤+𝑉^𝑗_eff(𝑤), (16) where 𝑉^𝑗_eff(𝑤)=𝑘−𝐴_𝑢(𝑤)²+(−1)^𝑗𝐴^′(𝑤)cosh²𝑤+(−1)^𝑗𝐴_𝑢(𝑤)−3−𝑘cosh𝑤sinh𝑤4cosh²1𝑤+4. (17) Thus, the transformed Dirac system can be given as 𝔥_𝑗𝜙_𝑗=𝐸²𝜙_𝑗,𝜙_𝑗=𝜌𝜓_𝑗. (18) We emphasize that the solutions 𝜙_𝑗,𝑛 have to satisfy ∫_Ω{|𝜙_1,𝑛(𝑤)|²+|𝜙_2,𝑛(𝑤)|²}𝑑𝑤<∞ for the physical reasons, and here 𝑛 is a quantum number. We will examine (18) in the following section using the choices of 𝐴_𝑢(𝑤).

3.1. Hyperbolic Model I

Let us choose 𝐴_𝑢(𝑤) as 𝐴_𝑢(𝑤)=𝐶₁sech²𝑤+𝐶₂tanh𝑤+𝐶₃, (19) where 𝐶₁,𝐶₂,𝐶₃ are constants; we may obtain 𝑉¹_eff(𝑤)=−𝐶₂+2𝐶₁𝐶₃−2𝐶₁𝑘+𝐶₃−𝑘²−12cosh²𝑤+𝐶²₁sech²𝑤+−𝐶₃+2𝐶₂𝐶₃+𝑘−2𝐶₂𝑘+𝐶cosh𝑤sinh𝑤₂₂−𝐶₂−14sinh²𝑤+𝐶₁1+2𝐶₂tanh𝑤. (20) Equating the coefficients of cosh𝑤sinh𝑤, cosh²𝑤, and sinh²𝑤 to zero, one can obtain 𝐶₂ and 𝐶₃ as 𝐶₂;𝐶₃=−12,12,12,32;(𝑘,−1+𝑘,1+𝑘,𝑘). (21) Thus, (20) becomes 𝑉¹_eff(𝑤)=𝐶²₁sech²𝑤+𝐶₁1+2𝐶₂+𝐶tanh𝑤₂−12²+2𝐶₁𝐶₃−1−𝑘2, (22) and also 𝑉²_eff(𝑤) turns into 𝑉²_eff(𝑤)=𝐶²₁sech²𝑤+𝐶₁−1+2𝐶₂tanh𝑤+2𝐶₂cosh²𝑤+2𝐶₃+𝐶−𝑘cosh𝑤sinh𝑤₂+12²−2𝐶₁𝐶₃−1−𝑘2. (23) As it is seen from (23), 𝑉²_eff(𝑤) is not a solvable potential model while (22) is known as Rosen-Morse II potential [54]. As a result, they are not shape invariant potentials but they share the same energy spectrum except the ground state. If we take 𝐶₂≠−1/2, this implies 𝐶₃=𝑘−1, 𝐶₃=𝑘+1, or 𝐶₃=𝑘. Next, we will come up with the whole solutions of the system (22). Hence, we use the transformations given in (18) 𝜙₁=sech²𝑤𝐺(𝑡),𝑡=tanh𝑤 (24) and we get 𝑑²𝐺𝑑𝑡²−4𝑡1−𝑡²𝑑𝐺+1𝑑𝑡1−𝑡²²×𝐸²−𝐶²₁+21−𝑡²−𝐶₁1+2𝐶₂𝑡−𝐶₂−12²−2𝐶₁𝐶₃+1−𝑘2𝐺(𝑡)=0. (25) We propose a solution which is 𝐺(𝑡)=(1−𝑡)^(𝑎−1)/2(1+𝑡)^(𝑏−1)/2𝐹(𝑡), where 𝐹(𝑡) is a polynomial that we look for and we obtain 1−𝑡²𝑑²𝐹𝑑𝑡²+(𝑏−𝑎−(𝑎+𝑏+2)𝑡)𝑑𝐹+1𝑑𝑡1−𝑡²𝐀𝑡²+𝐁𝑡+𝐂𝐹(𝑡)=0, (26) where 𝐀,𝐁,𝐂 are constants. Hence, the solutions of the system are given as 𝐸_1,𝑛1=±𝑅12+2𝐶₁𝑘−𝐶₃−𝐶₂−12²−−1+1−4𝐶²₁2−𝑛²−𝐶₁1+2𝐶₂/2²−1+1−4𝐶²₁/2−𝑛^21/2,𝜙_1,𝑛=𝑁₁(1−tanh𝑤)^{(−1+1−4𝐶21)/2}(1+tanh𝑤)^{𝐶1(1+2𝐶2)/2}×𝑃^{(−1+1−4𝐶21,𝐶1(1+2𝐶2}_𝑛⁾⁾(tanh𝑤), (27)(28) where the symbol 𝑃_𝑛^(⋅,⋅)(tanh𝑤) corresponds to the Jacobi polynomials. We also note that 𝐶₁<1/2 for the physical solutions. It is noted that 𝑉²_eff(𝑤) which is not a solvable model shares the same spectrum with 𝑉¹_eff(𝑤) except for the ground state [54] (Figure 1).

(a)

(b)

(c)

(d)

Figure 1 

The graphs of (a) component of the vector potential 𝐴𝑢(𝑤) in (19), effective models (b) 𝑉eff,1 (22) and (c) 𝑉eff,2 (23) for the values of 𝐶1=0.4, 𝐶2=0.5, 𝑅=1, 𝐶3=𝑘+1, and 𝑘=2, and (d) the energies ±𝐸1,𝑛 in (27) for 𝑘=200. There are bound states only for 𝑛=0,1,2 and 𝑉eff,2 is an increasing function in the positive domain while 𝑉eff,1 remains nearly constant. The scalar component of the vector potential 𝐴𝑢 is similar to 𝑉eff,1.

3.2. Hyperbolic Model II

If we set 𝐴_𝑢 as given below 𝐴_𝑢(𝑤)=𝐶₁sech²𝑤+𝐶₂sech²tanh𝑤𝑎₁tanh𝑤−𝑎₂+𝐶₃tanh𝑤+𝐶₄, (29) where we use constants 𝐶₁,𝐶₂,𝐶₃,𝐶₄, we may get 𝑉¹_eff(𝑤)=14−𝐶₃+2𝐶₁𝐶₄−2𝐶₁𝑘+−34+𝑘−𝐶₄²cosh²𝑤+𝐶²₁sech²𝑤+𝐶₃𝐶₃−1sinh²𝑤+𝑘−𝐶₄+2𝐶₃𝐶₄−2𝐶₃𝑘cosh𝑤sinh𝑤+𝐶₁1+2𝐶₃𝐶tanh𝑤−²sech²𝑤𝑎₁tanh𝑤−𝑎₂+𝑎₁𝐶₂sech²𝑤tanh𝑤𝑎₁tanh𝑤−𝑎₂²+𝐶²₂sech²𝑤tanh²𝑤𝑎₁tanh𝑤−𝑎₂²+2𝐶₂𝐶₄−𝑘tanh𝑤𝑎₁tanh𝑤−𝑎₂+2𝐶₁𝐶₂sech²𝑤tanh𝑤𝑎₁tanh𝑤−𝑎₂+𝐶₂1+2𝐶₃tanh²𝑤𝑎₁tanh𝑤−𝑎₂. (30) In [44], it is shown that a class of exactly solvable potentials, whose solutions correspond to the Jacobi-type 𝑋₁ exceptional orthogonal polynomials, can be generated. In their work, the eigenvalue equation is given by [44] 𝐻_eff𝜓(𝑥)=𝜀𝜓(𝑥),𝐻_eff=−Δ+𝑣_eff(𝑥), (31) where 𝜓(𝑥)=𝑓(𝑥)𝐹(𝑔(𝑥)) and 𝐹(𝑔(𝑥)) satisfies the differential equation which is 𝑑²𝐹𝑑𝑔²+𝑄𝑔𝑑𝐹𝑔𝑑𝑔+𝑅𝐹=0. (32) The authors choose 𝐹_𝑛∼𝑃_𝑛^(𝛼,𝛽)(𝑥),  𝑛=1,2,3,…,  𝛼,𝛽>−1,  𝛼≠𝛽, where 𝑃_𝑛^(𝛼,𝛽)(𝑥) is the Jacobi-type 𝑋₁ polynomial in Example 2 in their work and obtain 𝜀−𝑣_eff(𝑥)=𝑔^′𝜆𝐴₁𝑔+𝐴₂1−𝑔²+𝐴₃𝑔+𝐴₄1−𝑔²²+𝐴₅+𝐴𝛽−𝛼𝑔−𝛽+𝛼𝛽−𝛼𝑔−⁶𝛽+𝛼², (33) where 𝐴_𝑖,  𝑖=1,2,…,6 are constants in terms of 𝛼,𝛽 [44] and 𝑔(𝑥)=tanh𝑥 and 𝜆 is a constant. If 𝑔(𝑥) is used in (33), then we get 𝜀−𝑣_eff(𝑥)=𝐴₆sech²𝑥𝑎₁tanh𝑥−𝑎₂²+𝐴₅sech𝑥𝑎₁tanh𝑥−𝑎₂+𝐴₄cosh²𝑥+𝐴₃sinh𝑥cosh𝑥+𝐴₂+𝐴₁tanh𝑥. (34) If we compare (30) and (33) we see that 𝜆=1, 𝑎₁=𝛽−𝛼, and 𝑎₂=𝛽+𝛼 in our problem. The energy spectrum and the solutions of (31) are given in [44]. Hence, we continue to compare 𝜀−𝑣_eff(𝑥) and 𝐸²−𝑉¹_eff(𝑤) which may provide complete solutions of our system without solving a differential equation. Let us add and subtract the terms 𝐶₅sech²𝑤/(−𝑎2+𝑎1tanh𝑤)+𝐶₆ to the right-hand side of (30). Then, we shall obtain 𝑉¹_eff(𝑤)=14−𝐶₆+2𝐶₁𝐶₄−2𝐶₁𝑘−𝐶²₃+𝑘−𝐶₄²+𝐶₃−12²−1cosh²𝑤+𝑘−𝐶₄+2𝐶₃𝐶₄−2𝐶₃𝑘cosh𝑤sinh𝑤+𝐶₁1+2𝐶₃−𝐶tanh𝑤₂+𝐶₅sech²𝑤𝑎₁tanh𝑤−𝑎₂+𝑎²₂𝐶²₁−𝑎₁𝑎₂𝐶₁sech²𝑤𝑎₁tanh𝑤−𝑎₂²,𝑉²_eff(𝑤)=14−𝐶₃+2𝐶₁𝐶₄−2𝐶₁𝑘+𝑘−𝐶₄²−34cosh²𝑤+𝐶₃1+𝐶₃sinh²𝑤+𝐶₄−𝑘+2𝐶₃𝐶₄−2𝐶₃𝑘cosh𝑤sinh𝑤+𝐶₁−1+2𝐶₃𝑎tanh𝑤−¹𝐶₁sech²𝑤𝑎₁tanh𝑤−𝑎₂+𝑎²₁𝐶₁1+𝐶₁tanh𝑤sech²𝑤tanh𝑤𝑎₁tanh𝑤−𝑎₂²+2𝑎₁𝐶₁𝑘−𝐶₄tanh𝑤𝑎₁tanh𝑤−𝑎₂−2𝑎₁𝐶²₁sech²𝑤tanh𝑤𝑎₁tanh𝑤−𝑎₂+𝑎₁𝐶₁1−2𝐶₃tanh²𝑤𝑎₁tanh𝑤−𝑎₂. (35)(36) Here we may use some constraints on the parameters which may be expressed in terms of 𝐶₁, 𝐶₂=−𝑎₁𝐶₁,𝐶₃𝑎=−²¹−𝑎²₂−2𝑎₁𝑎₂𝑘2𝑎²₁−𝑎²₂,𝐶₄𝑎=−²²𝑘𝑎²₁−𝑎²₂,𝐶₅=−2𝑎₁𝐶₁𝐶₃,𝐶₆=2𝑎²₁𝑘𝐶₁𝑎²₁−𝑎²₂. (37) Under these constraints, 𝑉¹_eff can be obtained as given in (35). Now we can compare (34) and (35) using 𝐴_𝑖,  𝑖=1,2,…,6 given in [44]. These constants are given as [44] 𝐴₁=𝛽²−𝛼²,𝐴2𝛼𝛽₂=𝑛²+𝑛+1𝛽+𝛼−14𝛽+𝛼²+𝛽−2𝛽+𝛼−4²+𝛼²,𝐴2𝛼𝛽₃=𝛽²−𝛼²2,𝐴₄𝛽=−²+𝛼²−22,𝐴₅=𝛽+𝛼𝛽−𝛼²2𝛼𝛽,𝐴₆=−2𝛽−𝛼². (38) Using (34), (35), (37), and (38), we obtain 1𝛽=±11+𝑘,𝛼=±.1−𝑘 (39) The energy eigenvalues and corresponding wave functions of the system are given in [44]. Using the results of [44] we may give the spectrum of our system as 𝐸_1,𝑚1=±𝑅𝑚+𝛼+𝛽2𝑚+𝛼+𝛽+22+𝛽𝛼−𝛼²+𝛽²−24−𝑘²1+𝑘^221/2, (40) where 𝑚=𝑛−1,  𝑚=0,1,2,… and one can see that the inside of the square root is positive. And the solutions are given by [44] 𝜙_1,𝑚(𝑤)=𝑁_𝑚(1−tanh𝑤)^(𝛼+1)/2(1+tanh𝑤)^(𝛽+1)/2𝛼+𝛽+𝛼−𝛽tanh𝑤×𝑃^(𝛼,𝛽)_𝑚+1(tanh𝑤), (41) where 𝑁_𝑚 is the normalization constant. Hence, 𝑉²_eff(𝑤) also shares the same energy except for the ground state [54] (Figure 2), 𝐸_1,𝑚=𝐸_2,𝑚−1,𝑚=1,2,…. (42)

(a)

(b)

(c)

(d)

Figure 2 

The graphs of (a) component of the vector potential 𝐴𝑢(𝑤) in (29), effective models (b) 𝑉eff,1 (35) and (c) 𝑉eff,2 (36), and (d) the energy 𝐸1,𝑛 in (40). The effective potential well 𝑉eff,2 and 𝑉eff,1, and 𝐴𝑢(𝑤) have the same singularity point. Energy eigenvalues are increasing.

4. Conclusion

In this paper, we have studied the quantum behavior of a massless two-dimensional Dirac particle subjected to a hyperbolic magnetic field in the curved space which is a two-dimensional surface on the sphere. Using appropriate transformations, we have given the Dirac Hamiltonian system a pair of first order differential equations and, then, effective Dirac Hamiltonians are expressed. Two specific models are studied according to the special choices of 𝐴_𝑢(𝑤). In fact, 𝔥_𝑗 is the physical Hamiltonian of the system with a coefficient of the second order derivative, and cosh²𝑤 corresponds to the inverse of the mass function in the position dependent mass Hamiltonian theories. We have seen that this specific choice leads to two specific exactly solvable effective potential models. In the first example, the Rosen-Morse II effective potential is obtained under some parameter restrictions. Energy eigenvalues and corresponding eigenfunctions are obtained using polynomial solutions. In the latter case we have adapted our model Hamiltonian 𝔥_𝑗 to a general one given in Example 2 in [44] and the bound state solutions are written. The partner potential 𝑉²_eff is different and more general than those obtained in [44]. On the contrary to the results in [44], the partner potentials are not shape invariant in our case. We also note that we have obtained only a limited number of bound states for the Rosen-Morse II effective potential and when the radius of the sphere approaches infinity, the energy of the system becomes zero in each case.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.