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The Solutions of Dirac Equation on the Hyperboloid Under Perpendicular Magnetic Fields

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The Solutions of Dirac Equation on the Hyperboloid Under Perpendicular Magnetic Fields The solutions of Dirac equation on the hyperboloid under perpendicular magnetic fields Duygu Demir Kızılırmak1 and Şengül Kuru2 1 Department of Medical Services and Techniques, Ankara Medipol University, Ankara, Turkey 2 Department of Physics, Faculty of Science, Ankara University, Ankara, Turkey E-mail: 1 [email protected], 2 [email protected] Abstract In this study, firstly it is reviewed how the solutions of the Dirac-Weyl equation for a massless charge on the hyperboloid under perpendicular magnetic fields are obtained by using supersymmetric (SUSY) quantum mechanics methods. Then, the solutions of the Dirac equation for a massive charge under magnetic fields have been computed in terms of the solutions which were found before for the Dirac-Weyl equation. As an example, the case of a constant magnetic field on the hyperbolic surface for massless and massive charges has been worked out. Keywords: Dirac equation, Dirac-Weyl equation, supersymmetric methods, hyperbolic surface 1. Introduction While in non-relativistic quantum mechanics the spin-0 particles are described by the Schrödinger equation, in relativistic quantum mechanics the scalar particles are described by Klein-Gordon equation, the spin-½ particles by means of Dirac equation, and massless spin-½ particles by the Dirac-Weyl equation [1, 2]. Here, we will consider both Dirac and Dirac-Weyl equations. But we will see that in this process we must also solve some related effective Schrödinger equations. Graphene has received much attention in the last years not only in condensed matter but also in theoretical physics [3]. In graphene, charged quasi-particles are described by the Dirac-Weyl equation [4]. Within this context, it is difficult to confine such massless particles in graphene by applying electric fields because of the strong Klein tunnelling. However, this can be achieved by means of magnetic fields. So, it is quite important to know the behaviour of massless charged particles under magnetic fields. Besides graphene, carbon has many allotropes such as graphite, fullerenes, nanotubes, nanoribbons, nano- cones etc. [3]. They have different geometries, so it is quite relevant to adapt the Dirac-Weyl equation to them. Up to now many studies have been done on Dirac-Weyl equation not only in flat space but also in curved spaces [5-17]. Our group has characterized Dirac-Weyl equations on the plane, sphere, cylinder [5-7] and finally on hyperbolic surfaces [8]. We have solved these problems analytically by using simple SUSY methods like factorization or intertwining. In this work, our aim is to solve analytically the Dirac equation for the massive particles on the hyperbolic surface under perpendicular magnetic fields by means of the solution of massless Dirac-Weyl equations. Hence, we will investigate the relation of the spectrum, ground states, etc. between these two equations and also the importance of mass. The organization of this paper is as follows. In the second section, the Dirac-Weyl equation on the hyperbolic geometry under magnetic fields is reviewed briefly. In section 3, the bound state solutions of Dirac equation for massive particles under magnetic field on the hyperboloid are found by using the solutions for massless particles. An example is worked out to illustrate these results. Finally, this paper will end with some conclusions. 2. Dirac-Weyl equation on the hyperboloid First, we will review the method to obtain the Dirac-Weyl equation on a surface by restricting that of the flat ambient space as it was done in a preceding paper [8]. 1 2.1 Dirac-Weyl equation on the hyperboloid under perpendicular magnetic fields The Dirac-Weyl equation in (2+1) dimensions in Cartesian coordinates is 휕Ф(푥,푦,푧,푡) 푣 (휎⃗. 푝⃗)Ф(푥, 푦, 푧, 푡) = 푖ℏ , (1) 퐹 휕푡 휕 휕 휕 where ⃗σ⃗⃗ = (휎 , 휎 , 휎 ) are Pauli matrices, 푝⃗ = −푖ℏ(휕 , 휕 , 휕 ) ≔ −푖ℏ( , , ) is the three dimensional momentum operator 푥 푦 푧 푥 푦 푧 휕푥 휕푦 휕푧 and 푣퐹 is an effective Fermi velocity. If there is an external magnetic field, then the interaction between the Dirac electron and 푞 this field can be described by the minimal coupling rule where the momentum operator 푝⃗ is replaced by 푝⃗ − 퐴⃗. Here, 푞 = 푐 −푒 is the electron charge and 퐴⃗ is the vector potential. The magnetic field 퐵⃗⃗ is defined in terms of 퐴⃗ as usual, 퐴⃗ = (퐴푥, 퐴푦, 퐴푧), 퐵⃗⃗ = 훻⃗⃗푥 퐴⃗ . (2) 푖퐸푡 − If the stationary solution Ф(푥, 푦, 푧, 푡) = Ѱ(푥, 푦, 푧)푒 ℏ is replaced in equation (1), we get the time-independent Dirac-Weyl equation 푞 푣 [휎⃗. (푝⃗ − 퐴⃗)] Ѱ(푥, 푦, 푧) = 퐸 Ѱ(푥, 푦, 푧). (3) 퐹 푐 The two-sheeted hyperboloid directed along the z axis can be expressed by the equation 푥2 + 푦2 − 푧2 = −푟2 . The relations between hyperbolic (푟, 푢, 휙) and Cartesian (푥, 푦, 푧) coordinates (in the upper sheet) are given by 푥 = 푟 sinhu cos휙, 푦 = 푟 sinhu sin휙, 푧 = 푟 coshu (4) where 0 < 푢 < ∞, 0 ≤ 휙 < 2휋 and 0 < 푟 < ∞ [8]. In order to get the Dirac-Weyl equation on the hyperbolic surface, we take the appropriate Pauli matrices and restrict the equation in the ambient space to that of the surface by using suitable momentum operators [8, 18]. We will assume that the magnetic field is perpendicular to the surface of the hyperboloid, 푟 = 푅 = 푐표푛푠푡푎푛푡, thus, hereafter we use the following notation Ѱ(푅, 푢, 휙) ≔ Ѱ(푢, 휙). The vector potential is chosen in the form 퐴⃗ = 퐴(푢)휙̂ = 퐴(푢)(− sin 휙 , cos 휙 , 0). Therefore, this system has rotational symmetry around the z axis and the total angular momentum ℏ 퐽 = −푖ℏ휕 + 휎 (5) 푧 휙 2 푧 commutes with the Hamiltonian: [퐻, 퐽푧] = 0. So, (퐻, 퐽푧) have common eigenfunctions: 퐽푧Ѱ(푢, 휙) = 휆ℏ Ѱ(푢, 휙) where 휆 is half odd number. Then, if we substitute the two-component spinor by 1 푖(휆− )휙 푒 2 푓1(푢) 1 Ѱ(푢, 휙) = 푁 ( 푖(휆+ )휙 ) (6) 푒 2 푓2(푢) in the eigenvalue equation 퐻 Ѱ(푢, 휙) = ℇ Ѱ(푢, 휙) where ℇ = 퐸/ℏ푣퐹, we can separate the variable 휙 from the Dirac-Weyl equation. Thus, we have only one variable 푢 in the reduced equations. Then, following the steps given in [8] 휎푦 푓 (푢) − 푢 휓 (푢) 퐹(푢) = ( 1 ) = 푒 2 ( 1 ) (7) 푓2(푢) 휓2(푢) 푇 1 푇 1 and using the transformation (휓 (푢) , 휓 (푢)) = (푔 (푢) , 푖푔 (푢)) = 퐺(푢) , we obtain the eigenvalue matrix 1 2 √sinh푢 1 2 √sinh푢 differential equation [8]: 휆 푞 0 휕 + 푖 ( − 퐴(푢)) 푅 푢 푅 sinh푢 푐ℏ 푔 (푢) 푔 (푢) ( ) ( 1 ) = ℇ ( 1 ) . (8) 휆 푞 푖푔 (푢) 푖푔 (푢) 휕 − 푖 ( − 퐴(푢)) 0 2 2 푅 푢 푅 sinh푢 푐ℏ 퐸 Remark that the resulting matrix Hamiltonian is Hermitian and ℇ = . It includes the centrifugal term in 휆 and the magnetic ℏ푣퐹 potential 퐴(푢) through a minimal coupling. 2 2.2 The solutions of Dirac-Weyl equation by using supersymmetric methods Let us define the first order differential operators ± 퐿 = ∓휕푢 + 푊(푢) (9) where 푊(푢) is the superpotential given by 휆 푞푅 푊(푢) = − + 퐴(푢) . (10) sinh푢 푐ℏ Notice that 퐿± are adjoint operators. Taking into account these definitions, the matrix Hamiltonian can be rewritten in terms of 퐿± as + 0 −푖퐿 푔1(푢) 푔1(푢) ( − ) ( ) = 푅ℇ ( ) (11) 푖퐿 0 푖푔2(푢) 푖푔2(푢) thus, + − 퐿 푔2(푢) = 푅ℇ푔1(푢), 퐿 푔1(푢) = 푅ℇ푔2(푢) . (12) From the above equations, a pair of decoupled second order effective Schrödinger equations is obtained + − 퐻1푔1(푢) = 퐿 퐿 푔1(푢) = 휀푔1(푢), (13) − + 퐻2푔2(푢) = 퐿 퐿 푔2(푢) = 휀푔2(푢), (14) where 휀 = 푅2ℇ2. These two equations can be expressed together in matrix form: + − 퐿 퐿 0 푔1(푢) 푔1(푢) ( − +) ( ) = 휀 ( ). (15) 0 퐿 퐿 푖푔2(푢) 푖푔2(푢) The effective Hamiltonians 퐻1 and 퐻2 are called partner Hamiltonians 휕2 휕2 퐻 = − + 푉 (푢), 퐻 = − + 푉 (푢) . (16) 1 휕푢2 1 2 휕푢2 2 The potentials corresponding to these two effective Schrödinger Hamiltonians are defined in terms of 푊(푢), 2 ′ 2 ′ 푉1(푢) = 푊(푢) − 푊 (푢), 푉2(푢) = 푊(푢) + 푊 (푢). (17) These relations show that 퐻1 and 퐻2 are indeed supersymmetric partner Hamiltonians. The following intertwining relations are satisfied between them: − − + + 퐻2퐿 = 퐿 퐻1, 퐻1퐿 = 퐿 퐻2. (18) Let the discrete spectrum of Hamiltonian 퐻1 be the set of eigenvalues denoted by {휀1,푛}, 푛 = 0,1, … corresponding to the − set of eigenfunctions {푔1,푛}. If the ground state of 퐻1 is annihilated by 퐿 − 퐿 푔1,0 = 0 , (19) from (13) the ground state eigenvalue of 퐻1 is 휀1,0 = 0. We must check that 푔1,0 satisfies suitable boundary conditions and it is square-integrable in (0, ∞). Then, the discrete spectrum of 퐻2 contains the eigenvalues {휀2,푛−1} and the normalized eigenfunctions {푔2,푛−1} , which can be written as follows: 1 − 푔2,푛−1(푢) = 퐿 푔1,푛(푢), 휀1,푛 = 휀2,푛−1, 푛 = 1,2, … . (20) √휀1,푛 Finally, the eigenvalues of equation (15) are 휀0: = 휀1,0 = 0, 휀푛: = 휀1,푛 = 휀2,푛−1, 푛 = 1,2, … . (21) Taking into account the above results, the eigenfunctions corresponding to the excited energy states of matrix Hamiltonian given by (8) will be 3 ±푔1,푛(푢) 퐺±,푛(푢) = 푁 ( ), (22) 푖푔2,푛−1(푢) Where the signs are for positive or negative energies. Therefore, the eigenvalues of Dirac-Weyl equation (퐸 = ℏ푣퐹ℇ) are found in terms of the eigenvalues 휀 of the partner Hamiltonians given by (16) from the relation 휀 = 푅2ℇ2: 퐸 1 Ɛ±,푛 ≔ = ± √휀푛 , 푛 = 1,2, … .
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