Abstract

The exact solutions of the -dimensional Dirac equation on the torus and the new extension and generalization of the trigonometric Pöschl-Teller potential families in terms of the torus parameters are obtained. Supersymmetric quantum mechanics techniques are used to get the extended potentials when the inner and outer radii of the torus are both equal and inequal. In addition, using the aspects of the Lie algebraic approaches, the algebra is also applied to the system where we have arrived at the spectrum solutions of the extended potentials using the Casimir operator that matches with the results of the exact solutions.

1. Introduction

There are several potentials which are used in both theoretical and applied physics with exact solutions in nonrelativistic quantum mechanics. These potential classes can also give physical results in the successful union of quantum mechanics and special relativity where the Dirac equation has a successful explanation on the antimatter, spin, and the realistic behavior of atoms [1]. On the other hand, the gravitational field effects on some quantum mechanical systems have been studied as an exciting research field [2–4]. From the symmetries of the Dirac equation [5] and Hermiticity and uniqueness [6] to the factorization and pseudo-supersymmetry [7, 8], covariant form of the Dirac equation and its aspects are studied. The gravitational background can bring some mathematical difficulties through some geometries. One of them is the solutions of the wave equations on the torus geometry [9–12]. At the same time, in physical applications, such as graphene related ones, it is stated that the curvature of the material can change the electron density of the states. In [9], graphene nanoribbon along the surface of a torus is examined within the long wave approximation where the Dirac equation is solved approximately. Then, the recent study aims to bring a new viewpoint to the problem of the exact solutions for the Dirac equation on the torus which may lead to both applied and theoretical interest in the recent studies. When considering the works which are about the general theory of relativity and the quantum mechanics unification, there is a fact that the curvature of spacetime at the position of the atom can affect the spectrum. Therefore, the problem of electron and its perturbed energy spectrum owing to the gravitational field can bring more problems in the solutions through the geometries of the spacetime. Moreover, there is a class of potentials known as rationally extended potential models that have attracted much attention [13, 14]. The extension of the theory within the exceptional polynomials for the Dirac systems is first studied using the Darboux transformations [15]. Hence, this work also involves the rationally extended potentials which can be suitable for the Hamiltonian obtained after torus parametrization.

In this paper, Section 2 involves the relativistic quantum mechanical wave equation in the gravitational background for a massless fermion where the Fermi velocity is taken as a position dependent function [16]. Supersymmetric partner potentials are obtained for the transformed Dirac Hamiltonian. The spectrum and spinor solutions are given when the inner and outer radii of the torus are equal and inequal. In Section 3, we present the Lie algebraic computations for the Dirac Hamiltonians.

2. Dirac Equation on the Torus

Formulating the Dirac equation in curved spacetime, it is known that all metrics related by a general coordinate transformation are physically equivalent and physical observables in gravitation field should be invariant under general coordinate transformations. This is known as general covariance principle which necessitates transforming a tensor in the flat spacetime as a tensor under general transformations in the curved manifold. The covariant generalization of the Dirac equation to curved space was independently introduced by Weyl and by Fock [17, 18]. Then, the Dirac equation can be written in terms of vierbein fields and gravitational spin connection as [17] where is the spin connection and is the spinor which includes electron’s wave-functions near the Dirac point. The Dirac matrices in curved spacetime satisfy Here is the metric tensor and the tetrad(vierbein) frames field is defined as where . The metric for the torus surface is given by In the metric given above, the inner radius of the torus is , the outer radius is given by , and . We can use the spin connection formula which is where are the Christoffel symbols. In [9], the symbols were given in terms of the variable which corresponds to in our work. So one can obtain them as and accordingly, For the Dirac matrices, we will use Pauli matrices as , , and . We use (6) and (7) in (1) and we get [9] where . We note that is taken and the Fermi velocity may not be a constant [16]. Using , we obtain where we use for the sake of simplicity. Considering (9), the solutions may take the form Then we have Equation (12) can also be expressed as where We may get as follows: One can make the coefficient of zero and can be found as Then, using in (16), (13) becomes whereIn this work [19], it is shown that, with a superpotential which is a hyperbolic function then, the pair of potentials is given aswhere this potential is defined on the half line and is known as generalized Pöschl-Teller potential [20–22]. In [22], the authors removed the rational term in using a parameter condition . In this case, the potential is known as generalized Pöschl-Teller potential in the literature with the exact solutions. Now we will discuss the cases depending on the different choices of rational term in superpotential. Now we shall look at (18) to arrive at a solvable model given below. If we take where are real constants, hence, our function satisfying (22) can be found as where are constants and Now we will search for the solutions for our systems in case of the equal and inequal torus radii.

2.1. Equal Inner and Outer Radii ()

First we assume that the superpotential is a trigonometric function which is and we get the partner potentials It can be seen from (26) that, in case of an applied special condition , and , one can obtain system (20). In order to find a parameter condition on the rational term, we can simplify (26) as Then, we can find these conditions as Thus, we have which is known as trigonometric Pöschl-Teller potential. On the other hand, we can get the partner potential as Here, it is well known that (33) and (34) are isospectral partner potentials except the ground state. Comparing (33) and in (22), we obtain The solutions of are already known as [20] This result shows that when , . Keeping in mind the eigenvalue equation (17), we can get Then, the solutions are written as [20] In fact, the complete solutions are not given yet. The spinor solutions can be given as where is the normalization constant. The solutions corresponding to the ones for the partner potential can be found. The system can be summarized as follows: For (34), and it is known that the energy states of these Hamiltonians are given byAt the same time, we can find solutions using (41): When it comes to the solutions of (10) which shares the same energy with (9), we get The wavefunction mapping can be given as and we obtain where Then, we can get as where and are real constants and equating (47) and (48), is found to be If the parameters are chosen as and , then we get and using (29), and can be expressed in terms of the torus parameter and hence, one can get the supersymmetric partner potentials for which are :