Analytical Solutions of the 1 D Dirac Equation Using the Tridiagonal Representation Approach

This paper aims at extending our previous work on the solution of the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA). In the approach, we expand the spinor wavefunction in terms of suitable square integrable basis functions that support a tridiagonal matrix representation of the wave operator. This will transform the problem from solving a system of coupled first order differential equations to solving an algebraic three-term recursion relation for the expansion coefficients of the wavefunction. In some cases, solutions to this recursion relation can be related to well-known classes of orthogonal polynomials whereas in other situations solutions represent new class of polynomials. In this work, we will discuss various solvable potentials that obey the tridiagonal representation requirement with special emphasis on simple cases with spin-symmetric and pseudospin-symmetric potential couplings. We conclude by mentioning some potential applications in graphene.


Introduction
The basic equation of relativistic quantum mechanics was formulated by Paul Dirac in 1928 in a way consistent with special relativity [1].This equation describes the behavior of weakly coupled electrons at high speeds or strongly coupled electrons such as in the case of core electron states in heavy atoms.
Among the benefits of this relativistic formulation, it is the natural emergence of the electron spin and the prediction of the existence of an antiparticle partner to the electron, the positron, which was discovered experimentally few years later.
The physics and mathematics of the Dirac equation is very rich, illuminating and providing a theoretical framework for different physical phenomena that are not present in the nonrelativistic regime such as the Klein paradox [2].In addition, Dirac equation emerges in the study of the transport properties in graphene, which makes it important for future applications.Graphene is the first truly two dimensional system whose carriers exhibit a relativistic-like behavior.Electrons in graphene are described by a massless two dimensional relativistic Dirac equation that gives rise to a gapless energy dispersion near the K and K' points of the first Brillouin zone.So in this context, graphene represents a test bed for many relativistic phenomena such as Klein tunneling that could be observed with carrier having speeds one thousand time smaller than the speed of light [3] [4] [5].
Exact solutions of the Dirac equation with a given potential configuration are limited and not trivial [6] [7] [8] [9] [10] compared to the nonrelativistic Schrödinger equation.In fact, the Dirac Hamiltonian being a matrix in the spinor space allows for more structure in the potential interaction.The terminology given to relativistic problems such as the "Dirac-Coulomb", "Dirac-Oscillator", "Dirac-Morse", ••• etc. refers to the Dirac equation that reduces to an effective Schrödinger-like equation with the named potential for the large spinor component.Different approaches were developed to generate exact solutions to the Dirac equation such as supersymmetric quantum mechanics [11], Darboux transformation [12] and factorization method [13] to mention only few.
This paper is an expanded version of our letters [14] with further developments and applications in which we use the J-matrix inspired Tridiagonal Representation Approach (TRA).The basic idea of the approach is to write the spinor wavefunction as a bounded infinite series with respect to a suitably chosen square integrable basis function as is a set of expansion coefficients that are functions of the energy and potential parameters and is a complete set of spinor basis functions that carry only kinematic information.Using this form of the spinor wavefunction in the stationary wave equation, ( ) where H is the Dirac Hamiltonian and requiring that the matrix representation of the wave operator, ( ) , be tridiagonal and symmetric so that the action of the wave operator on the elements of the basis is allowed to take the general form ( ) . This requirement transforms the wave equation to the following three-term recursion relation for 0 Thus, the problem now reduces to solving this three-term recursion relation which is equivalent to solving the original wave equation.Of course, this equation can be solved in different ways in mathematics [16] [17].Sometimes solutions of Equation (1.1) can be written in a closed form by direct comparison to well-known orthogonal polynomials.However, in other cases this recursion relation does not correspond to any of the known orthogonal polynomials giving rise to new class of orthogonal polynomials.The challenge will then be to write these solutions in a closed form and find the properties of the associated orthogonal polynomials such as the weight function, generating function, spectrum formula, asymptotics, zeroes, etc.
In the following section, we introduce the general formulation of the problem and show how to calculate the matrix elements of the Dirac wave operator in a general basis.Then we consider two important choices of bases depending on the configuration space of the problem.One is written in terms of the Jacobi polynomials and the other is written in terms of Laguerre polynomials.In the third section, we present various examples of exactly solvable potentials with special focus on possible applications.We give our conclusions and discuss future work in the last section.

Theoretical Modeling
The most general form of 1D time-independent Dirac equation in the presence of scalar potential and pseudo-scalar potential

( )
W x can be written in the following form (in the relativistic units where ε is the energy and spinor wavefunction.The space component of the vector potential can be gauged away to simplify the problem using a unitary transformation provided that the phase function Λ(x) obey the relationship ( ) , hence from now on we set U(x) = 0 in our equations.
Our strategy then is to write the spinor components as an expansion over a complete basis set ( ) In the presence of symmetry, we get more solutions to Equation (2.1).These symmetries include the spin-symmetric coupling in which ( ) ( ) pseudospin symmetric coupling which requires ( ) ( ) , and the presence of the scalar potential alone, i.e. ( ) ( ) 0 V x W x = = .We have discussed all the possible symmetries in Section 3. In these symmetries, the problem reduces to solving an effective 1D Schrödinger-like equation which has been treated using the TRA in the past [18] [19] [20] [21], and very recently in [22].In what follows, we will discuss the situation in which there is no relationship between the potentials in Equation (2.1).
We start here by relating the spinor components using Equation (2.1), in absence of U(x), as follows: ( ) ( ) We refer sometimes to ( ) as the "larger" component due to the dominator in (2.3).The general case we refer to holds when we have V(x) alone or V(x) and S(x) together with or without W(x) with no symmetry between V(x) and S(x).To simplify the problem, we relate the corresponding basis functions through the kinetic balance relation [23]: where η is just a real dimensionless constant and m ε ≠ − which means that this solution will cover only the positive energy space.Using Equation (2.4) in (2.2) with the coordinate transformation ( ) x y x → , we can write the matrix elements of the wave operator, J, as follows: ( ) where the prime over the variable stands for the derivative with respect to x, i.e. d d The coordinate transformation is made such that we make the domain of the Hamiltonian compatible with the domain of the basis functions, see where is Jacobi polynomial of order n.The parameters { } , , , α β µ ν , with , 1 µ ν > − , are real numbers.
An interesting task for the motivated reader is to use these bases to verify that Equation (2.3) leads to Equation (2.4), i.e. the tridiagonal representation of , n m J cannot be made unless we use the kinetic balance relation.In this section, we do the calculations in Laguerre basis, while the calculations in Jacobi basis can be found in Appendix B. This gives the following form of the matrix elements: Using the properties of n L ν (A1), (A3), and (A4), we impose the following constraints: 2) The parameters are constrained to be 2 Thus, the tridiagonal form of , n m J becomes: ( 1) 1 The potentials that allow (2.10) to hold must be chosen such that: We will discuss all the possible symmetries in Section 3.

Results and Discussions
This section is divided into three parts organized as follows.In Section 3.1, we expose different results related to graphene.In Sections 3.2, we discuss the set of possible solvable potentials in presence of spin symmetries.Then we move to Section 3.3 to expose some results on the general case.

Scalar Potential
In this situation we consider , where y σ is the 2 × 2 Pauli matrix 0 , on Dirac Hamiltonian in Equation (2.1), this gives the following form of the wave equation: where F v is the Fermi speed, ( ) is the 2D momentum operator, e is the electron's charge, c is the speed of light, A is the two-vector potential, and E is the energy eigen- value.Now, choosing the z-axis normal to the graphene sheet, then the magnetic field could be generated from the two-vector potential in the Landau gauge , 0 , 0, 0, ( ) , d This gauge suggests that the spinor is separable as with the following maps: ( ) ( ) ( ) The constraint in Equation (3.1.4)allows us to break Dirac equation into two effective Schrödinger equations for each spinor component which we write down in compact form as follows: where , and . Consequently, we just need to solve the effective Schrödinger Equation (3.1.5)for any component and find the other spinor component using the relation in (3.1.4).However, we need to stress that each solution of (3.1.5)will cover part of the energy space complementary to the other one.Luckily, Schrödinger equation has been treated in the past, using the TRA, by different authors including Alhaidari and Bahlouli [15] [18] [19] [20] [21].We have tabulated few of the solvable potentials of Equation (3.1.5),which were treated by the TRA, in Table 2.We should point out that the situation with is mathematically similar to the previous case which results, again, in having two Schrödinger equations for each spinor component as in Equation (3.1.5)with ( ) ( ) . Next, wewill discuss different situations that will be useful for graphene system.
As a first example, we consider the following hyperbolic magnetic field barrier: ( ) ( ) where 0 B and α are constants.This case corresponds to m k = and ( ) Comparison to Schrödinger equation, this situation is equivalent to the following potential: Table 2. Some of the solvable potentials for Schrödinger equation which were obtained in the past using the TRA [14] [23] [28].
( ) This potential is called the hyperbolic Rosen-Morse potential which was treated using the TRA in [15].Using the energy spectrum of this potential (See Table 2), we write the energy spectrum of an electron in graphene in this hyperbolic barrier as follows: where ( ) ( ) . This result agrees with the result obtained in [3].The upper component of the spinor wavefunction is now written as [15]: where ( ) ) ( ) The lower spinor component can be easily calculated using Equation (3.1.4).
Another interesting example we mention here is the case when the magnetic barrier takes the following exponentially decaying form: ( ) ( ) This is simply the 1D Morse oscillator potential, up to a constant, which was treated using the TRA for Schrödinger equation by Alhaidari in [15].Using the results in [15], we write the energy eigenvalues for Dirac-Weyl equation as follows: , e e e The potential in (3.1.15)is the generalized Hulthén potential which was treated in the TRA in [15].Based on the results obtained in [15], we write the energy eigenvalue of Dirac-Weyl equation for this situation as follows: ( ) where ( ) . We leave it to the interested reader to calculate the lower component using Equation (3.1.4).Up to our knowledge, this situation was not treated in the past.Note that for very large values of x the barrier will behave similarly to the previously mentioned case.It is obvious that all of the previous systems have finitely many bound states as shown in the spectrum formulas.
We have also solved other interesting situations in which the magnetic field is constant, singular 2 1 x , and few other cases are summarized in Table 3.  3. List of magnetic field configurations in graphene with the energy eigenvalue and the supersymmetric potential (Schrödinger potentials) of each case [14].  2 These cases are very useful in nuclear physics [26] [27] [28].For the case when 0 ∆ = , we use Equation (2.1) to write an equation for the upper spinor component which reads:

Spin-Symmetry and Pseudo-Spin Symmetry
where ( ) . Thus, we need to solve Schrödinger Equation (3.2.1) for ψ + and then use (2.1) to compute ψ − .As discussed in previous section, we will rely on the obtained solutions of Schrödinger equation using the TRA, where we tabulated few of these solvable potentials in Table 2, to obtain solvable potentials in these cases.Similarly, one can follow the same procedure for 0 Σ = , which results in an effective Schrödinger equation for the spinor lower component which is shown below: where ( ) , and The effective Schrödinger equation in this case reads: ( ) where 2E m ε = − .This is equivalent to Schrödinger equation for the Har- monic oscillator with "frequency" . The basis components are written in Laguerre basis as , which was treated in the TRA, see [15], with 1 2 ν = ± .Using the energy spectrum for the Harmonic oscillator obtained in the TRA, we write the bound states spectrum formula for this potential configuration in Dirac equation as follows: It is well known that Laguerre and Hermite polynomials are related when 1 2 ν = ± , see [16] [17].The spinor wavefunc- tion lower component for this case reads where ( ) ( ) The spinor upper component can be easily evaluated using Equation (2.1).For applications, this situation can be modeled for an electron in graphene moving under the influence of linear electric and magnetic fields by considering the following map in Dirac-Weyl equation: Moreover, this case has also been studied in nuclear physics and our results match with what other authors obtained, see for example, [29] [30] [31].However, one can follow a similar procedure for the same oscillator potentials for spin-symmetric couplings, i.e.
for ( ) ( ) , with ( ) 0 W x = , and obtain the spectrum to be similar to (3.2.4) with m m ε ε − → + under the square root.Another example we would like to mention here is the case when we have spin-symmetric with ( ) ( ) ( ) This is a special form of Rosen-Morse potential which was treated in the TRA in [15].Using the spectrum formula for ( ) ss U x , we calculate the bound states spectrum formula of Dirac particle in this potential configuration to be: where ( ) ( ) The upper spinor component is written as below 1 tanh 1 tanh tanh , where ( ) . Thus, the condition in (3.2.7) requires m ε < , which means that this system has finitely many bound states.We leave it to the interested reader to calculate ( )

1).
One last case we would like to discuss in this section is when ( ) ( ) ( ) . The Schrödinger equation in the presence of sinusoidal potential was studied by Alhaidari and Bahlouli in [21].The spinor basis component is written in terms of Jacobi basis.By comparison, we write the J-matrix elements associated with this case as follows: ( ) where 2 2 1 4 µ ν = = . Using (3.2.9) in (1.1), we write the three-term recursion relation as: that can be evaluated exactly at any order n with initial conditions usually taken to be Unfortunately, exact solutions of (3.2.10)   cannot be written in a closed form as the recursion relation cannot be compared to any well-known class of orthogonal polynomials contrary to what we had in the previous examples.In fact, the solutions are referred to new polynomials which have been called "dipole polynomials" and have been found in different physical problems like electron in the dipole field and non-central potential problems [19] [20], see also Equation ( 8) in [32].Moreover, the eigenstates can be evaluated at any order and the energy eigenvalues can be computed numerically high accuracy.As an illustration, we have tabulated the lowest ten energy eigenvalues in Table 4.The spinor upper component for this system is written as , where are solutions to (3.2.10) and ( ) is written in terms of Jacobi basis below: ( ) 1 cos 1 cos cos where . Similarly, we can follow this procedure to find solutions for the pseudo-spin-symmetric coupling for the same sinusoidal potentials.For more solvable potentials in the presence of spin symmetries, we have tabulated more results in Table 5.

General Case
An interesting example we mention here is when , and ( ) which can be used for an electron in graphene moving under the influence of a , and Table 5.Few examples of solvable potentials in the spin-symmetric coupling with the bound states spectrum formula and spinor wavefunction upper component for each [14].One can obtain similar results in the pseudo-spin symmetry.
We will not be able exhaust all solvable potentials in this section, but one can follow the same procedure to obtain different solvable potentials like ( ) 0 e x V x V κ − = , ( ) ( ) , and others.We should point out here that we have used the kinetic balance relation [23] in our calculations which allow analytical solutions of the wave equation in the none simple cases.This relation is based on nonrelativistic approximation which usually gives one energy solution either the "positive energy sector" or the "negative energy sector".The interested reader is ad-vised to refer to [23] and references therein.

Conclusion and Future Recommendations
We have solved the one-dimensional Dirac equation using the Tridiagonal Representation Approach (TRA).This approach, even limited, provides a very easy and handy approach to find analytical solutions to a certain class of solvable potentials for the 1D Dirac Equation.In the presence of symmetry between the potential components in Dirac Equation, the problem can be reduced to solving an effective Schrodinger-like equation which was treated previously using the TRA [15] [18] [19] [20] [21].The solvable potential configurations we obtained have been discussed in details in Section 3. As a potential application of our analytical results, we have mentioned in Section 3 that some of our results can be used directly in graphene to treat electrons subject to electrostatic or magneto static (or both) barriers, a subject of major importance in recent graphene literature.Finally, we would like to express our interest in extending our approach to Dirac Equation in higher dimensions.

1 )
We use the coordinate transformation that satisfies e κ .This form is compatible with the Laguerre weight function in (2.6) and eases the measure transformation.
and 2 1 b β − =, to ensure the tridiagonal representation of the first matrix element in (2reals ρ and σ , to ensure the tridiagonal represen- tation of the last matrix element in (2.8).
behind this transformation is that Equation (3.1.1)is equivalent to Dirac-Weyl equation for an electron in graphene moving under the influence of an external magnetic field acting perpendicular to the plane of the graphene sheet.To see the correspondence, we write down the Dirac-Weyl equation which reads[24] [25] 1 : ).This will reduce Equation (3.1.2) to Equation(3.1.1) 1.5), we find that the Schrödinger potential reads: Our results in this example agree with previous findings[3].One last example we mention in this section is what we call the Hulthén barrier in which the magnetic field takes the following form: constants.Following the same procedure, this will be the situation when the scalar potential is ( ) supersymmetric potential (Schrödinger potential) now reads: refer to the spin-symmetric coupling the situation where we have 0 ∆ = , and the pseudo spin symmetric coupling for Table where ss C and ps C are constants for spin symmetry and pseudo spin symmetry, respectively.I. A. Assi, H. Bahlouli DOI: 10.4236/jamp.2017.5101722083 Journal of Applied Mathematics and Physics

3 . 2 )
by comparison with the three-term recursion relation of Meixner-Pollaczek polynomials (A13), we find that the solutions to Equation (3.3.2) are the normalized Meixner-Pollaczek polynomials.Using the infinite spectrum formula of these polynomials (A13), we obtain the following bound states spectrum for is the Meixner-Pollaczek polynomial of order n in ξ , ( ) ω ξ is its weight func-

Table 1 .
Few interesting coordinate transformations that are very useful in our approach which will be used to obtain different class of solvable potentials.

Table 4 .
The first ten positive and negative energy solutions to Equation(3.23).Here we took