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The solutions of the Alhaidari formalism of the Dirac equation for the gravitational plus exponential potential have been presented using the parametric Nikiforov-Uvarov method. The energy eigenvalues and the corresponding unnormalized eigenfunctions are obtained in terms of Laguerre polynomials.

The bound state solutions of the Dirac equation are only possible for some potentials of physical interest [1-5]. These solutions could be exact or approximate and they nornally contain all the necessary information for the quantum system. Quite recently, several authors have tried to solve the problem of obtaining exact or approximate solutions of the Dirac equation for a number of special potentials using different methods [6-20]. Some of these potentials are known to play very important roles in many fields of Physics such as Molecular Physics, Solid State and Chemical Physics [

The purpose of the present work is to present the solution of the Alhaidari formalism of the Dirac equation [

where is the displacement, is the momentum, is the mass, is gravitational acceleration and δ is an adjustable parameter. The GEP could be used to calculate the energy of a body falling under gravity from quantum mechanical point of view. Berberan-Santos et al. [

The Dirac equation for the lower and upper spinor components can be written as [

where is the rest mass, is the relativistic energy, and is the vector potential.

where and is a real parameter. The “” designate the upper and lower components respectively.

The Nikiforov-Uvarov (NU) method is based on the solutions of a generalized second-order linear differential equation with special orthogonal functions [

can be solved by this method. This can be done by transforming Equation (2) into an equation of hypergeometric type with appropriate coordinate transformation to get

To find the exact solution to Equation (3), we write as

Substitution of Equation (6) into Equation (5) yields Equation (7) of hypergeometric type as

In Equation (6), the wave function is defined as the logarithmic derivative [

with being at most first order polynomials. Also, the hypergeometric-type functions in Equation (7) for a fixed integer is given by the Rodrigue relation as

where is the normalization constant and the weight function must satisfy the condition

with

In order to accomplish the condition imposed on the weight function it is necessary that the polynomial be equal to zero at some point of an interval and its derivative at this interval at will be negative [

The function and the parameter required for the NU method are then defined as [

The values in Equation (13) are possible to evaluate if the expression under the square-root be square of polynomials. This is possible if and only if its discriminant is zero. Therefore, the new eigenvalue equation becomes [

A comparison between Equations (14) and (15) yields the energy eigenvalues.

Secondly, the parametric generalization of the NU method is expressed by the generalized hypergeometric-type equation [

Equation (16) is solved by comparing it with Equation (5) and the following polynomials are obtained:

Now, substituting Equation (17) into Equation (13) gives

where

The resulting value of in Equation (18) is obtained from the condition that the function under the square-root should be square of a polynomial and we get

where

The new for becomes

value becomes

Using Equation (11), we obtain

The physical condition for the bound state solution is and thus

With the aid of Equations (12) and (13), we obtain the energy equation as

The weight function is obtained from Equation (10) as

And together with Equation (9), we have

where

are the Jacobi polynomials. The second part of the wave function is obtained from Equation (6) as

where

Thus the total wave function becomes

where N_{n} is the normalization constant.

The potential in Equation (1) can be written as

where. We can also write Equation (33) as

On arranging Equation (34) we get our working potential as

The potential of Equation (35) can be used to solve various quantum mechanical equations including the Schrodinger equation (SE), Klein-Gordon equation (KG) and Dirac equation using the NU method for their exact solutions. Writing Equation (32) with the GEP we get

Ignoring all terms of the form with in Equation (36) as these will not affect the physics of the calculations, we write Equation (36) as

where we have used and transformation in Equation (36).

Comparing Equation (37) with Equation (16) yields the following parameters

Other coefficients are determined as

From Equation (16)

From Equation (18)

From Equation (22)

The negative derivative of Equation (42) then becomes

The new for the NU method is chosen as

For

Now using Equations (24), (38) and (39) we obtain the energy spectrum of the GEP as

where

The weight function is obtained from Equation (25) and the parameters of Equation (39) as

and using Equation (26) we get the wavefunction χ_{n}(s) as

where and is the Laguerre polynomial. From Equation (28) the wave function is

The unnormalized wave function is then obtained from Equation (30) as

where is the normalization constant.

In addition, the corresponding lower-spinor wave function is

In summary, we have obtained the energy eigenvalues and the corresponding un-normalized wavefunction using the parametric NU method for the Dirac equation with the gravitational plus exponential potential.

The authors wish to acknowledge Dr. A. N. Ikot of the Department of Physics, University of Uyo in Nigeria for some useful discussions during the preparation of this paper.