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An Algebraic Method for the Analytical Solutions of the Klein-Gordon equation for any angular momentum for some diatomic potentials

Analytical solutions of the Klein-Gordon equation are obtained by reducing the radial part of the wave equation to a standard form of a second order differential equation. Differential equations of this standard form are solvable in terms of hypergeometric functions and we give an algebraic formulation for the bound state wave functions and for the energy eigenvalues. This formulation is applied for the solutions of the Klein-Gordon equation with some diatomic potentials.


Introduction
Dirac equation and Klein-Gordon ( K-G) equations are used for the description of particle dynamics in relativistic quantum mechanics. Therefore, the search for the solutions of these equations has been an important research area and several methods were used to obtain solutions of these equations with various physical potentials. These methods include the Asymptotic Iteration method [1][2][3][4], the Nikiforov-Uvarov method [5][6][7], Supersymmetric method [8][9][10][11][12], Variational method [13,14], the Large-N Expansion method [15], Exact Quantization Rule [16][17][18]. The vector and scalar potential couplings are introduced into the relativistic wave equations by using the four-vector linear momentum and the mass terms which are present in the free form of these equations. Recently, there has been an interest in finding solutions of these equations with mixtures of scalar and vector potentials [4,19]. This is important for the studies of pseudospin symmetry in nuclear physics [20]. Unfortunately, these wave equations with some important potentials can be solved only for states with zero angular momentum [21][22][23]. In this work we study anyl state solutions for these potentials using appropriate approximations [24][25][26][27] for the centrifugal term .
The organization of the paper is as follows. In the next section we give a brief outline of our scheme. In section 3 we give several application of the scheme for the solution of the Klein-Gordon equation. We obtain the wave functions and write down equation for the energy eigenvalues for each application. Finally we summarize our conclusions in section 4.

A Brief Review of the Method
For many problems of quantum mechanics that can be solved analytically the radial wave equation reduces to the following second order differential equation   p . In Ref. 28 the details of these calculations are given and it is shown that in order to have bound state solutions the following set of conditions must be satisfied Using these equations for 1 q and 1 p one can rewrite Eq.(3b) as In addition one gets the following equation which gives the energy eigenvalues A similar analysis can be done for the 0 3 = c case. In this case Eq. (1) takes the following form (11) and its asymptotic form suggests the following factorization When we insert this into Eq. (11) we get Here ) ( 1 z A is a second order polynomial in the variable z and its coefficients are given in reference 28. We do not write these coefficients here since they will take large space. The condition on the physical bound states requires the following relations And equation (14) takes the following form (17) where n is a positive integer. From the conditions on the parameters it also follows that which is an equation for the energy eigenvalues. Eq. (17) In the following sections we give several applications of these results.

Applications
The time independent Klein-Gordon equation with a vector potential V and a scalar potential S can be written as ( When the potentials are spherically symmetric, the wave functions are written as ) , for the radial part of the wave functions.
We start with the potentials which lead to a radial equation given by Eq. (11). That is applications

Case 1: Coulomb Potential
Let us take the vector potential as and assume that the scalar potential is zero. This can be considered as the potential for the pionic atom. This allows us to write Eq. (21) as Comparing this equation with Eq. (1) we get Thus we must use Eqs. (15) and (16) Now we can determine the energy eigenvalues using Eq. (18) which gives Replacing the parameters and solving this equation for E we obtain We can choose the negative sign for the bound states. These are the energy levels for a pionic atom. In Ref. (33) K-G equation with Coulomb Potential is solved with a different method. The energy spectrum given by Eq. (24) agrees exactly with their result. It is also solved with the same potential in D dimensions in Ref. (54). Our results agree with this reference for D=3. The wave functions corresponding to Eq. (24) are given by Eq. (19) and can be written as where N is a constant. This wave function agrees with the wave function of reference (33) . If we take a Coulomb-like scalar potential S we can find the analytical solutions using the same method. The Mie potential is given by [35][36][37][38][39][40] ] 2 ) ( 2 Assuming that the vector and scalar potentials are equal in Eq. (21) which can be written as Comparing this with Eq. (11) we can write the parameters of this equation as follows Inserting these parameters into Eqs. (15) and (16) The corresponding wave functions are obtained from Eq. (19) as Using Eqs. (11,15,16) we find The wave functions are written using Eq.(19) as In Ref. (34) Assuming that the scalar potential is equal to vector potential, K-G equation given by Eq. (20) is separable and writing the wave function as for the radial wave function. Here λ is a constant which can be written in terms of the angular quantum number l . From Eq. ( 11) we find . Thus we must use Eqs. (15) and (16) for the calculation of respectively and also Eq. (14) for calculating k. We find The corresponding wave functions are This potential is defined as [21,[30][31][32][33] where 0 V is the depth of the potential, δ is the screening parameter and 0 ≠ q is the deformation parameter. For this potential we first write equation (21) as We first consider the zero angular momentum ( It is more convenient to use a new variable s which is defined as Eq. (40) has the form given in Eq.(1). Therefore we can write the parameters as We use Eqs . (5) and (6) for the calculation of 1 q and 1 p respectively and obtain , . Inserting these into Eq.(10) we find the following equation for the energy eigenvalues The corresponding wave functions are given by Eq. (9) as where H is given by When we consider an arbitrary angular momentum l we have to add the angular momentum term to Eq. (40). In order to find analytical solutions we will use the approximation [24][25][26][27] 2 . Therefore, we have to replace the parameters i Λ with i Γ in the equations for the energy levels and wave functions. When we compare our result with Ref. (31) where the K-G equation is solved in D-dimensions. We find that the wave functions are the same. For the energy eigenvalues they use a different set of parameters.

Case 6: Deformed Woods-Saxon Potential
This potential is given by [41][42][43][44][45][46] )] exp( 1 [ We compare this equation with Eq. (1) and obtain where H is defined by Eq. (7). For the energy eigenvalues we use Eq. ( 10) and write that equation as where we have defined X as The solution of this quadratic equation gives . Replacing the parameters we get the following equation for the energy eigenvalues The corresponding wave functions are given by Eq. (9) and can be expressed as To study the K-G equation for anyl state the centrifugal term must be added to Eq. (45). In order to find analytical solutions we use the approximate expansion [47][48][49][50]    The Generalized Pöschl-Teller potential [53][54][55][56][57][58][59] is defined as We will take this as the vector potential and assume that the scalar potential is equal to vector potential. First we define u(r)=R(r)r and insert this into Eq.
Here it is assumed that the scalar potential is equal to vector potential and angular momentum is zero. Comparing this with Eq. (1) we get To add the centrifugal term we use the approximation [59,60] ) ( sinh / / 1 in Eq. (53) for the energy levels and wave functions. We can compare these results with Ref. (57) . They have the same potential and they also use an approximation for the centrifugal term. There is an additional constant term in their approximation of the centrifugal term. When that constant is taken as zero, their results are also given by our Eq. (53). That is, our results for this application are consistent with Ref. (57).