Approximate Solutions of the Dirac Equation for the Rosen-Morse Potential in the Presence of the Spin-Orbit and Pseudo-Orbit Centrifugal Terms

In quantum mechanics, it is well known that the exact solutions play fundamental role, this is because, these solutions usually contain all the necessary information about the quantum mechanical model under investigation. In recent years, there has been a renewed interest in obtaining the solutions of the Dirac equations for some typical potentials under special cases of spin symmetry and pseudo-spin symmetry (Arima et al., 1969; Hecht and Adler, 1969). The idea about spin symmetry and pseudo-spin symmetry with the nuclear shell model has been introduced in 1969 by Arima et al. (1969) & Hecht and Adler (1969). This idea has been widely used in explaining a number of phenomena in nuclear physics and related areas. Spin and pseudo-spin symmetric concepts have been used in the studies of certain aspects of deformed and exotic nuclei (Meng & Ring, 1996; Ginocchio, 1997; Ginocchio & Madland, 1998; Alberto et al., 2001; 2002; Lisboa et al., 2004a; 2004b; 2004c; Guo et al., 2005a; 2005b; Guo & Fang, 2006; Ginocchio, 2004; Ginocchio, 2005a; 2005b). Spin symmetry (SS) is relevant to meson with one heavy quark, which is being used to explain the absence of quark spin orbit splitting (spin doublets) observed in heavy-light quark mesons (Page et al., 2001). On the other hand, pseudo-spin symmetry (PSS) concept has been successfully used to explain different phenomena in nuclear structure including deformation, superdeformation, identical bands, exotic nuclei and degeneracies of some shell model orbitals in nuclei (pseudo-spin doublets)(Arima, et al., 1969; Hecht & Adler, 1969; Meng & Ring, 1996; Ginocchio, 1997; Troltenier et al., 1994; Meng, et al., 1999; Stuchbery, 1999; 2002). Within this framework also, Ginocchio deduced that a Dirac Hamiltonian with scalar S(r) and vector V(r) harmonic oscillator potentials when V(r) = S(r) possesses a spin symmetry (SS) as well as a U(3) symmetry, whereas a Dirac Hamiltonian for the case of V(r) + S(r) = 0 or V(r) = −S(r) possesses a pseudo-spin symmetry and a pseudo-U(3) symmetry (Ginocchio, 1997; 2004; 2005a; 2005b). As introduced in nuclear theory, the PSS refers to a quasi-degeneracy of the single-nucleon doublets which can be characterized with the non-relativistic quantum mechanics (n, l, j = l+ 1 2 ) and (n − 1, l+ 2, j = l+ 3 2 ), where n, l and j are the single-nucleon radial, orbital and total angular momentum quantum numbers for a single particle, respectively (Arima et al., 1969; Hecht & Adler, 1969; Ginocchio, 2004; Approximate Solutions of the Dirac Equation for the Rosen-Morse Potential in the Presence of the Spin-Orbit and Pseudo-Orbit Centrifugal Terms

In this study, the Rosen-Morse potential is considered, due to the important applications of in atomic, chemical and molecular Physics as well (Rosen & Morse, 1932). This potential is very useful in describing interatomic interaction of the linear molecules. The Rosen-Morse potential is given as V(r)=−V 1 sech 2 αr + V 2 tanh αr, where V 1 and V 2 are the depth of the potential and α is the range of the potential, respectively. Thus, our aim is to employ the newly improved approximation scheme (or Pekeris-type approximation scheme) in order to obtain the PSS and SS solutions of the Dirac equations for the Rosen-Morse potential with the centrifugal term. This potential has been studied by various researchers in different applications (

Basic Equations for the upper-and lower-components of the Dirac spinors
In the case of spherically symmetric potential, the Dirac equation for fermionic massive spin− 1 2 particles interacting with the arbitrary scalar potential S(r) and the time-component V(r) of a four-vector potential can be expressed as (Greiner, 2000 where E is the relativistic energy of the system, M is the mass of a particle, P = −ih∇ is the momentum operator. α and β are 4 × 4 Dirac matrices, given as where I is the 2 × 2 identity matrix and σ i (i = 1, 2, 3) are the vector Pauli matrices. Following the procedure stated in (Greiner, 2000 ; κ = ±(j + 1 2 ), (4) where F nκ (r) and G nκ (r) are the radial wave functions of the upper and lower spinors components, respectively. Y ℓ jm (θ, φ) and Y ℓ jm are the spherical harmonic functions coupled to the total angular momentum j and its projection m on the z−axis. The orbital and pseudo-orbital angular momentum quantum numbers for SS (ℓ) and PSS (ℓ) refer to the upper (F nκ (r)) and lower (G nκ (r)) spinor components, respectively, for which ℓ(ℓ + 1)=κ(κ + 1) and ℓ(ℓ + 1)=κ(κ − 1). For the relationship between the quantum number κ to the quantum numbers for SS (ℓ) and PSS (ℓ) ( For comprehensive reviews, see Ginocchio (1997) and (2005b). On substituting equation (4) into equation (2), the two-coupled second-order ordinary differential equations for the upper and lower components of the Dirac wave function are obtained as follows: Eliminating F nκ (r) and G nκ (r) from equations (5) and (6), the following two Schrödinger-like differential equations for the upper and lower radial spinors components are obtained, respectively as: where Δ(r)=V(r) − S(r) and Σ(r)=V(r)+S(r) are the difference and the sum of the potentials V(r) and S(r), respectively.

Spin symmetry solutions of the Dirac equation with the Rosen-Morse potential with
arbitrary κ In equation (9), we adopt the choice of Σ(r)=2V(r) → V(r) as earlier illustrated by Alhaidari et al. (2006), which enables us to reduce the resulting solutions into their non-relativistic limits under appropriate transformations, that is, Using the centrifugal term approximation in equation (13) and introducing a new variable of the form z = e −2αr in equation (9), the following equation for the upper component spinor F nκ (r) is obtained as: where The upper component spinor F nκ (z) has to satisfy the boundary conditions, F nκ (z)=0a t z → 0(r → ∞) and F nκ (z)=1atz → 1(r → 0). Then, the function F nk (z) can be written as where and On substituting equation (18) into equation (16) with equations (17), (19) and (20) , the second-order differential equation is obtained as whose solutions are the hypergeometric functions (Gradshteyn & Ryzhik, 2007), its general form can be expressed as in which the first term can be expressed as: where The hypergeometric function f nκ (z) can be reduced to polynomial of degree n, whenever either a or b equals to a negative integer −n. This implies that the hypergeometric function f nκ (z) given by equation (23) can only be finite everywhere unless Using equations (17), (19) and (20) in equation (25), an explicit expression for the energy eigenvalues of the Dirac equation with the Rosen-Morse potential under the spin symmetry condition is obtained as: It is observed that, the spin symmetric limit leads to quadratic energy eigenvalues. Hence, the solution of equation (26)   negative energy eigenvalues exist in the spin limit. Therefore, in the spin limit, only positive energy eigenvalues are chosen for the spin symmetric limit. Using equations (18) to (25), the radial upper component spinor can be obtained as N nκ is the normalization constant which can be determined by the condition that By making use of the equation (23) and the following integral (see formula (7.512.12) in Gradshteyn & Ryzhik (2007)): where (Pochhammer symbol). In order to find the lower component spinor, the recurrence relation of the hypergeometric function (Gradshteyn & Ryzhik, 2007) is used to evaluate equation (10) and this is obtained as

Pseudopin symmetry solutions of the Dirac equation with the Rosen-Morse potential with arbitrary κ
In the case of pseudospin symmetry, that is, the difference as in equation (11).
dΣ(r) dr = 0o r Σ(r)=Constant = C ps , and taking into consideration the choice of Δ(r)=2V(r) → V(r) as earlier illustrated by Alhaidari et al. (2006). Then, With the pseudo-centrifugal approximation in equation (13) and substituting z = −e −2αr , then, the following equation for the lower component spinor G nκ (r) is obtained as: where With boundary conditions in the previous subsection, then, writing the function G nκ (z) as where and On substituting equation (35) into equation (33) and using equations (34), (36) and (37), equation (33) becomes whose solutions are the hypergeometric functions (Gradshteyn & Ryzhik, 2007), its general form can be expressed as in which the first term can be expressed as: Also, in the similar fashion as obtained in the case of the spin symmetry condition, an explicit expression for the energy eigenvalues of the Dirac equation with the Rosen-Morse potential under the pseudospin symmetry is obtained as: It is observed that, the pseudospin symmetric limit leads to quadratic energy eigenvalues. Therefore, the solution of equation (42) consists of positive and negative energy eigenvalues for each n and κ. Since, it has been shown that there are only negative energy eigenvalues and no bound positive energy eigenvalues exist in the pseudospin limit (Ginocchio, 2005). Therefore, in the pseudospin limit, only negative energy eigenvalues are chosen. The radial lower component spinor can be obtained by considering equations (35)-(41) as N nκ is the normalization constant which can be determined by the condition that ∞ 0 | G nκ (r) | 2 dr = 1 and by making use of the equations (23) and (28), we have where A nκ = 3 F 2 (2β + k, −n, n + 2(1 + β + q); k + 2(β + q + 3 2 );2β + 1; 1) and (x) a = Γ(x+a)  Similarly, by using equation (12) F nκ (r) can also be obtained as It is pertinent to note that, the negative energy solution for the pseudospin symmetry can be obtained directly from the positive energy solution of the spin symmetry using the parameter mapping (Berkdemir & Cheng, 2009;Ikhdair, 2010):

Remarks
In this work, solutions of some special cases are studied:

s-wave solutions:
Our results include any arbitrary κ values, therefore, there is need to investigate if our results will give similar results for s-wave for the spin symmetry when κ = −1orℓ = 0 and for the pseudospin when κ = 1orℓ = 0. For the SS, κ = −1 (or ℓ = 0) in equation (26) gives where For the PSS, κ = 1 (or ℓ = 0) in equation (42) gives where The

Solutions for the standard Eckart potential:
By setting V 1 = −V 1 and V 2 = −V 2 in equation (1), we have the standard Eckart potential. The energy eigenvalues for the SS and the PSS are given, respectively as: and where q 2 and q 2 are obtained, respectively as: The corresponding upper and lower component spinors for the SS and the PSS can easily be obtained from equations (27), (31), (43) and (45).

Solutions of the PT-Symmetric Rosen-Morse potential:
The choice of V 2 = iV 2 in equation (1) For a given potential V(r),ifV(−r)=V * (r) (or V(η − r)=V * (r)) exists, then, the potential V(r) is said to be PT-Symmetric. Here, P denotes the parity operator (space reflection, P : r → −r,orr → η − r) and T denotes the time reversal operator (T : i →−i). For the case of the SS and the PSS solutions of this PT-Symmetric version of the Rosen-Morse potential, the energy eigenvalue equations are: respectively. q and q have their usual values as in equations (19) and (36), the corresponding upper and lower component spinors for the SS and the PSS can be obtained directly from equations (27), (31), (43) and (45).

The relativistic bound state solutions of the Rosen-Morse potential with the centrifugal term
The Klein-Gordon and the Dirac equations describe relativistic particles with zero or integer and 1/2 integral spins, respectively ( Landau & Lifshift 1999;Merzbacher, 1998;Greiner, 2000;Alhaidari et al., 2006;Dong, 2007). However, the exact solutions are only possible for a few simple systems such as the hydrogen atom, the harmonic oscillator, Kratzer potential and pseudoharmonic potential.  In the recent years, some researchers have used the Pekeris-type approximation scheme for the centrifugal term to solve the relativistic equations to obtain the ℓ or κ− wave energy equations and the associated wave functions of some potentials. These include: Morse potential (Bayrak et al., 2010), hyperbolical potential (Wei & Liu, 2008), Manning-Rosen potential (Wei & Dong, 2010), Deng-Fan oscillator (Dong, 2011).
In the context of the standard function analysis approach, the approximate bound state solutions of the arbitrary ℓ-state Klein-Gordon and κ-state Dirac equations for the equally mixed Rosen-Morse potential will be obtained by introducing a newly improved approximation scheme to the centrifugal term.

Approximate bound state solutions of the Klein-Gordon equation for the Rosen-Morse
potential for ℓ = 0 The time-independent Klein-Gordon equation with the scalar S(r) and vector V(r) potentials is given as (Landau & Lifshift, 1999;Merzbacher, 1998;Greiner, 2000;Alhaidari et al., 2006): where M,h and c are the rest mass of the spin-0 particle, Planck's constant and velocity of the light, respectively. For spherical symmetrical scalar and vector potentials, putting where Y ℓ,m (θ, φ) is the spherical harmonic function, we obtain the radial Klein-Gordon equation as U n,ℓ (r)+ 1 We are considering the case when the scalar and vector potentials are equal (that is, S(r)= V(r)), coupled with the resulting simplification in the solution of the relativistic problems as discussed by Alhaidari et al., 2006, we have U n,ℓ (r)+ 1 This equation cannot be solved analytically for the Rosen-Morse potential with ℓ = 0, unless, we introduce the approximation scheme (earlier discussed in this chapter) to the centrifugal term. With this approximation scheme, and the potential in (1) together with the transformation z = −e −2αr in equation (75), we have z 2 U n,ℓ (z)+zU n,ℓ (z) (1−z) 2 U n,ℓ (z)=0, (76) In the similar manner, the energy equation of the arbitrary ℓ-state Klein-Gordon equation with equal scalar and vector potentials of the Rosen-Morse potential is obtained as follows: where The associated wave function can be expressed as where and N n,ℓ is the normalization constant which can easily be determined in the usual manner.

Approximate bound state solutions of the Dirac equation for the Rosen-Morse potential
for any κ In this subsection, we consider equations (2), (3) and (4), and on re-writing equations (5) and (6) for the case of equal scalar and vector, i. e. V(r)=S(r), we have the following two coupled differential equations: With the substitution of equation (82) into equation (83) and taking into consideration the suggestion of Alhaidari et al., (2006), a Schrödinger-like equation for the arbitrary spin-orbit coupling quantum number κ is obtained as

Conclusions
The approximate analytical solutions of the Dirac equation with the Rosen-Morse potential with arbitrary κ under the pseudospin and spin symmetry conditions have been studied, the standard function analysis approach has been adopted. The Pekeris-type approximation 462 Theoretical Concepts of Quantum Mechanics www.intechopen.com scheme (a newly improved approximation scheme) has been used for the centrifugal (or pseudo centrifugal) term in order to solve for any values of κ. Under the PSS and SS conditions, the energy equations, the upper-and the lower-component spinors for the Rosen-Morse potential for any κ have been obtained. The solutions of some special cases are also considered and the energy equations with their associated spinors for the PSS and SS are obtained, these include: (i) the s-state solution, (ii) the standard Eckart potential, (iii) the PT-Symmetric Rosen-Morse potential, (iv) the reflectionless-type potential, (v) the non-relativistic limit. Also, in the context of the standard function analysis approach, the approximate bound state solutions of the arbitrary ℓ-state Klein-Gordon and κ-state Dirac equations for the equally mixed Rosen-Morse potential are obtained by introducing a newly improved approximation scheme to the centrifugal term. The approximate analytical solutions with the Dirac-Rosen-Morse potential for any κ or ℓ have been obtained. The upper-and lowercomponent spinors have been expressed in terms of the hypergeometric functions (or Jacobi polynomials). The approximate analytical solutions obtained in this study are the same with other results available in the literature.

Acknowledgments
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