Project supported by the Higher Education Project (Grant No. 698/UN27.11/PN/2015).
The Dirac equation for Eckart potential and trigonometric Manning Rosen potential with exact spin symmetry is obtained using an asymptotic iteration method. The combination of the two potentials is substituted into the Dirac equation, then the variables are separated into radial and angular parts. The Dirac equation is solved by using an asymptotic iteration method that can reduce the second order differential equation into a differential equation with substitution variables of hypergeometry type. The relativistic energy is calculated using Matlab 2011. This study is limited to the case of spin symmetry. With the asymptotic iteration method, the energy spectra of the relativistic equations and equations of orbital quantum number l can be obtained, where both are interrelated between quantum numbers. The energy spectrum is also numerically solved using the Matlab software, where the increase in the radial quantum number n r causes the energy to decrease. The radial part and the angular part of the wave function are defined as hypergeometry functions and visualized with Matlab 2011. The results show that the disturbance of a combination of the Eckart potential and trigonometric Manning Rosen potential can change the radial part and the angular part of the wave function.
In 1928 Dirac investigated the relativistic wave equation covariance of the Schrodinger equation and proposed a matrix
The Dirac equation for the case of exact spin symmetric occurs when the difference in the magnitude betwen the repulsive vector potential and the attractive scalar potential is zero and the sum of the magnitudes of the repulsive vector potential and attractive scalar potential is equal to a given potential. The exact pseudo-spin symmetry occurs when the sum of the magnitude of the repulsive vector potential and the attractive scalar potential is zero and the difference between the vector potential and scalar potential is equal to a given potential, which is central or non-central. [ 4 ]
Solutions of the Dirac equation for some typical potentials under special cases of spin symmetry and pseudospin symmetry have been investigated. For spin symmetry κ ( κ + 1) = l ( l + 1) that gives κ = l = j + 1/2 for κ > 0 and κ = −( l + 1) = −( j + 1/2) for κ < 0, where κ is the eigenvalue of the spin orbit coupling operator, l is orbital quantum number, and j is the total angular momentum quantum number. For the pseudospin symmetry case, κ ( κ − 1) = l̃ ( l̃ + 1) that gives κ = l̃ + 1 = j + 1/2 for κ > 0 and κ = − l̃ = −( j + 1/2) for κ < 0, where l̃ is the quantum number of the pseudospin orbital. In nuclear physics, spin symmetry and pseudospin symmetry concepts have been used to study the aspect of deformed and super deformation nuclei. The concept of spin symmetry has been applied to the spectra of meson and antinucleon. [ 5 ]
Some researchers have investigated the Dirac equation by using a variety of potentials and different methods, such as the spin symmetry in the antinucleon spectrum and tensortype Coulomb potential with spin–orbit number κ in a state of spin symmetry and p-spin symmetry, [ 6 ] bound states of the Dirac equation with position-dependent mass for the Eckart potential, [ 7 ] the exact solution of Klein–Gordon with the Pöschl–Teller double-ring-shaped Coulomb potential, [ 8 ] the exact solution of the Dirac equation for the Coulomb potential plus NAD potential by using the Nikorov–Uvarov method, [ 9 ] the potential Deng–Fan and the Coulomb potential tensor using the asymptotic iteration method (AIM), [ 10 ] the potential Poschl–Teller plus the Manning Rosen radial section with the hypergeometry method, [ 11 ] the solution of Klein–Gordon equation for Hulthen non-central potential in radial part with Romanovski polynomial [ 12 ] and the solution of the Schrodinger equation with the Hulthen plus Manning– Rosen potential, [ 13 ] the Scarf potential with the new tensor coupling potential for spin and pseudospin symmetries using Romanovski polynomials, [ 14 ] for the q-deformed hyperbolic Poschl–Teller potential and the trigonometric Scarf II noncentral potential by using AIM, [ 15 ] eigensolutions of the deformed Woods–Saxon potential via AIM, [ 16 ] approximate solutions of the Klein Gordon equation with an improved Manning Rosen potential in D -dimensions using SUSYQM, [ 17 ] and eigen spectra of the Dirac equation for a deformed Woods–Saxon potential via the similarity transformation. [ 18 ]
In this study, we will solve the Dirac equation for the Eckart potential plus the trigonometric Manning Rosen potential in a state of spin symmetry using an asymptotic iteration method. The asymptotic iteration method is a method of solving the second-order differential equations. [ 20 ]
The rest of this paper is organized as follows. The asymptotic iteration method will be briefly reviewed in Section 2. In Section 3, we review the Eckart potential and trigonometric Manning Rosen potential, give a brief introduction to the Dirac equation with equal scalar and vector potentials, and apply the separation of variables in spherical coordinates. In Section 4, we solve the radial and angular parts of the Dirac equation by using the modified Eckart potential combined with trigonometric Manning Rosen potential and obtain the relativistic energy spectrum and wavefunction via the asymptotic iteration method. In Section 5, we present graphically some wavefunctions of the Dirac equation, present several relativistic energy spectra, and discuss some consequences of the results obtained. Finally, some conclusions are drawn from the present study in Section 6.
This method is used to solve differential equation in the following form:
Eigen values can be obtained using the following equation: [ 19 ]
While eigen functions of Eq. (
where α can be solved by applying the asymptotic iteration k
Rewriting the second-order differential of Eq. (
The general formula for the exact solution of y n ( x ) is given by
C′ is the normalized constant, and 2 F 1 is the hipergeometry function. [ 20 ]
The Dirac equation with scalar potential S ( r ) and vector potential V (
where E is the relativistic energy,
In the Pauli–Dirac representation, let
For spin symmetry, equation (
Substituting Eq. (
In spherical coordinates, the Eckart potential combined with trigonometric Manning Rosen potential is defined as
Substitute Eq. (
then we will have
Separating the variables in Eq. (
The Dirac spin may be written according to the upper f nκ (
where F n r ( r ) is the Dirac spin component, G n r ( r ) is the Dirac pseudospin component,
By substituting Eq. (
Eliminating F n r ( r ) and G n r ( r ) in Eqs. (
κ is the quantum number of the spin orbital, l is the orbital quantum number, and l̃ the quantum number of the pseudospin orbital.
In the case of exact spin symmetry d Δ ( r )/d r =
We assume that Σ ( r ) is the Eckart potensial, which is defined as
and we use
as the centrifugal term. The radial Dirac equation takes the following form:
By substituting variable coth ( αr ) = 1 − 2 z into Eq. (
into Eq. (
where − B s + E s = 4 δ 2 and B s − E s = 4 γ 2 .
By comparing this equation with Eq. (
Combining these results with the quantization condition given by Eq. (
where ɛ n r = A s /4.
When the above expressions are generalized into
the eigenvalues turn into
By substituting Eqs. (
where n r is the radial quantum number ( n r = 0,1,2…), l is orbital quantum number which is obtained from the angular part solution. The radial wavefunction can be obtained by using Eqs. (
From Eq. (
By substituting Eq. (
where z = [1-coth ( αr )]/2, so
where C n r is the radial normalization constant, 2 F 1 is the hypergeometry function, and (2 δ + 1) n r is a Pochamer symbol.
For the angular part in Eq. (
and the simplified equation is
and simplying it, equation (
with l being the orbital quantum number.
From Eq. (
To have this eigen value of the equation, further iterations λ k and s k , where k is the stated iteration, by using Eq. (
When the above expressions are generalized
Energy eigen value can be obtained by using the generalized Δ k , which yields
where l is the orbital quantum number and n l is the angular quantum number.
Angular wavefunction can be obtained by using Eqs. (
From Eq. (
Substituting Eq. (
where z = [1 + i cot ( θ )]/2, so the angular part wavefunction can be obtained as follows:
with C n l being the angular normalization constant.
In this section, we discuss several results obtained in the previous section. From the energy eigen value in Eq. (
By varying parameters corresponding to values of δ and γ , some of the radial wavefunctions are listed in Table
For the angular solution of the wavefunction in Eq. (
In this paper, we study the Dirac equation for particle spin-1/2 in the Eckart potential combined with the trigonometric Manning Rosen potential in condition that the scalar potential equals the vector potential. The radial part of the spinor wavefunction is obtained approximately from Eq. (