Research Article | Open Access
Scattering and Bound State Solutions of the Yukawa Potential within the Dirac Equation
In the presence of spin symmetry case, we obtain bound and scattering states solutions of the Dirac equation for the equal scalar and vector Yukawa potentials for any spin-orbit quantum number . The approximate analytical solutions are presented for the bound and scattering states and scattering phase shifts.
For studying the quantum mechanical systems, it is necessary to pay attention to two points. These two points are to study bound states to take the necessary information about the system under consideration and also solving scattering states for a system under the effect of a potential. Solving both of these problems gives us complete information about a quantum mechanical system under consideration.
The solutions of scattering and/or bound state problem have been investigated for the well-known potentials by applying different methods [1–10]. The analytical scattering state solution of the-wave Schrödinger Equation for the Eckart potential has been obtained in . The solution of the Schrödinger equation for the modified Morse potential has been studied by Wei and Chen . Rojas and Villalba have found the solutions of the Klein-Gordon equation for one-dimensional Wood-Saxon potential by hypergeometric functions . The exact solutions of scattering state have been studied for the-wave Schrödinger equation with the Manning-Rosen potential by using standard method . Low momentum scattering states of the Dirac equation have been studied in . Properties of scattering state solutions of the Klein-Gordon equation Coulomb scalar plus vector potential have been studied in .
In this work, we have studied bound state and scattering state of the Dirac equation with the Yukawa potential. The Dirac equation describes the particle dynamics in the relativistic quantum mechanics [17, 18]. Thus, solving the Dirac equation is very significant in describing the nuclear shell structure [19, 20]. Also the Yukawa potential has many applications in different areas of physics like high-energy physics  and atomic, molecular, and plasma physics .
This paper is organized as follows. In Section 2, we briefly introduce the Dirac equation with scalar and vector potential with any spin-orbit quantum number . The bound and scattering states of the Yukawa potential within the Dirac equation are presented in Section 3. Finally, our concluding remarks are given in Section 4.
2. Dirac Equation with Scalar and Vector Potential
The Dirac equation with scalar and vector potential ( and ) is  where is the relativistic energy of the system and is the three-dimensional momentum operator. and are the usual Dirac matrices given as where is theunitary matrix and the three Pauli matrices are given as where is the orbital angular momentum of the spherical nucleons and the total angular momentum operator and spin-orbit commute with Dirac Hamiltonian. The eigenvalues of spin-orbit coupling operator are and for unaligned spin and the aligned spin , respectively. Thus, in the Pauli-Dirac representation, where is the upper component and is the lower component of the Dirac spinors. and are spin and pseudospin spherical harmonics and is the projection of the angular momentum on the-axis. Substituting (4) into (1), one obtains two coupled differential equations for the upper and the lower radial wave functions as follows: where Solving (5) leads to a second-order Schrödinger-like differential equation for the upper and the lower components of the Dirac wavefunctions as follows: where and.
3. Dirac Equation with the Yukawa Potential
3.1. Bound State Solutions
For the bound state, by taking the following variable (forandfor) one can obtain (13) in the following form: Taking the form of the wave function and substituting this equation into (15), one gets a hypergeometric-type equation as follows : where Comparing (17) with the hypergeometric equation of the form  we can obtain the wavefunction as the hypergeometric function: where Then, with (14) and (16), we have the upper-spinor component of wavefunction for the Dirac equation with the Yukawa potential as follows: where is the normalization constant. With (9), the lower-spinor component can be obtained as Therefore, Finally, the spinor wave function under the condition of equal scalar and vector potentials with (4), (7b), (22), and (24) becomes By considering the finiteness of the solution, the quantum condition is given by It is the energy eigenvalue equation of bound state for the upper component of the Dirac equation with the Yukawa potential.
3.2. Scattering State
Now, we turn to solve (10) for scattering state. For this purpose, we use a new variable as follows: and obtain Considering the boundary condition of the scattering state, we take the following trial wavefunction: And inserting this equation into (28), we obtain the following equation: where This equation is a hypergeometric type. Thus, (29) is as follows: ( is the normalization constant) that is, the upper component of wavefunction. The lower component is And therefore, total wavefunction of scattering state with (4), (7b), (32), and (33) is where According to (26), we obtain the following form of energy eigenvalue equation for scattering states: Now, by finding the asymptotic form of (32) for large , we try to obtain the scattering phase shifts. For this purpose, we use the following property of the hypergeometric function: Therefore, we obtain By using this definition the upper component of wavefunction is as follows: Now, with (38) and (39) in the limitwe have Then The general boundary condition of the scattering state wave function on the “ scale” is Therefore, we have Thus, we obtain the scattering phase shifts as follows: Then we have the scattering phase shifts of the upper component of wavefunction as For the lower component, with (34), we have Then we have where Thus with (43), (44), (45), and (49), we have the following: And finally we have Therefore, the scattering phase shifts of two components are equal but in a constant coefficient.
We have studied the Dirac equation with the Yukawa potential and have obtained bound and scattering states of this problem. The energy eigenvalues, eigenstates, and scattering phase shifts have been presented. The numerical results of the energy eigenvalues of this work have been compared with the ones obtained in the literature in Table 1.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors thank the anonymous referees for the valuable comments and suggestions.
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