Scattering and Bound State Solutions of the Yukawa Potential within the Dirac Equation

For studying the quantummechanical 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 l-wave Schrödinger Equation for the Eckart potential has been obtained in [11]. The solution of the Schrödinger equation for the modified Morse potential has been studied by Wei and Chen [12]. Rojas and Villalba have found the solutions of the Klein-Gordon equation for one-dimensional Wood-Saxon potential by hypergeometric functions [13]. The exact solutions of scattering state have been studied for the s-wave Schrödinger equation with the Manning-Rosen potential by using standard method [14]. Low momentum scattering states of the Dirac equation have been studied in [15]. Properties of scattering state solutions of the Klein-Gordon equation Coulomb scalar plus vector potential have been studied in [16]. 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 [21] and atomic, molecular, and plasma physics [22]. 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 κ. 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.


Introduction
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][2][3][4][5][6][7][8][9][10]. The analytical scattering state solution of the -wave Schrödinger Equation for the Eckart potential has been obtained in [11]. The solution of the Schrödinger equation for the modified Morse potential has been studied by Wei and Chen [12]. Rojas and Villalba have found the solutions of the Klein-Gordon equation for one-dimensional Wood-Saxon potential by hypergeometric functions [13]. The exact solutions of scattering state have been studied for the -wave Schrödinger equation with the Manning-Rosen potential by using standard method [14]. Low momentum scattering states of the Dirac equation have been studied in [15]. Properties of scattering state solutions of the Klein-Gordon equation Coulomb scalar plus vector potential have been studied in [16].
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 [21] and atomic, molecular, and plasma physics [22]. 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.
where is the relativistic energy of the system and = − ∇ is the three-dimensional momentum operator. and are the usual 4 × 4 Dirac matrices given as where is the 2 × 2 unitary matrix and the three 2 × 2 Pauli matrices are given as where is the orbital angular momentum of the spherical nucleons and the total angular momentum operator and spin-orbit = ( ⋅ + 1) commute with Dirac Hamiltonian. The eigenvalues of spin-orbit coupling operator are = ( + (1/2)) > 0 and = −( + (1/2)) < 0 for unaligned spin = − (1/2) and the aligned spin = + (1/2), respectively. Thus, in the Pauli-Dirac representation, ( , ) and̃( , ) 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 ( − 1) =̃(̃+ 1) and ( + 1) = ( + 1). When scalar potential ( ) is equal to the vector potential ( ), (7b) becomes and with (5), we have These are two equations with equal scalar and vector potential that we have used in this work.

Dirac Equation with the Yukawa Potential
According to (8), we have The Yukawa potential is where is the screening parameter and is the strength of the potential [21,23]. Instead of the centrifugal term in (10), the following approximation has been used in recent years: that is, it has a good accuracy for small values of the potential parameter [24,25]. Substituting (11) and (12) into (10), we obtain the following form of the wave equation: Advances in High Energy Physics 3

Bound State Solutions.
For the bound state, by taking the following variable ( → 1 for → 0 and → 0 for → ∞) 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 [26]: Comparing (17) with the hypergeometric equation of the form [26] (1 − ) 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 lowerspinor component can be obtained as Therefore, 4

Advances in High Energy Physics
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.

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: 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 r, 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 limit → ∞ we have Then The general boundary condition of the scattering state wave function on the "( 1 /2 ) 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 , = 2 ( + 1) + arg Γ (2 ) − arg Γ ( + − √ 2 2 + 2 + 4 + ( + )) − arg Γ ( + + √ 2 2 + 2 + 4 + ( + )) .
For the lower component, with (34), we have Therefore, the scattering phase shifts of two components are equal but in a constant coefficient.

Conclusions
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.