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

Abstract

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 𝜅. The approximate analytical solutions are presented for the bound and scattering states and scattering phase shifts.

1. 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 [1–10]. The analytical scattering state solution of the𝑙-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 the𝑠-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.

2. Dirac Equation with Scalar and Vector Potential

The Dirac equation with scalar and vector potential (𝑆(𝑟) and 𝑉(𝑟)) is [ℏ=𝑐=1] 𝛼⋅𝑝+𝛽(𝑀+𝑆(𝑟))𝜓(𝑟)=(𝐸−𝑉(𝑟))𝜓(𝑟), (1) where𝐸 is the relativistic energy of the system and 𝑃=−𝑖∇ is the three-dimensional momentum operator. 𝛼 and 𝛽 are the usual 4×4 Dirac matrices given as 𝛼=0110,𝛽=𝐼00−𝐼,𝑖=1,2,3, (2) where𝐼 is the2×2unitary matrix and the three 2×2 Pauli matrices 𝜎_𝑖 are given as 𝜎₁=0110,𝜎₂=0−𝑖−𝑖0,𝜎₃=100−1, (3) where 𝐿 is the orbital angular momentum of the spherical nucleons and the total angular momentum operator𝐽 and spin-orbit 𝐾=(𝜎⋅𝐿+1) commute with Dirac Hamiltonian. The eigenvalues of spin-orbit coupling operator are 𝑘=(𝑗+(1/2))>0 and 𝑘=−(𝑗+(1/2))<0 for unaligned spin 𝑗=𝑙−(1/2) and the aligned spin 𝑗=𝑙+(1/2), respectively. Thus, in the Pauli-Dirac representation, 𝜓_𝑛𝑘(𝑟)=𝑓_𝑛𝑘(𝑟)𝑔_𝑛𝑘(𝑟)=𝐹_𝑛𝑘(𝑟)𝑟𝑌^𝑙_𝑗𝑚𝑖𝐺𝜃,𝜑_𝑛𝑘(𝑟)𝑟𝑌𝑙_𝑗𝑚̃𝜃,𝜑, (4) where 𝑓_𝑛𝑘(𝑟) is the upper component and 𝑔_𝑛𝑘(𝑟) is the lower component of the Dirac spinors. 𝑌^𝑙_𝑗𝑚(𝜃,𝜑) and 𝑌𝑙_𝑗𝑚̃(𝜃,𝜑) are spin and pseudospin spherical harmonics and 𝑚 is the projection of the angular momentum on the𝑧-axis. Substituting (4) into (1), one obtains two coupled differential equations for the upper and the lower radial wave functions as follows: 𝑑+𝑘𝑑𝑟𝑟𝐹_𝑛𝑘(𝑟)=𝑀+𝐸_𝑛𝑘−Δ(𝑟)𝐺_𝑛𝑘(𝑟),𝑑−𝑘𝑑𝑟𝑟𝐺_𝑛𝑘(𝑟)=𝑀−𝐸_𝑛𝑘+Σ(𝑟)𝐹_𝑛𝑘(𝑟), (5) where Δ(𝑟)(𝑟)(𝑟),Σ(𝑟)(𝑟)(𝑟).=𝑉−𝑆=𝑉+𝑆 (6) Solving (5) leads to a second-order Schrödinger-like differential equation for the upper and the lower components of the Dirac wavefunctions as follows: 𝑑²𝑑𝑟²−𝑘(𝑘−1)𝑟²𝐺_𝑛𝑘(𝑟)+−𝑀+𝐸_𝑛𝑘−Δ(𝑟)𝑀−𝐸_𝑛𝑘(𝑟)−((𝑟))+Σ𝑑Σ/𝑑𝑟)(𝑑/𝑑𝑟−𝑘/𝑟𝑀−𝐸_𝑛𝑘(𝑟)𝐺+Σ^𝑛𝑘(𝑟)𝑑=0,₂𝑑𝑟²−𝑘(𝑘+1)𝑟²𝐹_𝑛𝑘(𝑟)+−𝑀+𝐸_𝑛𝑘−Δ(𝑟)𝑀−𝐸_𝑛𝑘(𝑟)+((𝑟))+Σ𝑑Δ/𝑑𝑟)(𝑑/𝑑𝑟+𝑘/𝑟𝑀+𝐸_𝑛𝑘(𝑟)𝐹−Δ^𝑛𝑘(𝑟)=0, (7a)(7b)where 𝑘(𝑘−1)=𝑙̃(𝑙̃+1) and𝑘(𝑘+1)=𝑙(𝑙+1).

When scalar potential𝑆(𝑟)is equal to the vector potential 𝑉(𝑟), (7b) becomes 𝑑²𝑑𝑟²−𝑘(𝑘+1)𝑟²−2𝐸_𝑛𝑘+𝑀𝑉(𝑟)𝐹_𝑛𝑘(𝑟)=𝑀²−𝐸²_𝑛𝑘𝐹_𝑛𝑘(𝑟) (8) and with (5), we have 𝐺_𝑛𝑘(𝑟)=1𝑀+𝐸_𝑛𝑘𝑑+𝑘𝑑𝑟𝑟𝐹_𝑛𝑘(𝑟). (9) These are two equations with equal scalar and vector potential that we have used in this work.

3. Dirac Equation with the Yukawa Potential

According to (8), we have 𝑑²𝑑𝑟²−𝑘(𝑘+1)𝑟²−2𝐸_𝑛𝑘+𝑀𝑉(𝑟)+𝐸²_𝑛𝑘−𝑀²𝐹_𝑛𝑘(𝑟)=0. (10) The Yukawa potential is 𝑉(𝑟)𝐴=−𝑟𝑒^−𝛼𝑟, (11) where 𝛼 is the screening parameter and𝐴 is the strength of the potential [21, 23].

Instead of the centrifugal term in (10), the following approximation has been used in recent years: 1𝑟²≈4𝛼²𝑒^−2𝛼𝑟1−𝑒^{−2𝛼𝑟2}; (12) 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: 𝑑²𝑑𝑟²−4𝛼²𝑘()𝑒𝑘+1^−2𝛼𝑟1−𝑒^{−2𝛼𝑟2}+4𝐴𝛼𝐸_𝑛𝑘×𝑒+𝑀_−2𝛼𝑟1−𝑒^−2𝛼𝑟+𝐸²_𝑛𝑘−𝑀²𝐹_𝑛𝑘(𝑟)=0. (13)

3.1. Bound State Solutions

For the bound state, by taking the following variable (𝑧→1for𝑟→0and𝑧→0for𝑟→∞) 𝑧=𝑒^−2𝛼𝑟 (14) one can obtain (13) in the following form: 𝑧()𝑑1−𝑧²𝑑𝑧²+()𝑑1−𝑧+1𝑑𝑧𝑧()×()𝐴1−𝑧−𝑘𝑘+1𝑧+𝛼𝐸_𝑛𝑘+𝑀𝑧−𝑧²+𝐸²_𝑛𝑘−𝑀²4𝛼²(1−𝑧)²𝐹_𝑛𝑘(𝑧)=0. (15) Taking the form of the wave function 𝐹_𝑛𝑘(𝑧)=𝑧^𝜇(1−𝑧)^𝜈𝑓_𝑛𝑘(𝑧) (16) and substituting this equation into (15), one gets a hypergeometric-type equation as follows [26]: 𝑧(1−𝑧)𝑓^′′_𝑛𝑘(𝑧)+2𝜇+1−2𝜇+2𝜈+1𝑧𝑓^′_𝑛𝑘(𝑧)+𝜇()𝜇−11−𝑧𝑧()𝑧−2𝜇𝜈+𝜈𝜈−1()1−𝑧+𝜇1−𝑧𝑧()1−𝜈−𝑘𝑘+1()+𝐴1−𝑧𝛼𝐸_𝑛𝑘+𝐸+𝑀_2𝑛𝑘−𝑀²4𝛼²(1−𝑧)𝑧𝑓_𝑛𝑘(𝑧)=0, (17) where 𝜇=𝑖𝐸²_𝑛𝑘−𝑀²4𝛼²,𝜈(𝜈−1)=𝑘(𝑘+1)(𝜈>0). (18) Comparing (17) with the hypergeometric equation of the form [26] 𝑧(1−𝑧)𝑓^′′_𝑛𝑘(𝑧)+[𝑐−(𝑎+𝑏+1)𝑧]𝑓^′_𝑛𝑘(𝑧)−𝑎𝑏𝑓_𝑛𝑘(𝑧)=0 (19) we can obtain the wavefunction as the hypergeometric function: 𝑓_𝑛𝑘(𝑧)=₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;𝑧, (20) where 𝑎_𝑛𝑘=𝜇+𝜈+𝐴𝛼𝐸_𝑛𝑘−𝐸+𝑀^2𝑛𝑘−𝑀²4𝛼²,𝑏_𝑛𝑘=𝜇+𝜈−𝐴𝛼𝐸_𝑛𝑘−𝐸+𝑀^2𝑛𝑘−𝑀²4𝛼²,𝑐_𝑛𝑘=2𝜇+1. (21) Then, with (14) and (16), we have the upper-spinor component of wavefunction for the Dirac equation with the Yukawa potential as follows: 𝐹_𝑛𝑘(𝑟)=𝐶_𝑛𝑘𝑒^{−2𝛼𝜇𝑟}1−𝑒^{−2𝛼𝑟𝜈}×₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;𝑒^−2𝛼𝑟, (22) where 𝐶_𝑛𝑘 is the normalization constant. With (9), the lower-spinor component can be obtained as 𝐺_𝑛𝑘(𝑟)=𝐶_𝑛𝑘𝑀+𝐸_𝑛𝑘𝑑+𝑘𝑑𝑟𝑟𝑒^{−2𝛼𝜇𝑟}1−𝑒^{−2𝛼𝑟𝜈}×₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;𝑒^−2𝛼𝑟. (23) Therefore, 𝐺_𝑛𝑘(𝑟)=𝐶_𝑛𝑘𝑀+𝐸_𝑛𝑘𝑘𝑒^{−2𝛼𝜇𝑟}𝑟1−𝑒^{−2𝛼𝑟𝜈}×₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;𝑒^−2𝛼𝑟−2𝛼𝜇𝑒^{−2𝛼𝜇𝑟}1−𝑒^{−2𝛼𝑟𝜈}×₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;𝑒^−2𝛼𝑟+2𝛼𝜈𝑒^{−2𝛼(𝜇+1)𝑟}1−𝑒^{−2𝛼𝑟𝜈−1}×₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;𝑒^−2𝛼𝑟+𝑎_𝑛𝑘𝑏_𝑛𝑘𝑐_𝑛𝑘𝑒^{−2𝛼𝜇𝑟}1−𝑒^{−2𝛼𝑟𝜈}×₂𝐹₁𝑎_𝑛𝑘+1,𝑏_𝑛𝑘+1,𝑐_𝑛𝑘+1;𝑒^−2𝛼𝑟. (24) Finally, the spinor wave function under the condition of equal scalar and vector potentials with (4), (7b), (22), and (24) becomes Ψ(𝑟)=𝐹_𝑛𝑘(𝑟)𝑟𝑌^𝑙_𝑗𝑚𝑖𝐺𝜃,𝜑_𝑛𝑘(𝑟)𝑟𝑌_𝑗𝑙_𝑚̃=𝐶𝜃,𝜑_𝑛𝑘𝑟1−𝑖𝑀+𝐸_𝑛𝑘(𝑟)̂ℜ(𝑟)𝑒𝜎⋅^{−2𝛼𝜇𝑟}1−𝑒^{−2𝛼𝑟𝜈}×𝑌^𝑙_𝑗𝑚𝜃,𝜑₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;𝑒^−2𝛼𝑟,ℜ(𝑟)=𝑘𝑟−2𝛼𝜇+2𝛼𝜈𝑒^−2𝛼𝑟1−𝑒^{−2𝛼𝑟−1}+𝑎_𝑛𝑘𝑏_𝑛𝑘𝑐_𝑛𝑘²𝐹₁𝑎_𝑛𝑘+1,𝑏_𝑛𝑘+1,𝑐_𝑛𝑘+1;𝑒^−2𝛼𝑟₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;𝑒^−2𝛼𝑟. (25) By considering the finiteness of the solution, the quantum condition is given by 𝜇+𝜈−𝐴𝛼𝐸_𝑛𝑘−𝐸+𝑀^2𝑛𝑘−𝑀²4𝛼²=−𝑛,𝑛=0,1,2,3. (26) 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: 𝑦=1−𝑒^−2𝛼𝑟,𝑦=1−𝑧,𝑟⟶∞,𝑦⟶1,𝑟⟶0,𝑦⟶0 (27) and obtain 𝑦𝑑1−𝑦²𝑑𝑦²𝑑−𝑦()1𝑑𝑦−𝑘𝑘+1𝑦+𝐴𝛼𝐸_𝑛𝑘+𝑦+𝑀4𝛼²𝐸1−𝑦^2𝑛𝑘−𝑀²𝐹_𝑛𝑘𝑦=0. (28) Considering the boundary condition of the scattering state, we take the following trial wavefunction: 𝐹_𝑛𝑘𝑦=𝑦^𝜂1−𝑦^𝛽𝑓_𝑛𝑘𝑦. (29) And inserting this equation into (28), we obtain the following equation: 𝑦1−𝑦𝑓^′′_𝑛𝑘𝑦+2𝜂−2𝜂+2𝛽+1𝑦𝑓^′_𝑛𝑘𝑦+𝜂𝜂−11−𝑦𝑦𝑦−2𝛽𝜂+𝛽𝛽−1𝑦1−𝑦−𝜂+𝛽()11−𝑦−𝑘𝑘+1𝑦+𝐴𝛼𝐸_𝑛𝑘+1+𝑀4𝛼²𝐸²_𝑛𝑘−𝑀²𝑦𝑓1−𝑦^𝑛𝑘𝑦=0, (30) where 𝛽=𝑖𝐸²_𝑛𝑘−𝑀²4𝛼²().,𝜂𝜂−1=𝑘𝑘+1𝜂>0 (31) This equation is a hypergeometric type. Thus, (29) is as follows: (𝑁_𝑛𝑘 is the normalization constant) 𝐹_𝑛𝑘𝑦=𝑁_𝑛𝑘𝑦^𝜂1−𝑦^𝛽₂𝐹₁𝑎,𝑏,𝑐;𝑦 (32) that is, the upper component of wavefunction. The lower component is 𝐺_𝑛𝑘(𝑟)=𝑁_𝑛𝑘𝑀+𝐸_𝑛𝑘𝑑+𝑘𝑑𝑟𝑟𝑒^{−2𝛼𝛽𝑟}1−𝑒^{−2𝛼𝑟𝜂}×₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;1−𝑒^−2𝛼𝑟. (33) And therefore, total wavefunction of scattering state with (4), (7b), (32), and (33) is 𝜓(𝑟)=𝐹_𝑛𝑘(𝑟)𝑟𝑌^𝑙_𝑗𝑚𝑖𝐺𝜃,𝜑_𝑛𝑘(𝑟)𝑟𝑌_𝑗𝑙_𝑚̃=𝑁𝜃,𝜑_𝑛𝑘𝑟1−𝑖𝐸_𝑛𝑘(𝑟)̂ℜ(𝑟)+𝑀𝜎⋅×𝑒^{−2𝛼𝛽𝑟}1−𝑒^{−2𝛼𝑟𝜂}𝑌^𝑙_𝑗𝑚×𝜃,𝜑₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;1−𝑒^−2𝛼𝑟, (34) where ℜ(𝑟)=𝑘𝑟−2𝛼𝛽+2𝛼𝜂𝑒^−2𝛼𝑟1−𝑒^{−2𝛼𝑟−1}+𝑎_𝑛𝑘𝑏_𝑛𝑘𝑐_𝑛𝑘ײ𝐹₁𝑎_𝑛𝑘+1,𝑏_𝑛𝑘+1,𝑐_𝑛𝑘+1;1−𝑒^−2𝛼𝑟₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;1−𝑒^−2𝛼𝑟,𝑎_𝑛𝑘=𝜂+𝛽+𝐴𝛼𝐸_𝑛𝑘−𝐸+𝑀^2𝑛𝑘−𝑀²4𝛼²,𝑏_𝑛𝑘=𝜂+𝛽−𝐴𝛼𝐸_𝑛𝑘−𝐸+𝑀^2𝑛𝑘−𝑀²4𝛼²,𝑐_𝑛𝑘=2𝜂. (35) According to (26), we obtain the following form of energy eigenvalue equation for scattering states: 𝜂+𝛽−𝐴𝛼𝐸_𝑛𝑘−𝐸+𝑀^2𝑛𝑘−𝑀²4𝛼²=−𝑛. (36) 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: ₂𝐹₁()=Γ(𝑐)Γ()𝑎,𝑏,𝑐;𝑥𝑐−𝑎−𝑏Γ()Γ()×𝑐−𝑎𝑐−𝑏₂𝐹₁()+()𝑎,𝑏;𝑎+𝑏−𝑐+1;1−𝑥1−𝑥^{𝑐−𝑎−𝑏}Γ(𝑐)Γ(𝑎+𝑏−𝑐)Γ(𝑎)Γ(𝑏)×₂𝐹₁(𝑐−𝑎,𝑐−𝑏;𝑐−𝑎−𝑏+1;1−𝑥₂),𝐹₁(𝑎,𝑏,𝑐,0)=1. (37)(38) Therefore, we obtain ₂𝐹₁𝑎,𝑏,𝑐;1−𝑒^−2𝛼𝑟=Γ(𝑐)Γ(𝑐−𝑎−𝑏)Γ()Γ()×𝑐−𝑎𝑐−𝑏₂𝐹₁𝑎,𝑏;𝑎+𝑏−𝑐+1;𝑒^−2𝛼𝑟+𝑒^{−2𝛼𝑟𝑐−𝑎−𝑏}Γ(𝑐)Γ(𝑎+𝑏−𝑐)Γ(𝑎)Γ(𝑏)×₂𝐹₁𝑐−𝑎,𝑐−𝑏;𝑐−𝑎−𝑏+1;𝑒^−2𝛼𝑟. (39) By using this definition 𝛽=𝑖𝑘₁,𝑘₁=𝐸²_𝑛𝑘−𝑀²4𝛼² (40) the upper component of wavefunction is as follows: 𝐹_𝑛𝑘(𝑟)=𝑁_𝑛𝑘1−𝑒^{−2𝛼𝑟𝜂}𝑒^{−2𝑖𝑘1𝛼𝑟}×₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;1−𝑒^−2𝛼𝑟. (41) Now, with (38) and (39) in the limit𝑟→∞we have 𝐹_𝑛𝑘(𝑟)_𝑟→∞⟶𝑁_{𝑛𝑘𝑟→∞}1−𝑒^{−2𝛼𝑟𝜂}𝑒^{−2𝑖𝑘1𝛼𝑟}×Γ𝑐_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘+Γ𝑐_𝑛𝑘Γ𝑎_𝑛𝑘+𝑏_𝑛𝑘−𝑐_𝑛𝑘Γ𝑎_𝑛𝑘Γ𝑏_𝑛𝑘𝑒^{4𝑖𝑘1𝛼𝑟}. (42) Then 𝐹_𝑛𝑘(𝑟⟶∞)⟶𝑁_𝑛𝑘Γ𝑐_𝑛𝑘×Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘×𝑒^{−𝑖(2𝑘1𝛼𝑟−𝛿)}+𝑒^{𝑖(2𝑘1𝛼𝑟−𝛿)}=2𝑁_𝑛𝑘Γ𝑐_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘×cos2𝑘₁𝛼𝑟−𝛿=2𝑁_𝑛𝑘Γ𝑐_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘×sin𝛿−2𝑘₁𝜋𝛼𝑟+2. (43) The general boundary condition of the scattering state wave function on the “(𝑘₁/2𝜋)  scale” is 𝜑(𝑟)=2sin𝑘₁𝜋𝑟−2𝑙+𝛿_𝑙. (44) Therefore, we have 𝐹_𝑛𝑘(𝑟⟶∞)⟶2𝑁_𝑛𝑘Γ𝑐_𝑛𝑘×Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘×sin2𝛼𝑘₁𝜋𝑟+2Γ𝑐+arg^𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘. (45) Thus, we obtain the scattering phase shifts as follows: 𝛿_{𝑙,𝑛𝑘}=𝜋2(𝑙+1)+argΓ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘−argΓ𝑐_𝑛𝑘−𝑎_𝑛𝑘−argΓ𝑐_𝑛𝑘−𝑏_𝑛𝑘. (46) Then we have the scattering phase shifts of the upper component of wavefunction as 𝛿_{𝑙,𝑛𝑘}=𝜋2(𝑙+1)+argΓ2𝛽−argΓ𝜂+𝛽−2𝜂²+𝛽²𝐴+4𝜂𝛽+𝛼𝐸_𝑛𝑘+𝑀−argΓ𝜂+𝛽+2𝜂²+𝛽²𝐴+4𝜂𝛽+𝛼𝐸_𝑛𝑘+𝑀. (47) For the lower component, with (34), we have 𝐺_𝑛𝑘(𝑟)_𝑟→∞⟶𝑁_𝑛𝑘𝐸_𝑛𝑘+𝑀_𝑟→∞ℜ(𝑟)1−𝑒^{−2𝛼𝑟𝜂}×𝑒^{−2𝑖𝑘1𝛼𝑟}Γ𝑐_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘+Γ𝑐_𝑛𝑘Γ𝑎_𝑛𝑘+𝑏_𝑛𝑘−𝑐_𝑛𝑘Γ𝑎_𝑛𝑘Γ𝑏_𝑛𝑘×𝑒^{4𝑖𝑘1𝛼𝑟}limℜ(𝑟_𝑟→∞)=lim_𝑟→∞𝑘𝑟−2𝛼𝛽+2𝛼𝜂𝑒^−2𝛼𝑟×1−𝑒^{−2𝛼𝑟−1}+𝑎_𝑛𝑘𝑏_𝑛𝑘𝑐_𝑛𝑘ײ𝐹₁𝑎_𝑛𝑘+1,𝑏_𝑛𝑘+1,𝑐_𝑛𝑘+1;1−𝑒^−2𝛼𝑟₂𝐹₁𝑎_𝑛𝑘,𝑏_𝑛𝑘,𝑐_𝑛𝑘;1−𝑒^−2𝛼𝑟𝑎=−2𝛼𝛽+^𝑛𝑘𝑏_𝑛𝑘𝑐_𝑛𝑘. (48) Then we have 𝐺_𝑛𝑘(𝑟⟶∞)⟶𝐻_𝑛𝑘Γ𝑐_𝑛𝑘×Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘×𝑒^{−𝑖(2𝑘1𝛼𝑟−𝛿)}+𝑒^{𝑖(2𝑘1𝛼𝑟−𝛿)}, (49) where 𝐻_𝑛𝑘=𝑁_𝑛𝑘𝐸_𝑛𝑘𝑎+𝑀−2𝛼𝛽+^𝑛𝑘𝑏_𝑛𝑘𝑐_𝑛𝑘. (50) Thus with (43), (44), (45), and (49), we have the following: 𝐺_𝑛𝑘(𝑟⟶∞)⟶2𝐻_𝑛𝑘Γ𝑐_𝑛𝑘×Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘×sin2𝛼𝑘₁𝜋𝑟+2Γ𝑐+arg^𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘Γ𝑐_𝑛𝑘−𝑎_𝑛𝑘Γ𝑐_𝑛𝑘−𝑏_𝑛𝑘. (51) And finally we have 𝛿_{𝑙,𝑛𝑘}=𝜋2(𝑙+1)+argΓ𝑐_𝑛𝑘−𝑎_𝑛𝑘−𝑏_𝑛𝑘−argΓ𝑐_𝑛𝑘−𝑎_𝑛𝑘−argΓ𝑐_𝑛𝑘−𝑏_𝑛𝑘. (52) Therefore, the scattering phase shifts of two components are equal but in a constant coefficient.

4. 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.

Table 1 

Energy eigenvalues of the Yukawa potential for different values of 𝑛 and 𝑘 (in ℏ=𝑐=1 unit) for 𝑀=0.5𝑓 m⁻¹, 𝐴=0.1, and 𝛼=0.01.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors thank the anonymous referees for the valuable comments and suggestions.