Advances in High Energy Physics

Volume 2014 (2014), Article ID 867483, 7 pages

http://dx.doi.org/10.1155/2014/867483

Research Article

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

F. Pakdel,¹ A. A. Rajabi,¹ and M. Hamzavi²

¹Deparment of Physics, University of Shahrood, Shahrood, Iran

²Deparment of Physics, University of Zanjan, Zanjan, Iran

Received 16 November 2013; Accepted 23 January 2014; Published 17 March 2014

Academic Editor: Shi-Hai Dong

Copyright © 2014 F. Pakdel et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP³.

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 . 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 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.

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.

When scalar potentialis 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.

3. 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:

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 [26]:

where

Comparing (17) with the hypergeometric equation of the form [26]

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.

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 and (in unit) for  m⁻¹, , and .

,
,

,

1, 1

2
−0.4996006388
0, −1

1
−0.4999001896

2, 2

3
−0.4991009891
1, −2

2
−0.4996006388

3, 3

4
−0.4984006400
2, −3

3
−0.4991009891

4, 4

5
−0.4974987420
3, −4

4
−0.4984006400

1, 1

2
−0.4991009891
0, −1

1
−0.4996006388

2, 2

3
−0.4984006400
1, −2

2
−0.4991009891

3, 3

4
−0.4974987420
2, −3

3
−0.4984006400

4, 4

5
−0.4963942080
3, −4

4
−0.4974987420

1, 1

2
−0.4984006400
0, −1

1
−0.4991009891

2, 2

3
−0.4974987420
1, −2

2
−0.4984006400

3, 3

4
−0.4963942080
2, −3

3
−0.4974987420

4, 4

5
−0.4950856720
3, −4

4
−0.4963942080

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.

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