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Advances in High Energy Physics
Volume 2012 (2012), Article ID 707041, 17 pages
Dirac Equation under Scalar, Vector, and Tensor Cornell Interactions
1Physics Department, Shahrood University of Technology, Shahrood 3619995161, Iran
2Young Researchers Club, Garmsar Branch, Islamic Azad University, Garmsar, Iran
3Computer Engineering Department, Shahrood University of Technology, Shahrood, Iran
Received 4 July 2012; Revised 27 August 2012; Accepted 29 August 2012
Academic Editor: S. H. Dong
Copyright © 2012 H. Hassanabadi 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.
Spin and pseudospin symmetries of Dirac equation are solved under scalar, vector, and tensor interactions for arbitrary quantum number via the analytical ansatz approach. The spectrum of the system is numerically reported for typical values of the potential parameters.
No doubt, the Cornell potential (alternatively called Funnel in the literature) is, if not the best, among the most appealing interactions in particle physics. The Cornell potential contains a confining term besides the Coulomb interaction and has successfully accounted for the particle physics data . Unfortunately, to our best knowledge, the potential does not possess exact solutions under all common equations of quantum mechanics, that is, the nonrelativistic Schrödinger equation, and relativistic Dirac, Klein-Gordon, Proca, and Duffin-Kemmer-Petiau (DKP) equations. Here, we focus on the relativistic symmetries of Dirac equation, that is, spin and pseudospin symmetries, which provide a reliable theoretical basis for hadronic and nuclear spectroscopy [2–12]. This has motivated many studies under various interactions within the past two decades ([13–18] and many references therein). Nevertheless, none of these papers has investigated the symmetry limits under the Cornell potential. This is definitely due to the complicated nature of the resulting differential equation which cannot be solved by common analytical techniques of quantum mechanics such as the supersymmetry quantum mechanics (SUSYQ), Lie groups, Nikiforov-Uvarov (NU) technique, and point canonical transformations, In our study, we make use of the ansatz approach to deal with this complicated equation. A survey on the application of this technique to other wave equations including Schrödinger, spinless-Salpeter, Dirac, Klein-Gordon, and DKP equations can be found in [19–27]. We organize the study as follows. In the first step, we review the most essential equations of the symmetry limits. We next propose a physical ansatz solution to the equation and, in a systematic manner, calculate the spectrum of the system for any arbitrary state. To provide a better understanding of the solutions, we provide some numerical data for the spectrum as well.
2. Dirac Equation Including Tensor Coupling
In spherical coordinates, Dirac equation with both scalar potential and vector potential is expressed as [2, 3] where is the relativistic energy of the system; and are the Dirac matrices and stands for the momentum operator. For a particle in a spherical field, the total angular momentum operator and spin-orbit matrix operator , where and are, respectively, the Pauli matrix and orbital angular momentum, commute with the Dirac Hamiltonian. The eigenvalues of are for the aligned spin (, etc.) and for the unaligned spin (, etc.). The complete set of the conservative quantities can be chosen as . As shown in , the Dirac spinor is considered as where and are the radial wave functions of the upper and lower components, respectively, and and , respectively, stand for spin and pseudospin spherical harmonics coupled to the angular momentum . is the projection of the angular momentum on the -axis. The orbital angular momentum quantum numbers and refer to the upper and lower components, respectively. The quasidegenerate doublet structure can be expressed in terms of pseudospin angular momentum and pseudoorbital angular momentum , which is defined as for aligned spin and for unaligned spin . As shown in [2, 3], substitution of (2.2) into (2.1) yields the following two-coupled differential equations: which simply give where, as the notation indicates, and .
2.1. Pseudospin Symmetry Limit
Under the condition of the pseudospin symmetry, or equivalently . [2, 3]. We choose as the Cornell potential: For the tensor term, we consider the Cornell potential: Substitution of these two terms in into (2.5) gives where and for and , respectively.
2.2. Spin Symmetry Limit
Substitution of the latter in (2.4) gives where and for and , respectively.
3. The Ansatz Solution
3.1. Solution of the Pseudospin Symmetry Limit
In the previous section, we obtained a Schrödinger-like equation of the form where Equation (3.1) fails to admit exact analytical solutions. Therefore, we follow the ansatz approach with the starting square: where By substitution of and into (3.3), we find Here, we consider the case . From (3.4)–(3.6) we find By comparing the corresponding powers of (3.1) and (3.7), we have where . Actually, to have well-behaved solutions of the radial wave function at boundaries, namely, the origin and the infinity, we need to take from (3.8) as Form (3.2), (3.8), the ground-state energy satisfies or which is more compactly written as where the parameter of potential (2.6) from (3.8) should satisfy the following restriction: From (3.3), (3.4), and (3.8), the upper and lower components of the wave function are
For the first node (), using and from (3.5), we arrive at Here, the consequent relations between the potential parameters and the coefficients , , , and are By solving the above equations one can find and as
The energy eigenvalues therefore are where the parameter from (3.8) is For the upper and lower components of the wave function we thus have
Following the analytic iteration procedures for the second node with and as defined in (3.6), the relations between the potential parameters and the coefficients , , , , and are The coefficients and are found from the constraint relations [20, 27–29]: Therefore, the energy eigenvalue in this case is and the lower component of the wave function is
3.2. Solution of the Spin Symmetry Limit
In this case, our ordinary differential equation is with which cannot be solved by our common exact analytical techniques. Let us propose the ansatz solution: where By substitution of and into (3.27), we find For the case of , from (3.26)–(3.29), we find By comparing the corresponding powers of (3.27) and (3.33), we have where . To have physically acceptable solutions, we pick up the value By considering (3.26), (3.32), the first node eigenvalue satisfies or equivalently where the parameter of potential (2.9a) should satisfy the restriction From (3.27), (3.28), and (3.32), the upper and lower components of the wave function are Secondly, for the first node , using and from (3.28)–(3.30), our resulting equation is The relations between the potential parameters and the coefficients , and are where and are found from (3.41) as [20, 27–29] or which determine the corresponding energy as with The upper and lower components of the wave function are then simply found to be
For the second node , we choose and as defined in (3.28) and (3.29). The relations between the potential parameters and the coefficients , and are where The energy eigenvalue therefore is For the upper component of the wave function, we have We have given some numerical values of the energy eigenvalues in Tables 1, 2, 3, 4, 5, and 6 for various states. For the final point, we wish to emphasize on the degeneracy-removing role of the (Cornell) tensor potential. As we already know, for vanishing tensor interaction , the pseudospin doublets, that is, states with quantum numbers and are degenerate. The degenerate states in the spin doublets are those with quantum numbers and , where ,and are the radial, the orbital, and the total angular momentum quantum numbers, respectively (see Tables 1 and 2). Our numerical data reveals that, in the pseudospin symmetry limit, the degenerate states for are for , for , for , for , and so forth. For spin symmetry limit and one can clearly see that the degeneracy occurs in for , for , for , for , and so forth.
Because of the established roles of the Cornell potential and spin, and pseudospin symmetries in nuclear and hadrons spectroscopy, we solved the Dirac equation under these symmetry limits for vector, scalar, and tensor interactions of Cornell-type. In our calculations, on the one hand, due to the failure of other common analytical techniques, and, on the other hand, the better insight which analytical techniques provide us in comparison with their numerical counterparts, we used the quasianalytical ansatz approach. By proposing novel physical solutions and after lengthy calculations, we could find the arbitrary-state solutions. Our results clearly show the degeneracy-removing role of the tensor term and provide the requisite understanding of the solutions for possible further studies. Both the energy spectrum and the eigenfunctions can be used in related systems after the proper phenomenological fits done.
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