Dirac Equation under Scalar, Vector, and Tensor Cornell Interactions

Abstract

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.

1. Introduction

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 [1]. 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 𝑆(𝑟) and vector potential 𝑉(𝑟) is expressed as [2, 3] ^→𝛼⋅^→𝑝+𝛽(𝑀+𝑆(𝑟))−𝑖𝛽^→𝛼⋅𝑟𝑈̂(𝑟)𝜓^→𝑟=[𝐸−𝑉(𝑟)]𝜓^→𝑟, (2.1) where 𝐸 is the relativistic energy of the system; 𝛼 and 𝛽 are the 4×4 Dirac matrices and ^→𝑝=−𝑖^→∇ stands for the momentum operator. For a particle in a spherical field, the total angular momentum operator ^→𝐽 and spin-orbit matrix operator 𝐾=(^→𝜎⋅^→𝐿+1), where 𝜎 and 𝐿 are, respectively, the Pauli matrix and orbital angular momentum, commute with the Dirac Hamiltonian. The eigenvalues of 𝐾 are??𝜅=−(𝑗+1/2) for the aligned spin (𝑠_1/2,𝑝_3/2, etc.) and 𝜅=(𝑗+1/2) for the unaligned spin (𝑝_1/2,𝑑_3/2, etc.). The complete set of the conservative quantities can be chosen as (𝐻,𝐾,𝐽²,𝐽_𝑧). As shown in [13], the Dirac spinor is considered as 𝜓_𝑛𝑘^→𝑟=𝑓_𝑛𝑘^→𝑟𝑔_𝑛𝑘^→𝑟=𝐹_𝑛𝑘^→𝑟𝑟𝑌^ℓ_𝑗𝑚𝑖𝐺𝜃,𝜑_𝑛𝑘→𝑟𝑟𝑌ℓ_𝑗𝑚̃𝜃,𝜑, (2.2) where 𝐹_𝑛𝑘(^→𝑟) and 𝐺_𝑛𝑘(^→𝑟) are the radial wave functions of the upper and lower components, respectively, and 𝑌^ℓ_𝑗𝑚(𝜃,𝜑) and 𝑌ℓ_𝑗𝑚̃(𝜃,𝜑), respectively, stand for spin and pseudospin spherical harmonics coupled to the angular momentum 𝑗. 𝑚??is the projection of the angular momentum on the 𝑧-axis. The orbital angular momentum quantum numbers ℓ and ℓ̃ refer to the upper and lower components, respectively. The quasidegenerate doublet structure can be expressed in terms of pseudospin angular momentum 𝑠=1/2 and pseudoorbital angular momentum ℓ̃, which is defined as ℓ̃=ℓ+1 for aligned spin 𝑗=ℓ̃−1/2 and ℓ̃=ℓ−1 for unaligned spin 𝑗=ℓ̃+1/2. As shown in [2, 3], substitution of (2.2) into (2.1) yields the following two-coupled differential equations: 𝑑+𝜅𝑑𝑟𝑟−𝑈(𝑟)𝐹_𝑛𝑘(𝑟)=𝑀+𝐸_𝑛𝑘−𝑉(𝑟)+𝑆(𝑟)𝐺_𝑛𝑘(𝑟),𝑑−𝜅𝑑𝑟𝑟+𝑈(𝑟)𝐺_𝑛𝑘(𝑟)=𝑀−𝐸_𝑛𝑘+𝑉(𝑟)+𝑆(𝑟)𝐹_𝑛𝑘(𝑟), (2.3) which simply give 𝑑²𝑑𝑟²−𝜅(𝜅+1)𝑟²+2𝜅𝑟𝑈(𝑟)−𝑑𝑈(𝑟)𝑑𝑟−𝑈²(𝑟)+𝑑Δ(𝑟)/𝑑𝑟𝑀+𝐸_𝑛𝜅(𝑟)𝑑−Δ+𝜅𝑑𝑟𝑟−𝑈(𝑟)𝐹_𝑛𝜅(𝑟)=𝑀+𝐸_𝑛𝜅−Δ(𝑟)𝑀−𝐸_𝑛𝜅+Σ(𝑟)𝐹_𝑛𝜅(𝑟),𝑑²𝑑𝑟²−𝜅(𝜅−1)𝑟²+2𝜅𝑟𝑈(𝑟)+𝑑𝑈(𝑟)𝑑𝑟−𝑈²(𝑟)−𝑑Σ(𝑟)/𝑑𝑟𝑀−𝐸_𝑛𝜅(𝑟)𝑑+Σ−𝜅𝑑𝑟𝑟+𝑈(𝑟)𝐺_𝑛𝜅(𝑟)=𝑀+𝐸_𝑛𝜅−Δ(𝑟)𝑀−𝐸_𝑛𝜅+Σ(𝑟)𝐺_𝑛𝜅(𝑟), (2.4)(2.5) where, as the notation indicates, Δ(𝑟)=𝑉(𝑟)−𝑆(𝑟) and Σ(𝑟)=𝑉(𝑟)+𝑆(𝑟).

2.1. Pseudospin Symmetry Limit

Under the condition of the pseudospin symmetry, 𝑑Σ(𝑟)/𝑑𝑟=0 or equivalently Σ(𝑟)=𝐶_𝑔ps=Const. [2, 3]. We choose Δ(𝑟) as the Cornell potential: Δ(𝑟)=𝑎_ps𝑏𝑟+^ps𝑟. (2.6) For the tensor term, we consider the Cornell potential: 𝑈(𝑟)𝐵=𝐴𝑟+𝑟. (2.7) Substitution of these two terms in into (2.5) gives 𝑑²𝑑𝑟²+1𝑟²−𝜅(𝜅−1)+2𝜅𝐵−𝐵−𝐵²−𝐴²𝑟²+𝑎_ps𝑀−𝑎_ps𝐸^𝑝𝑠_𝑛𝜅+𝑎_ps𝐶_ps𝑟+1𝑟𝑏_ps𝑀−𝑏_ps𝐸^ps_𝑛𝜅+𝑏_ps𝐶_ps−𝑀²−𝑀𝐶_ps+𝐸^ps_𝑛𝜅²−𝐸^ps_𝑛𝜅𝐶_ps+2𝜅𝐴+𝐴−2𝐴𝐵×𝐺^𝑝𝑠_𝑛𝜅(𝑟)=0, (2.8) where 𝜅=−ℓ̃ and 𝜅=ℓ̃+1 for 𝜅<0 and 𝜅>0, respectively.

2.2. Spin Symmetry Limit

In the spin symmetry limit 𝑑Δ(𝑟)/𝑑𝑟=0 or Δ(𝑟)=𝐶_𝑔𝑠=const. [2, 3]. As the previous section, we considerΣ(𝑟)=𝑎_𝑠𝑏𝑟+^𝑠𝑟,𝑈(𝑟)𝐵=𝐴𝑟+𝑟. (2.9a)(2.9b)

Substitution of the latter in (2.4) gives 𝑑²𝑑𝑟²+1𝑟²−𝜅(𝜅+1)+2𝜅𝐵+𝐵−𝐵²−𝐴²𝑟²+−𝑎_𝑠𝑀−𝑎_𝑠𝐸^𝑠_𝑛𝜅+𝑎_𝑠𝐶_𝑠𝑟+1𝑟−𝑏_𝑠𝑀−𝑏_𝑠𝐸^𝑠_𝑛𝜅+𝑏_𝑠𝐶_𝑠−𝑀²+𝑀𝐶_𝑠+𝐸^ps_𝑛𝜅²−𝐸^𝑠_𝑛𝜅𝐶_𝑠+2𝜅𝐴−𝐴−2𝐴𝐵𝐹^𝑠_𝑛𝜅(𝑟)=0, (2.10) where 𝜅=ℓ and 𝜅=−ℓ−1 for 𝜅<0 and 𝜅>0, 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 𝑑²𝐺^ps_𝑛𝜅(𝑟)𝑑𝑟²+𝜀^ps_𝑛𝜅−𝑎^ps₁𝑟²−𝑏₁^𝑝𝑠𝑐𝑟+^ps1𝑟+𝑑^ps₁−𝜅(𝜅−1)𝑟²𝐺^ps_𝑛𝜅(𝑟)=0, (3.1) where 𝜀^ps_𝑛𝜅=−𝑀²−𝑀𝐶_ps+𝐸^ps_𝑛𝜅²−𝐸^ps_𝑛𝜅𝐶_ps𝑎+2𝜅𝐴+𝐴−2𝐴𝐵,_ps1=𝐴²,𝑏^ps₁=−𝑎_ps𝑀+𝑎_ps𝐸^ps_𝑛𝜅−𝑎_ps𝐶_ps,𝑐^ps₁=𝑏_ps𝑀−𝑏_ps𝐸^ps_𝑛𝜅+𝑏_ps𝐶_ps,𝑑^ps₁=2𝜅𝐵−𝐵−𝐵². (3.2) Equation (3.1) fails to admit exact analytical solutions. Therefore, we follow the ansatz approach with the starting square: 𝐺^ps_𝑛𝜅(𝑟)=𝑓^ps_𝑛(𝑟)exp𝑔^ps_𝜅(𝑟), (3.3) where 𝑓^ps_𝑛(𝑟)=1,if𝑛=0,_𝑛∏_𝑖=1𝑟−𝛼^𝑛_𝑖𝑔,if𝑛≥1,_ps𝜅(𝑟)1=−2𝛼^ps𝑟²−𝛽^ps𝑟+𝛿^pslnr,𝛼^ps>0,𝛽^ps>0. (3.4)(3.5) By substitution of 𝑓_𝑛(𝑟) and 𝑔_𝜅(𝑟) into (3.3), we find 𝐺^ps′′_𝑛𝜅(𝑟)=𝑔^ps_𝜅^′′(𝑟)+𝑔^ps_𝜅^′2(𝑟)+𝑓^ps_𝑛^′′(𝑟)+2𝑔^ps_𝜅^′(𝑟)𝑓^ps_𝑛^′(𝑟)𝑓^ps_𝑛(𝑟)𝐺^ps_𝑛𝜅(𝑟). (3.6) Here, we consider the case 𝑛=0. From (3.4)–(3.6) we find 𝐺^ps_0𝜅^′′(𝑟)=𝛼^ps2𝑟²+2𝛼^ps𝛽^ps𝑟−𝛼^ps1+2𝛿^ps+0+𝛽^ps2−2𝛽^ps𝛿^ps+0𝑟+𝛿^ps𝛿^ps−1𝑟²𝐺^ps_0𝜅(𝑟). (3.7) By comparing the corresponding powers of (3.1) and (3.7), we have 𝛼^ps=𝑎^ps₁,𝑎^ps₁𝛽>0,_ps=𝑏^ps₁2𝑎^ps₁,𝑐^ps₁=2𝛽^ps𝛿^ps,𝛿+0_ps=121±(2𝜅−1)²−4𝑑^ps₁=121±𝑘^ps₁,𝜀^ps_0𝜅=𝛼^ps1+2𝛿^ps+0−𝛽^ps2, (3.8) where 𝑘^ps₁=(2𝜅−1)²−4𝑑^ps₁. Actually, to have well-behaved solutions of the radial wave function at boundaries, namely, the origin and the infinity, we need to take 𝛿 from (3.8) as 𝛿^ps=121+𝑘^ps₁. (3.9) Form (3.2), (3.8), the ground-state energy satisfies−𝑀²−𝑀𝐶_ps+𝐸^ps_0𝜅²−𝐸^ps_0𝜅𝐶_𝑝𝑠+2𝜅𝐴+𝐴−2𝐴𝐵−𝛼^ps1+2𝛿^ps+0+𝛽^ps2=0 (3.10a) or 𝐸^ps_0𝜅²−𝐸^ps_0𝜅𝐶_ps−𝑀²−𝑀𝐶_ps+2𝜅𝐴+𝐴−2𝐴𝐵−𝑎^ps₁2+(2𝜅−1)²−4𝑑^ps₁+𝑏^ps₁²4𝑎^ps₁=0, (3.10b)which is more compactly written as 𝐸^ps±_0𝜅=12𝐶_ps±𝐶²_ps−8𝜅𝐴−4𝐴+8𝐴𝐵+4𝑀²+4𝑀𝐶_ps+8𝑎^ps₁+4𝑎^ps₁4𝜅²−4𝜅+1−4𝑑^ps₁−4𝛽^ps21/2,(3.35), (3.11) where the parameter 𝑎_ps of potential (2.6) from (3.8) should satisfy the following restriction: 𝑎_ps=−2𝑏_ps𝑎^ps₁1+(2𝜅−1)²−4𝑑^ps₁=−2𝑏_ps𝑎^ps₁1+𝑘^ps₁. (3.12) From (3.3), (3.4), and (3.8), the upper and lower components of the wave function are𝑔_𝜅(𝑟)1=−2𝑎^ps₁𝑟²−𝑏^ps₁2𝑎^ps₁1𝑟+21+(2𝜅−1)²−4𝑑^ps₁𝐺ln𝑟,_ps0𝜅(𝑟)=𝑁_0𝜅𝑟^{(1/2)(1+(2𝜅−1)2−4𝑑ps1)}−1exp2𝑎^ps₁𝑟²−𝑏^ps₁2𝑎^ps₁𝑟𝐹_ps0𝜅,(𝑟)=1𝑀−𝐸^ps_0𝜅+𝐶_ps𝑑−𝜅𝑑𝑟𝑟+𝑈(𝑟)𝐺^ps_0𝜅(𝑟). (3.13a)(3.13b)(3.13c)

For the first node (𝑛=1), using 𝑓₁(𝑟)=(𝑟−𝛼₁⁽¹⁾) and 𝑔_𝜅(𝑟) from (3.5), we arrive at 𝑎^ps₁𝑟²+𝑏^ps₁𝑐𝑟−^ps1𝑟+𝜅(𝜅−1)−𝑑^ps₁𝑟²−𝜀^ps_1𝜅=−𝛼^ps−𝛿^ps𝑟²+𝛼^ps2𝑟²+𝛽^ps2+𝛿^ps2𝑟²+2𝛼^ps𝛽^ps𝑟−2𝛿^ps𝛼^ps−2𝛽^ps𝛿^ps𝑟+−2𝛼^ps𝑟−2𝛽^ps+2𝛿^ps/𝑟𝑟−𝛼₁⁽¹⁾. (3.14) Here, the consequent relations between the potential parameters and the coefficients 𝛼^ps, 𝛽^ps, 𝛿^ps, and 𝛼₁⁽¹⁾ are 𝛼^ps=𝑎^ps₁,𝛽^ps=𝑏^ps₁2𝑎^ps₁,𝛿^ps=121+𝑘^ps₁,𝜀^ps_1𝜅=𝛼^ps1+2𝛿^ps+1−𝛽^ps2,𝑐^ps₁−2𝛽^ps𝛿^ps𝛼₁⁽¹⁾=2𝛿^ps,𝑐^ps₁−2𝛽^ps𝛿^ps+1=2𝛼^ps𝛼₁⁽¹⁾. (3.15) By solving the above equations one can find 𝑐^ps₁ and 𝛼₁⁽¹⁾ as𝑐^ps₁=𝑏^ps₁2𝑎^ps₁(2𝜅−1)²−4𝑑^ps₁+2+𝑏^ps₁²4𝑎^ps₁+2𝑎^ps₁1+(2𝜅−1)²−4𝑑^ps₁,𝛼^ps𝛼₁⁽¹⁾²+𝛽^ps𝛼₁⁽¹⁾−𝛿^ps=0,→𝛼₁⁽¹⁾=12−𝛽^ps+𝛽^ps2+4𝛼^ps𝛿^ps𝛼^ps=−𝑏^ps₁4𝑎^ps₁+𝑏^ps₁²16𝑎^ps₁²+1+(2𝜅−1)²−4𝑑^ps₁2𝑎^ps₁. (3.16a)(3.16b)

The energy eigenvalues therefore are 𝐸^ps±_1𝜅=12𝐶_ps±𝐶²_ps−8𝜅𝐴−4𝐴+8𝐴𝐵+4𝑀²+4𝑀𝐶_ps+16𝑎^ps₁+4𝑎^ps₁4𝜅²−4𝜅+1−4𝑑^ps₁−4𝛽^ps21/2, (3.17) where the parameter 𝑎_ps from (3.8) is 𝑎_𝑝𝑠=2𝑎^ps₁2𝑐^ps₁+𝑐^ps₁𝑘^ps₁+𝑐^ps₁²+6𝑎^ps₁+14𝑎^ps₁𝑘^ps₁+10𝑎^ps₁𝑘^ps₁²+2𝑎^ps₁𝑘^ps₁³−3𝑀−4𝑀𝑘^ps₁−𝑀𝑘^ps₁²+3𝐸^ps_1𝜅+4𝐸^ps_1𝜅𝑘^ps₁+𝐸^ps_1𝜅𝑘^ps₁²−3𝐶_ps−4𝐶_ps𝑘^ps₁−𝐶_ps𝑘^ps₁². (3.18) For the upper and lower components of the wave function we thus have𝐺^ps_1𝜅(𝑟)=𝑁_1𝜅𝑟−𝛼₁⁽¹⁾𝑟^{(1/2)(1+(2𝜅−1)2−4𝑑ps1)}−1exp2𝑎^ps₁𝑟²−𝑏^ps₁2𝑎^ps₁𝑟𝐹_ps1𝜅,(𝑟)=1𝑀−𝐸^ps_1𝜅+𝐶_ps𝑑−𝜅𝑑𝑟𝑟+𝑈(𝑟)𝐺^ps_1𝜅(𝑟). (3.19a)(3.19b)

Following the analytic iteration procedures for the second node (𝑛=2) with 𝑓₂(𝑟)=(𝑟−𝛼₁⁽²⁾)(𝑟−𝛼₂⁽²⁾) and 𝑔_𝜅(𝑟) as defined in (3.6), the relations between the potential parameters and the coefficients 𝛼^ps, 𝛽^ps, 𝛿^ps, 𝛼₁⁽²⁾, and 𝛼₂⁽²⁾are 𝛼^ps=𝑎^ps₁,𝛽^ps=𝑏^ps₁2𝑎^ps₁,𝛿^ps=121+𝑘^ps₁,𝜀^ps_2𝜅=𝛼^ps1+2𝛿^ps+2−𝛽^ps2,𝑐^ps₁−2𝛽^ps𝛿^ps+2=2𝛼^ps2_𝑖=1𝛼_𝑖⁽²⁾,𝑐^ps₁−2𝛽^ps𝛿^ps2_𝑖<𝑗𝛼_𝑖⁽²⁾𝛼_𝑗⁽²⁾=2𝛿^ps2_𝑖=1𝛼_𝑖⁽²⁾,𝑐^ps₁−2𝛽^ps𝛿^ps+1²_𝑖=1𝛼_𝑖⁽²⁾=4𝛼^ps2_𝑖<𝑗𝛼_𝑖⁽²⁾𝛼_𝑗⁽²⁾+22𝛿^ps+1. (3.20) The coefficients 𝛼₁⁽²⁾ and 𝛼₂⁽²⁾ are found from the constraint relations [20, 27–29]: 𝛼^ps2_𝑖=1𝛼_𝑖⁽²⁾²+𝛽^ps2_𝑖=1𝛼_𝑖⁽²⁾−2𝛿^ps𝛿+1=0,_ps2𝑖=1𝛼_𝑖⁽²⁾²−𝛽^ps2_𝑖=1𝛼_𝑖⁽²⁾+1²_𝑗<𝑘𝛼_𝑗⁽²⁾𝛼_𝑘⁽²⁾−2𝛼^ps2_𝑗<𝑘𝛼_𝑗⁽²⁾²𝛼_𝑘⁽²⁾²=0. (3.21)(3.22) Therefore, the energy eigenvalue in this case is 𝐸^ps±_2𝜅=12𝐶_ps±𝐶²_ps−8𝜅𝐴−4𝐴+8𝐴𝐵+4𝑀²+4𝑀𝐶_ps+24𝑎^ps₁+4𝑎^ps₁4𝜅²−4𝜅+1−4𝑑^ps₁−4𝛽^ps21/2, (3.23) and the lower component of the wave function is 𝐺^ps_2𝜅(𝑟)=𝑁²_2𝜅𝑖=1𝑟−𝛼_𝑖⁽²⁾𝑟^{(1/2)(1+(2𝜅−1)2−4𝑑ps1)}−1exp2𝑎^ps₁𝑟²−𝑏^ps₁2𝑎^ps₁𝑟. (3.24)

3.2. Solution of the Spin Symmetry Limit

In this case, our ordinary differential equation is 𝑑²𝐹^𝑠_𝑛𝜅(𝑟)𝑑𝑟²+𝜀^𝑠_𝑛𝜅−𝑎^𝑠₁𝑟²−𝑏^𝑠₁𝑐𝑟+^𝑠1𝑟+𝑑^𝑠₁−𝜅(𝜅+1)𝑟²𝐹^𝑠_𝑛𝜅(𝑟)=0, (3.25) with 𝜀^𝑠_𝑛𝜅=−𝑀²+𝑀𝐶_𝑠+𝐸^𝑠_𝑛𝜅²−𝐸^𝑠_𝑛𝜅𝐶_𝑠𝑎+2𝜅𝐴−𝐴−2𝐴𝐵,_𝑠1=𝐴²,𝑏^𝑠₁=𝑎_𝑠𝑀+𝑎_𝑠𝐸^𝑠_𝑛𝜅−𝑎_𝑠𝐶_𝑠,𝑐^𝑠₁=−𝑏_𝑠𝑀−𝑏_𝑠𝐸^𝑠_𝑛𝜅+𝑏_𝑠𝐶_𝑠,𝑑^𝑠₁=2𝜅𝐵+𝐵−𝐵², (3.26) which cannot be solved by our common exact analytical techniques. Let us propose the ansatz solution: 𝐹^𝑠_𝑛𝜅(𝑟)=𝑓^𝑠_𝑛(𝑟)exp𝑔^𝑠_𝜅(𝑟), (3.27) where 𝑓^𝑠_𝑛(𝑟)=1,if𝑛=0,_𝑛∏_𝑖=1𝑟−𝛼^𝑛_𝑖𝑔,if𝑛≥1,_𝑠𝜅(𝑟)1=−2𝛼^𝑠𝑟²−𝛽^𝑠𝑟+𝛿^𝑠ln𝑟,𝛼^𝑠>0,𝛽^𝑠>0. (3.28)(3.29) By substitution of 𝑓_𝑛(𝑟) and 𝑔_𝜅(𝑟) into (3.27), we find 𝐹^𝑠_𝑛𝜅^′′(𝑟)=𝑔^𝑠_𝜅^′′(𝑟)+𝑔^𝑠_𝜅^′2(𝑟)+𝑓^𝑠_𝑛^′′(𝑟)+2𝑔^𝑠_𝜅^′(𝑟)𝑓^𝑠_𝑛^′(𝑟)𝑓^𝑠_𝑛(𝑟)𝐹^𝑠_𝑛𝜅(𝑟). (3.30) For the case of 𝑛=0, from (3.26)–(3.29), we find 𝐹^𝑠_𝑛𝜅^′′(𝑟)=𝛼^𝑠2𝑟²+2𝛼^𝑠𝛽^𝑠𝑟−𝛼^𝑠1+2𝛿^𝑠+0+𝛽^𝑠2−2𝛽^𝑠𝛿^𝑠+0𝑟+𝛿^𝑠𝛿^𝑠−1𝑟²𝐹^𝑠_𝑛𝜅(𝑟). (3.31) By comparing the corresponding powers of (3.27) and (3.33), we have 𝛼^𝑠=𝑎^𝑠₁,𝛽^𝑠=𝑏^𝑠₁2𝑎^𝑠₁,𝑎^𝑠₁𝑐>0,_𝑠1=2𝛽^𝑠𝛿^𝑠,𝛿+0_𝑠=121±(2𝜅+1)²−4𝑑^𝑠₁=121±𝑘^𝑠₁,𝜀^𝑠_0𝜅=𝛼1+2𝛿^𝑠+0−𝛽^𝑠2, (3.32) where 𝑘^𝑠₁=(2𝜅+1)²−4𝑑^𝑠₁. To have physically acceptable solutions, we pick up the value 𝛿^𝑠=121+𝑘^𝑠₁. (3.33) By considering (3.26), (3.32), the first node eigenvalue satisfies −𝑀²+𝑀𝐶_𝑠+𝐸^𝑠_𝑛𝜅²−𝐸^𝑠_𝑛𝜅𝐶_𝑠+2𝜅𝐴−𝐴−2𝐴𝐵−𝛼^𝑠1+2𝛿^𝑠+0+𝛽^𝑠2=0, (3.34) or equivalently 𝐸^𝑠±_0𝜅=12𝐶_𝑠±𝐶²_𝑠+4𝑀²−4𝑀𝐶_𝑠−8𝜅𝐴+4𝐴+8𝐴𝐵+8𝑎^𝑠₁+4𝑎^𝑠₁4𝜅²+4𝜅+1−4𝑑^𝑠₁−4𝛽^𝑠21/2, (3.35) where the parameter 𝑎_𝑠 of potential (2.9a) should satisfy the restriction 𝑎_𝑠=−2𝑏_𝑠𝑎^𝑠₁1+𝑘^𝑠₁. (3.36) From (3.27), (3.28), and (3.32), the upper and lower components of the wave function are 𝑔_𝜅(𝑟)1=−2𝑎^𝑠₁𝑟²−𝑏^𝑠₁2𝑎^𝑠₁1𝑟+21+(2𝜅+1)²−4𝑑^𝑠₁𝐹ln𝑟,_𝑠0𝜅(𝑟)=𝑁_0𝜅𝑟^{(1/2)(1+(2𝜅+1)2−4𝑑𝑠1)}−1exp2𝑎^𝑠₁𝑟²−𝑏^𝑠₁2𝑎^𝑠₁𝑟𝐺_𝑠0𝜅,(𝑟)=1𝑀+𝐸^𝑠_0𝜅−𝐶_𝑠𝑑+𝜅𝑑𝑟𝑟−𝑈(𝑟)𝐹^𝑠_0𝜅(𝑟). (3.37)(3.38)(3.39) Secondly, for the first node (𝑛=1), using 𝑓₁(𝑟)=(𝑟−𝛼₁⁽¹⁾) and 𝑔_𝜅(𝑟) from (3.28)–(3.30), our resulting equation is 𝑎^𝑠₁𝑟²+𝑏^𝑠₁𝑐𝑟−^𝑠1𝑟+−𝑑^𝑠₁+𝜅(𝜅+1)𝑟²−𝜀^𝑠_𝑛𝜅=−𝛼^𝑠−𝛿^𝑠𝑟²+𝛼^𝑠2𝑟²+𝛽^𝑠2+𝛿^𝑠2𝑟²+2𝛼^𝑠𝛽^𝑠𝑟−2𝛿^𝑠𝛼^𝑠−2𝛽^𝑠𝛿^𝑠𝑟+−2𝛼^𝑠𝑟−2𝛽^𝑠+2𝛿^𝑠/𝑟𝑟−𝛼₁⁽¹⁾. (3.40) The relations between the potential parameters and the coefficients 𝛼^s,𝛽^s,𝛿^s, and 𝛼₁⁽¹⁾ are 𝛼^𝑠=𝑎^𝑠₁,𝛽^𝑠=𝑏^𝑠₁2𝑎^ps₁,𝑎^𝑠₁>0,𝛿^𝑠=121+(2𝜅+1)²−4𝑑^𝑠₁=121+𝑘^𝑠₁,𝜀^𝑠_1𝜅=𝛼^𝑠1+2𝛿^𝑠+1−𝛽^𝑠2,𝑐^𝑠₁−2𝛽^𝑠𝛿^𝑠𝛼₁⁽¹⁾=2𝛿^𝑠,𝑐^𝑠₁−2𝛽^𝑠𝛿^𝑠+1=2𝛼^𝑠𝛼₁⁽¹⁾, (3.41) where 𝑐^𝑠₁ and 𝛼₁⁽¹⁾ are found from (3.41) as [20, 27–29]𝑐^𝑠₁=𝑏^𝑠₁2𝑎^𝑠₁(2𝜅+1)²−4𝑑^𝑠₁+2+𝑏^𝑠₁²4𝑎^𝑠₁+2𝑎^𝑠₁1+(2𝜅+1)²−4𝑑^𝑠₁,𝛼^𝑠𝛼₁⁽¹⁾²+𝛽^𝑠𝛼₁⁽¹⁾−𝛿^𝑠=0, (3.42a)(3.42b) or 𝛼₁⁽¹⁾=12−𝛽^𝑠+𝛽^𝑠2+4𝛼^𝑠𝛿^𝑠𝛼^𝑠=−𝑏^𝑠₁4𝑎^𝑠₁+𝑏^𝑠₁²16𝑎^𝑠₁²+1+(2𝜅+1)²−4𝑑^𝑠₁2𝑎^𝑠₁, (3.42c)which determine the corresponding energy as 𝐸^𝑠±_1𝜅=12𝐶_𝑠±𝐶²_𝑠+4𝑀²−4𝑀𝐶_𝑠−8𝜅𝐴+4𝐴+8𝐴𝐵+16𝑎^𝑠₁+4𝑎^𝑠₁4𝜅²+4𝜅+1−4𝑑^𝑠₁−4𝛽^𝑠21/2, (3.43) with 𝑎_𝑠=2𝑎^𝑠₁2𝑐^𝑠₁+𝑐^𝑠₁𝑘^𝑠₁+𝑐^𝑠₁²+6𝑎^𝑠₁+14𝑎^𝑠₁𝑘^𝑠₁+10𝑎^𝑠₁𝑘^𝑠₁²+2𝑎^𝑠₁𝑘^𝑠₁³3𝑀+4𝑀𝑘^𝑠₁+𝑀𝑘^𝑠₁²+3𝐸^𝑠_1𝜅+4𝐸^𝑠_1𝜅𝑘^𝑠₁+𝐸^𝑠_1𝜅𝑘^𝑠₁²−3𝐶_𝑠−4𝐶_𝑠𝑘^𝑠₁−𝐶_𝑠𝑘^𝑠₁². (3.44) The upper and lower components of the wave function are then simply found to be𝐹^𝑠_1𝜅(𝑟)=𝑁_1𝜅𝑟−𝛼₁⁽¹⁾𝑟^{(1/2)(1+(2𝜅+1)2−4𝑑𝑠1)}−1exp2𝑎^𝑠₁𝑟²−𝑏^𝑠₁2𝑎^𝑠₁𝑟𝐺_𝑠1𝜅,(𝑟)=1𝑀+𝐸^𝑠_1𝜅−𝐶_𝑠𝑑+𝜅𝑑𝑟𝑟−𝑈(𝑟)𝐹^𝑠_1𝜅(𝑟). (3.45a)(3.45b)

For the second node (𝑛=2), we choose 𝑓₂(𝑟)=(𝑟−𝛼₁⁽²⁾)(𝑟−𝛼₂⁽²⁾) and 𝑔_𝜅(𝑟) as defined in (3.28) and (3.29). The relations between the potential parameters and the coefficients 𝛼^𝑠,𝛽^𝑠,𝛿^𝑠,𝛼₁⁽²⁾, and 𝛼₂⁽²⁾ are 𝛼^𝑠=𝑎^𝑠₁,𝛽^𝑠=𝑏^𝑠₁2𝑎^𝑠₁,𝛿^𝑠=121+𝑘^𝑠₁,𝜀^𝑠_2𝜅=𝛼^𝑠1+2𝛿^𝑠+2−𝛽^𝑠2,𝑐^𝑠₁−2𝛽^𝑠𝛿^𝑠+2=2𝛼^𝑠2_𝑖=1𝛼_𝑖⁽²⁾,𝑐^𝑠₁−2𝛽^𝑠𝛿^𝑠2_𝑖<𝑗𝛼_𝑖⁽²⁾𝛼_𝑗⁽²⁾=2𝛿^𝑠2_𝑖=1𝛼_𝑖⁽²⁾,𝑐^𝑠₁−2𝛽^𝑠𝛿^𝑠+1²_𝑖=1𝛼_𝑖⁽²⁾=4𝛼²_𝑖<𝑗𝛼_𝑖⁽²⁾𝛼_𝑗⁽²⁾+22𝛿^𝑠+1, (3.46) where 𝛼^𝑠2_𝑖=1𝛼_𝑖⁽²⁾²+𝛽^𝑠2_𝑖=1𝛼_𝑖⁽²⁾−2𝛿^𝑠𝛿+1=0,_𝑠2𝑖=1𝛼_𝑖⁽²⁾²−𝛽^𝑠2_𝑖=1𝛼_𝑖⁽²⁾+1²_𝑗<𝑘𝛼_𝑗⁽²⁾𝛼_𝑘⁽²⁾−2𝛼^𝑠2_𝑗<𝑘𝛼_𝑗⁽²⁾²𝛼_𝑘⁽²⁾²=0. (3.47)(3.48) The energy eigenvalue therefore is 𝐸^𝑠±_2𝜅=12𝐶_𝑠±𝐶²_𝑠+4𝑀²−4𝑀𝐶_𝑠−8𝜅𝐴+4𝐴+8𝐴𝐵+24𝑎^𝑠₁+4𝑎^𝑠₁4𝜅²+4𝜅+1−4𝑑^𝑠₁−4𝛽^𝑠21/2. (3.49) For the upper component of the wave function, we have 𝐹^𝑠_2𝜅(𝑟)=𝑁²_2𝜅𝑖=1𝑟−𝛼_𝑖⁽²⁾𝑟^{(1/2)(1+(2𝜅+1)2−4𝑑𝑠1)}−1exp2𝑎^𝑠₁𝑟²−𝑏^𝑠₁2𝑎^𝑠₁𝑟. (3.50) 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 (𝐴=𝐵=0), the pseudospin doublets, that is, states with quantum numbers (𝑛,𝑙,𝑗=𝑙+1/2) and (𝑛−1,𝑙+2,𝑗=𝑙+3/2) are degenerate. The degenerate states in the spin doublets are those with quantum numbers (𝑛,ℓ,𝑗=ℓ+1/2) and (𝑛,ℓ,𝑗=ℓ−1/2), where 𝑛,𝑙,and 𝑗 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 (𝐴=𝐵=0) are 𝑛𝑠_1/2,(𝑛−1)𝑑_3/2 for ℓ̃=1(ℓ=0), 𝑛𝑝_3/2,(𝑛−1)𝑓_5/2 for ℓ̃=2(ℓ=1), 𝑛𝑑_5/2,(𝑛−1)𝑔_7/2 for ℓ̃=3(ℓ=2), 𝑛𝑓_7/2,(𝑛−1)ℎ_9/2 for ℓ̃=4(ℓ=3), and so forth. For spin symmetry limit and (𝐴=𝐵=0) one can clearly see that the degeneracy occurs in (𝑛𝑝_1/2,𝑛𝑝_3/2) for ℓ=1, (𝑛𝑑_3/2,𝑛𝑑_5/2) for ℓ=2,(𝑛𝑓_5/2,𝑛𝑓_7/2) for ℓ=3, (𝑛𝑔_7/2,𝑛𝑔_9/2) for ℓ=4, and so forth.

Table 1 

Bound state for the pseudospin symmetry 𝑏=1, 𝑀=1?fm^-1, 𝐶_ps=0?fm^-1.

Table 2 

Bound state for the spin symmetry 𝑏=1, 𝑀=1?fm^-1, 𝐶_𝑠=0?fm^-1.

Table 3 

Energies in the pseudospin symmetry Limit for 𝐴=0.5, 𝐵=0.2, 𝑏=1, 𝑀=1?fm^-1.

Table 4 

Energies in the pseudospin symmetry limit for 𝐴=0.5, 𝐵=0.2, 𝑏=1, 𝐶_ps=0fm⁻¹.

Table 5 

Energies in the spin symmetry limit for 𝐴=0.5, 𝐵=0.2, 𝑏=1, 𝑀=1?fm^-1.

Table 6 

Energies in the spin symmetry limit for 𝐴=0.5, 𝐵=0.2, 𝑏=1, 𝐶_𝑠=0fm⁻¹.

4. Conclusion

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.