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Approximate bound state solutions of Dirac equation with Hulth\'{e}n potential including Coulomb-like tensor potential

Sameer M. Ikhdair, Ramazan Sever

arXiv:1001.4327v2math-phmath.MPquant-ph

TL;DR

The paper addresses bound-state solutions of the Dirac equation for Hulthén scalar and vector potentials with Coulomb-like tensor coupling under spin and pseudospin symmetry. It applies a parametric NU method to derive approximate energy equations and spinor wavefunctions, including zero-tensor and non-relativistic limits.

  • Problem

    The work investigates Dirac bound states for Hulthén scalar and vector potentials with Coulomb-like tensor coupling and arbitrary κ under spin and pseudospin symmetry.

    Source: §I · pp. 3–4

  • Method

    A parametric generalization of the NU method and an approximation for the tensor-generated centrifugal term yield closed-form approximate solutions under both symmetries.

    Source: §I · p. 3, §IV · p. 17

  • Results

    The authors obtain approximate energy equations and corresponding wavefunctions, with zero-tensor and non-relativistic limits also considered.

    Source: §IV · p. 17

  • Takeaways & Limitations

    A parametric transformation reduces the Dirac solutions to Schrödinger solutions.

    Source: §I · p. 4

    The centrifugal-term approximation is stated to be valid for small screening parameters, and its treatment is limited to very few lowest energy states.

    Source: A. Spin symmetric bound state solution · p. 10

Abstract

from arXiv · show

We solve the Dirac equation approximately for the attractive scalar $S(r)$ and repulsive vector $V(r)$ Hulth\'{e}n potentials including a Coulomb-like tensor potential with arbitrary spin-orbit coupling quantum number $\kappa .$ In the framework of the spin and pseudospin symmetric concept, we obtain the analytic energy spectrum and the corresponding two-component upper- and lower-spinors of the two Dirac particles by means of the Nikiforov-Uvarov method in closed form. The limit of zero tensor coupling and the non-relativistic solution are obtained. The energy spectrum for various levels is presented for several $\kappa $ values under the condition of exact spin symmetry in the presence or absence of tensor coupling.

I. INTRODUCTION

The introduction situates spin and pseudospin symmetry in Dirac systems, reviews prior potential solutions, and motivates approximate Hulthén bound states with Coulomb-like tensor coupling for arbitrary κ.

I. INTRODUCTION

  • Spin symmetry corresponds to a constant difference potential, while pseudospin symmetry corresponds to a constant sum potential; exact zero difference or sum yields Dirac-Hamiltonian SU(2) symmetries.Pseudospin symmetry is approximate in real nuclei, with its quality depending on pseudo-centrifugal and pseudospin-orbit potentials.Source: §I · p. 2
  • Earlier work addressed Dirac solutions for several potentials and κ values, including the authors’ NU-method studies of generalized Woods-Saxon and Rosen-Morse potentials.The paper motivates the Hulthén case with tensor coupling as a further analytically tractable setting.Source: §I · p. 3
  • The paper targets analytic bound-state solutions for scalar and vector Hulthén potentials with Coulomb-like tensor coupling and arbitrary κ under spin or pseudospin symmetry.It also considers zero-tensor and non-relativistic limits.Source: §I · pp. 3–4
  • The Coulomb-like tensor interaction generates a new spin-orbit centrifugal term and may provide a different form of spin-orbit coupling with possible physical applications.The introduced approximation scheme handles centrifugal and pseudo-centrifugal terms for arbitrary κ.Source: §I · pp. 3–4
  • The Hulthén potential is Coulomb-like near the origin but decays exponentially at large distances, giving it less bound-state capacity than the Coulomb potential.The potential has applications across nuclear, atomic, molecular, and chemical physics.Source: §I · p. 4

II. THE NIKIFOROV-UVAROV METHOD

The section outlines the Nikiforov-Uvarov method for second-order hypergeometric-type equations and extends it parametrically to exponential-type potentials.

II. THE NIKIFOROV-UVAROV METHOD

  • The NU method solves second-order differential equations of hypergeometric type using special orthogonal functions.Its polynomial solutions are expressed through a Rodrigues relation and a weight function.Source: §II · pp. 5–6
  • The method recasts the equation into hypergeometric-type form by choosing a factorization and polynomial coefficients with specified degrees.The section states that σ and the transformed eσ are at most quadratic, while eτ is linear.Source: §II · pp. 5–6
  • The eigenvalue condition links λ and λ_n, with the NU construction selecting solutions through a zero-discriminant condition and a family indexed by n.The stated family uses n=0, 1, 2, … .Source: §II · p. 6
  • A parametric generalization uses a change of variable z=z(r) for central and non-central exponential-type potentials, yielding analytic polynomials and constants for radial equations.Its parameters are determined during the solution procedure and the expressions are collected in Appendix A.Source: §II · p. 6

III. DIRAC EQUATION WITH A TENSOR COUPLING

The section formulates the spherical Dirac problem with scalar, vector, and tensor potentials, then derives coupled radial equations and symmetry-specific second-order equations for the spinor components.

III. DIRAC EQUATION WITH A TENSOR COUPLING

  • The spherical Dirac Hamiltonian includes attractive scalar S(r), repulsive vector V(r), and tensor U(r) potentials, with radial upper and lower spinor components.The radial functions are subject to vanishing boundary conditions at the origin and infinity.Source: §III · pp. 7–8
  • The spin-orbit quantum number κ labels aligned and unaligned spin states, and determines the orbital and pseudo-orbital angular momenta of the spinor components.For a central field, κ is an eigenvalue of the spin-orbit matrix operator that commutes with the Dirac Hamiltonian.Source: §III · p. 7
  • Substitution of the spinor form yields coupled radial Dirac equations; eliminating either component gives second-order equations for the upper and lower radial functions.The difference and sum potentials are Δ(r)=V(r)−S(r) and Σ(r)=V(r)+S(r).Source: §III · p. 8
  • With Hulthén sum or difference potentials and Coulomb-like tensor coupling, the equations are exactly solvable for the stated special κ cases because of the spin-orbit centrifugal term.The text gives κ=0,−1 and κ=0,1 for the respective equations.Source: §III · p. 9

A. Spin symmetric bound state solution

For spin symmetry, the paper treats the Hulthén potential with a Coulomb-like tensor interaction using the Nikiforov–Uvarov method and a centrifugal-term approximation. It derives energy levels and upper- and lower-spinor wavefunctions for arbitrary κ.

A. Spin symmetric bound state solution

  • The centrifugal-term approximation is intended for small screening parameter δ, corresponding to large r_0 or r ≪ r_0.The paper notes that the singular orbital term limits this approximation to very few of the lowest-energy states.Source: A. Spin symmetric bound state solution · p. 10
  • The energy spectrum is given by a transcendental condition, including the relation (E_nκ − M)(E_nκ + M − C_s) = d_0(κ + H)(κ + H + 1).The treatment requires the bound-state parameter λ_nκ to be positive; n counts radial-wavefunction nodes.Source: A. Spin symmetric bound state solution · p. 11
  • The paper also gives zero-tensor-coupling and non-relativistic limits of the spin-symmetric spectrum.The zero-tensor limit is compared with a Schrödinger-equation result for the Hulthén potential.Source: A. Spin symmetric bound state solution · p. 13
  • The radial solution yields an upper-spinor wavefunction and a separately obtained lower component G_nκ(r).The upper radial function is also rewritten using Jacobi polynomials.Source: A. Spin symmetric bound state solution · pp. 13–14
  • With the usual approximation, the reported spin-unaligned states are degenerate for H ≠ 0 and H = 0, while the new approximation removes those degeneracies.For H ≠ 0, the listed np_1/2, nd_3/2, and nf_5/2 energies are higher than their H = 0 counterparts.Source: A. Spin symmetric bound state solution · p. 15

B. Pseudospin symmetric bound state solutions

The pseudospin-symmetric Hulthén problem is treated with a Coulomb-like tensor interaction, yielding an energy-spectrum equation and conditions for its solutions. Parameter transformations connect this case to the spin-symmetric solution, and a specified limit recovers the Klein–Gordon solution.

B. Pseudospin symmetric bound state solutions

  • The pseudospin setup takes the Hulthén potential for Δ(r), with Σ(r) constant; the radial equation reduces to a Schrödinger-type equation.The energy-equation derivation also specifies a largest integer n_max less than r_0.Source: B. Pseudospin symmetric bound state… · pp. 15–16
  • The energy spectrum is determined by a transcendental equation obtained by inserting the model parameters into the spectrum relation.The pseudospin parameter must be real and positive for a real solution.Source: B. Pseudospin symmetric bound state… · p. 16
  • The pseudospin energy must satisfy E_nκ > M + C_s, and the energy relation obeys (E_nκ + M)(E_nκ − M − C_ps) = d_0(κ + H)(κ + H − 1).The energy spectrum depends on n and κ through κ(κ − 1).Source: B. Pseudospin symmetric bound state… · p. 16
  • The Klein–Gordon solution is recovered when C_ps = H = d_0 = 0 and the stated parameter substitutions are applied.The pseudospin solution is obtained from the spin-symmetric one through parameter transformations, including swapping the upper and lower spinors and changing the sign of energy.Source: B. Pseudospin symmetric bound state… · pp. 16–17

IV. CONCLUDING REMARKS

The work derives approximate analytic energy equations and wavefunctions for the Hulthén potential coupled to a Coulomb-like tensor under spin and pseudospin symmetry. The solutions recover the usual Hulthén Dirac results when tensor coupling is absent and admit a non-relativistic limit under exact spin symmetry.

IV. CONCLUDING REMARKS

  • The Coulomb-like tensor interaction adds a centrifugal-like term, which is approximated to obtain closed-form solutions for the energy equation and wavefunctions.The wavefunctions are expressed using Jacobi polynomials or hypergeometric functions.Source: §IV · p. 17
  • Under exact spin symmetry, the relativistic solution reduces to its non-relativistic limit through an appropriate parameter mapping.Source: §IV · p. 17
  • With zero tensor coupling, the results reduce to the usual Hulthén Dirac solutions for arbitrary spin-orbit quantum number κ.Source: §IV · pp. 17–18

Appendix A: Parametric Generalization of the NU Method

The appendix lays out the parametric generalization of the Nikiforov–Uvarov method by specifying the polynomials, energy equation, wavefunctions, and relevant constants. Its wavefunctions are given in Jacobi-polynomial or hypergeometric form, subject to stated parameter constraints.

Appendix A: Parametric Generalization of the NU Method

  • The parametric NU formulation provides key polynomials, a general energy equation, general wavefunctions, and relevant constants.Source: Appendix A · p. 19
  • The appendix presents the essential polynomials and associated energy and wavefunction expressions for exponential potentials.It describes the formulation as parameterized for any arbitrary exponential potential.Source: Appendix A · p. 19
  • The general wavefunctions use Jacobi polynomials and can also be expressed with the hypergeometric function 2F1.The stated domain requires R_ν > −1, R_μ > −1, and x ∈ [−1, 1] for the Jacobi polynomials; the hypergeometric form specifies z ∈ [0, 1/c_3].Source: Appendix A · p. 19

Appendix B: Normalization of the radial wavefunctions

The radial-wavefunction normalization constant is obtained from the normalization condition after a coordinate transformation. The integral calculation uses a cited integral formula with stated parameter conditions.

Appendix B: Normalization of the radial wavefunctions

  • The normalization constant N_nκ is calculated from the radial normalization condition using x = 1 − 2e^(−r/r_0).Source: Appendix B · p. 20
  • The normalization derivation assumes λ_nκ > 0 and η_κ ≥ 1.Source: Appendix B · p. 20
  • The calculation uses an integral formula valid for R_ν > 0 and R_μ > −1.Source: Appendix B · p. 20
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