Electromagnetic response of confined Dirac particles

Paris, Mark W.

doi:10.1103/physrevc.68.025201

Cited by 14 publications

(18 citation statements)

References 15 publications

(30 reference statements)

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“…Paris [14] and by Soares de Castro and Franklin [15], who discussed analytic solutions. We, however, have adopted a numerical approach and, in this paper, we discuss how to go about solving these coupled ODE's numerically.…”

Section: Introductionmentioning

confidence: 99%

Solving the radial Dirac equations: a numerical odyssey

Silbar¹,

Goldman²

2010

Eur. J. Phys.

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We discuss, in a pedagogical way, how to solve for relativistic wave functions from the radial Dirac equations. After an brief introduction, in Section II we solve the equations for a linear Lorentz scalar potential, V s (r), that provides for confinement of a quark. The case of massless u and d quarks is treated first, as these are necessarily quite relativistic. We use an iterative procedure to find the eigenenergies and the upper and lower component wave functions for the ground state and then, later, some excited states. Solutions for the massive quarks (s, c, and b) are also presented. In Section III we solve for the case of a Coulomb potential, which is a timelike component of a Lorentz vector potential, V v (r). We re-derive, numerically, the (analytically well-known) relativistic hydrogen atom eigenenergies and wave functions, and later extend that to the cases of heavier one-electron atoms and muonic atoms. Finally, Section IV finds solutions for a combination of the V s and V v potentials. We treat two cases. The first is one in which V s is the linear potential used in Sec. II and V v is Coulombic, as in Sec. III. The other is when both V s and V v are linearly confining, and we establish when these potentials give a vanishing spin-orbit interaction (as has been shown to be the case in quark models of the hadronic spectrum).

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Section: Introductionmentioning

confidence: 99%

Solving the radial Dirac equations: a numerical odyssey

Silbar¹,

Goldman²

2010

Eur. J. Phys.

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“…One such calculation was made by Paris [13] in a massless confined Dirac particle, in which V v = V s . It would be interesting to see how a Klein-Gordon particle would behave under the same potentials.…”

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confidence: 99%

Spin and pseudospin symmetries and the equivalent spectra of relativistic spin-1/2 and spin-0 particles

2007

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We show that the conditions which originate the spin and pseudospin symmetries in the Dirac equation are the same that produce equivalent energy spectra of relativistic spin-1/2 and spin-0 particles in the presence of vector and scalar potentials. The conclusions do not depend on the particular shapes of the potentials and can be important in different fields of physics. When both scalar and vector potentials are spherical, these conditions for isospectrality imply that the spin-orbit and Darwin terms of either the upper component or the lower component of the Dirac spinor vanish, making it equivalent, as far as energy is concerned, to a spin-0 state. In this case, besides energy, a scalar particle will also have the same orbital angular momentum as the (conserved) orbital angular momentum of either the upper or lower component of the corresponding spin-1/2 particle. We point out a few possible applications of this result. When describing some strong interacting systems it is often useful, because of simplicity, to approximate the behavior of relativistic spin-1/2 particles by scalar spin-0 particles obeying the Klein-Gordon equation. An example is the case of relativistic quark models used for studying quark-hadron duality because of the added complexity of structure functions of Dirac particles as compared to scalar ones. It turns out that some results (e.g., the onset of scaling in some structure functions) almost do not depend on the spin structure of the particle [1]. In this work we will give another example of an observable, the energy, whose value may not depend on the spinor structure of the particle, i.e., whether one has a spin-1/2 or a spin-0 particle. We will show that when a Dirac particle is subjected to scalar and vector potentials of equal magnitude, it will have exactly the same energy spectrum as a scalar particle of the same mass under the same potentials. As we will see, this happens because the spin-orbit and Darwin terms in the second-order equation for either the upper or lower spinor component vanish when the scalar and vector potentials have equal magnitude. It is not uncommon to find physical systems in which strong interacting relativistic particles are subject to Lorentz scalar potentials (or position-dependent effective masses) that are of the same order of magnitude of potentials which couple to the energy (time components of Lorentz four-vectors). For instance, the scalar and vector (hereafter meaning time-component of a four-vector potential) nuclear mean-field potentials have opposite signs but similar magnitudes, whereas relativistic models of mesons with a heavy and a light quark, like D-or B-mesons, explain the observed small spin-orbit splitting by having vector and scalar potentials with the same sign and similar strengths [2].It is well-known that all the components of the free Dirac spinor, i.e., the solution of the free Dirac equation, satisfy the free Klein-Gordon equation. Indeed, from the free Dirac equationone getswhere use has been made of the relation γ µ γ ν ∂ µ ∂...

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“…Note that we use different scalar and vector potentials, in contrast to [21,22], where V s = V v is used to simplify the calculations.…”

Section: The Modelmentioning

confidence: 99%

Modeling quark-hadron duality in polarization observables

Jeschonnek

Orden

2005

Phys. Rev. D

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We apply a model for the study of quark-hadron duality in inclusive electron scattering to the calculation of spin observables. The model is based on solving the Dirac equation numerically for a scalar confining linear potential and a vector color Coulomb potential. We qualitatively reproduce the features of quark-hadron duality for all potentials considered, and discuss the onset of scaling and duality for the responses, spin structure functions, and polarization asymmetries. Duality may be applied to gain access to kinematic regions which are hard to access in deep inelastic scattering, namely for $x_{Bj} \to 1$, and we discuss which observables are most suitable for this application of duality

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Electromagnetic response of confined Dirac particles

Cited by 14 publications

References 15 publications

Solving the radial Dirac equations: a numerical odyssey

Solving the radial Dirac equations: a numerical odyssey

Spin and pseudospin symmetries and the equivalent spectra of relativistic spin-1/2 and spin-0 particles

Modeling quark-hadron duality in polarization observables

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