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A. S. de Castro

Publications and source records attributed to A. S. de Castro.

At least 19 recordsLinked to original sources
arXiv2026

A generalized harmonic oscillator problem for a spin-1/2 fermion

The exact bound-state wavefunctions and the corresponding energy equation are calculated for a new generalized harmonic oscillator problem describing a spin-1/2 fermion in 3+1-dimensions, involving scalar, vector, and tensor couplings acting simultaneously within a particular plane of motion. For the scalar and vector coupling, singular harmonic oscillator shapes are considered, such that the singular term is needed to allow analytical solutions for the wavefunctions while preserving binding under adequate conditions. For the tensor sector,the Dirac oscillator potential is employed, which adds another independent binding mechanism to the problem. The exact bound-state solutions are computed by specifically tuning coefficients for an appropriate pair of \textit{Ansätze} for the radial functions, which leads to wavefunctions in terms of generalized Laguerre polynomials. Although the energy equation cannot provide a general expression for the energy spectrum, specific constraints on the quantum numbers as simple functions of the external potential parameters can be derived for it, which determines the conditions for bound solutions to exist, and of what type: particle, antiparticle or both. It is shown that the results can be simply mapped to the spherically symmetric analogue problem, and this is used to show that the general result encompass several previous particular cases of spherically symmetric harmonic oscillator problems in the Dirac equation available in the literature.

quant-ph↗
arXiv2026

A generalized Coulomb problem for a spin-1/2 fermion

We study the Dirac equation in 3+1 dimensions with a general combination of scalar, vector and tensor interactions with arbitrary strengths, all of them described by central Coulomb potentials acting on a particular plane of motion. For the tensor coupling a constant term is also included, since this gives rise to an effective Coulomb potential, which is necessary for the formation of bound states in a pure tensor coupling configuration. The exact bound-state solutions for this generalized Coulomb problem are computed by exploiting the freedom in choosing the coefficients of the \textit{Ansätze} for the radial functions, which leads to wave functions in terms of generalized Laguerre polynomials. From the quantization condition, the exact energy spectrum is also determined and its dependence on the parameters of the potentials is discussed. We show that similar features of the equations for the problem in the plane and the spherically symmetric problem allow a simple and direct mapping between them, thereby providing the solution to the spherical Coulomb problem. Our results are validated by showing that the solutions correctly encompass several previous solutions available in the literature for particular cases of this problem, for which we further develop the analysis of the parameters. We also derive two new particular cases not yet reported in the literature: the case of breaking of spin and pseudospin symmetries by the addition of a Coulomb plus constant tensor potential and the problem of a scalar plus tensor Coulomb potentials.

quant-ph↗
arXiv2024

Symmetry generators and quantum numbers for fermionic circularly symmetric motion

The planar dynamics of spin-1/2 quantum relativistic particles is important for several physical systems. In this paper we derive, by a simple method, the generators for the continuous symmetries of the 3+1 Dirac equation for planar motion, when there is circular symmetry, i.e., the interactions depend only on the radial coordinate. We consider a general set of potentials with different Lorentz structures. These generators allow for several minimal complete sets of commuting observables and their corresponding quantum numbers. We show how they can be used to label the general eigenspinors for this problem. We also derive the generators of the spin and pseudospin symmetries for this planar Dirac problem, which arise when the vector and scalar potentials have the same magnitude and tensor potential and the space components of the four-vector potential are absent. We investigate the associated energy degeneracies and compare them to the known degeneracies in the spherically symmetric 3+1 Dirac equation.

quant-ph↗
arXiv2019

Pure Coulomb tensor interaction in the Dirac equation

In this work we compute bound solutions for particles and anti-particles of the Dirac equation for a pure tensor radial Coulomb potential plus a constant. We find that the binding depends on the sign the tensor constant potential, and allows only bound solutions for a certain sign and magnitude of the $κ$ quantum number, which is related to the spin-orbit coupling in the Dirac equation. This relation is reversed for anti-particle solutions. On the other hand, the Coulomb tensor potential, although not biding, changes the range of $κ$ values for which there are bound solutions.

quant-ph↗
arXiv2017

Solutions of the three-dimensional radial Dirac equation from the Schrödinger equation with one-dimensional Morse potential

New exact analytical bound-state solutions of the radial Dirac equation in 3+1 dimensions for two sets of couplings and radial potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of equations that can be transformed into polynomial equations. Several analytical results found in the literature, including the Dirac oscillator, are obtained as particular cases of this unified approach.

hep-th↗
arXiv2017

New solutions of the D-dimensional Klein-Gordon equation via mapping onto the nonrelativistic one-dimensional Morse potential

New exact analytical bound-state solutions of the D-dimensional Klein-Gordon equation for a large set of couplings and potential functions are obtained via mapping onto the nonrelativistic bound-state solutions of the one-dimensional generalized Morse potential. The eigenfunctions are expressed in terms of generalized Laguerre polynomials, and the eigenenergies are expressed in terms of solutions of irrational equations at the worst. Several analytical results found in the literature, including the so-called Klein-Gordon oscillator, are obtained as particular cases of this unified approach

hep-th↗
arXiv2016

Fermions in a mixed vector-scalar double-step potential via continuous chiral transformation

The behaviour of fermions in the background of a double-step potential is analyzed with a general mixing of scalar and vector couplings via continuous chiral-conjugation transformation. Provided the vector coupling does not exceed the scalar coupling, a Sturm-Liouville approaching for the double-step potential shows that the transmission coefficient exhibits oscillations and that a finite set of intrinsically relativistic bound-state solutions might appear as poles of the transmission amplitude in a strong coupling regime. An isolated bound-state solution resulting from coupled first-order equations might also come into sight. It is also shown that all those possible bound solutions disappear asymptotically as one approaches the conditions for the realization of the so-called spin and pseudospin symmetries in a four-dimensional space-time. Furthermore, we show that due to the additional mass acquired by the fermion from the scalar background the high localization of the fermion in an extreme relativistic regime does not violate the Heisenberg uncertainty principle.

hep-th↗
arXiv2016

A large class of bound-state solutions of the Schrödinger equation via Laplace transform of the confluent hypergeometric equation

It is shown that analytically soluble bound states of the Schrödinger equation for a large class of systems relevant to atomic and molecular physics can be obtained by means of the Laplace transform of the confluent hypergeometric equation. It is also shown that all closed-form eigenfunctions are expressed in terms of generalized Laguerre polynomials. The generalized Morse potential is used as an illustration.

quant-ph↗
arXiv2015

Transmission coefficient and two-fold degenerate discrete spectrum of spin-1 bosons in a double-step potential

The scattering of spin-1 bosons in a nonminimal vector double-step potential is described in terms of eigenstates of the helicity operator and it is shown that the transmission coefficient is insensitive to the choice of the polarization of the incident beam. Poles of the transmission amplitude reveal the existence of a two-fold degenerate spectrum. The results are interpreted in terms of solutions of two coupled effective Schrödinger equations for a finite square well with additional $δ$-functions situated at the borders.

hep-th↗
arXiv2015

Relativistic Coulomb scattering of spinless bosons

The relativistic scattering of spin-0 bosons by spherically symmetric Coulomb fields is analyzed in detail with an arbitrary mixing of vector and scalar couplings. It is shown that the partial wave series reduces the scattering amplitude to the closed Rutherford formula exactly when the vector and scalar potentials have the same magnitude, and as an approximation for weak fields. The behavior of the scattering amplitude near the conditions that furnish its closed form is also discussed. Strong suppressions of the scattering amplitude when the vector and scalar potentials have the same magnitude are observed either for particles or antiparticles with low incident momentum. We point out that such strong suppressions might be relevant in the analysis of the scattering of fermions near the conditions for the spin and pseudospin symmetries. From the complex poles of the partial scattering amplitude the exact closed form of bound-state solutions for both particles and antiparticles with different scenarios for the coupling constants are obtained. Perturbative breaking of the accidental degeneracy appearing in a pair of special cases is related to the nonconservation of the Runge-Lenz vector.

nucl-th↗
arXiv2014

Corroborating the equivalence between the Duffin-Kemmer-Petiau and the Klein-Gordon and Proca equations

It is shown that the Hamiltonian version of the Duffin-Kemmer-Petiau theory with electromagnetic coupling brings about a source term at the current. It is also shown that such a source term disappears from the scenario if one uses the correct physical form for the Duffin-Kemmer-Petiau field, regardless the choice for representing the Duffin-Kemmer-Petiau matrices. This result is used to fix the ambiguity in the electromagnetic coupling in the Duffin-Kemmer-Petiau theory. Moreover, some widespread misconceptions about the Hermiticity in the Duffin-Kemmer-Petiau theory are discussed.

hep-th↗
arXiv2014

Scattering and bound states of fermions in a mixed vector-scalar smooth step potential

The scattering of a fermion in the background of a smooth step potential is considered with a general mixing of vector and scalar Lorentz structures with the scalar coupling stronger than or equal to the vector coupling. Charge-conjugation and chiral-conjugation transformations are discussed and it is shown that a finite set of intrinsically relativistic bound-state solutions appears as poles of the transmission amplitude. It is also shown that those bound solutions disappear asymptotically as one approaches the conditions for the realization of the so-called spin and pseudospin symmetries in a four-dimensional space-time.

hep-th↗
arXiv2013

Stationary states of fermions in a sign potential with a mixed vector-scalar coupling

The scattering of a fermion in the background of a sign potential is considered with a general mixing of vector and scalar Lorentz structures with the scalar coupling stronger than or equal to the vector coupling under the Sturm-Liouville perspective. When the vector coupling and the scalar coupling have different magnitudes, an isolated solution shows that the fermion under a strong potential can be trapped in a highly localized region without manifestation of Klein's paradox. It is also shown that the lonely bound-state solution disappears asymptotically as one approaches the conditions for the realization of spin and pseudospin symmetries.

hep-th↗
arXiv2013

Harmonic oscillator: an analysis via Fourier series

The Fourier series method is used to solve the homogeneous equation governing the motion of the harmonic oscillator. It is shown that the general solution to the problem can be found in a surprisingly simple way for the case of the simple harmonic oscillator. It is also shown that the damped harmonic oscillator is susceptible to the analysis.

physics.gen-ph↗
arXiv2013

Missing solution in a Cornell potential

Missing bound-state solutions for fermions in the background of a Cornell potential consisting of a mixed scalar-vector-pseudoscalar coupling is examined. Charge-conjugation operation, degeneracy and localization are discussed.

hep-th↗