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arXiv:nucl-th/0006081 · Journal of Physics A Mathematical and General

On aq-analogue of the spin-orbit coupling

M. Micu, Fl. Stancu

Published Aug 3, 2000• 0 citations
arXivHTMLPDF

Abstract

Based on the tensor method, a q -analogue of the spin-orbit coupling is introduced in a q -deformed Schrödinger equation, previously derived for a central potential. Analytic expressions for the matrix elements of the representations j = ℓ±½ are derived. The spectra of the harmonic oscillator and the Coulomb potential are calculated numerically as a function of the deformation parameter, without and with the spin-orbit coupling. The harmonic oscillator spectrum presents strong analogies with the bound spectrum of a Woods-Saxon potential customarily used in nuclear physics. The Coulomb spectrum simulates relativistic effects. The addition of the spin-orbit coupling reinforces this picture.

DOI
10.1088/0305-4470/33/32/307

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Based on the tensor method, a q -analogue of the spin-orbit coupling is introduced in a q -deformed Schrödinger equation, previously derived for a central potential. Analytic expressions for the matrix elements of the representations j = ℓ±½ are derived. The spectra of the harmonic oscillator and the Coulomb potential are calculated numerically as a function of the deformation parameter, without and with the spin-orbit coupling. The harmonic oscillator spectrum presents strong analogies with the bound spectrum of a Woods-Saxon potential customarily used in nuclear physics. The Coulomb spectrum simulates relativistic effects. The addition of the spin-orbit coupling reinforces this picture.
We propose a q-deformation of the su(2)-invariant Schrodinger equation of a spinless particle in a central potential, which allows us not only to determine a deformed spectrum and the corresponding eigenstates, as in other approaches, but also to calculate the expectation values of some physically-relevant operators. Here we consider the case of the isotropic harmonic oscillator and of the quadrupole operator governing its interaction with an external field. We obtain the spectrum and wave functions both for q in R+ and generic q in S1, and study the effects of the q-value range and of the arbitrariness in the suq(2) Casimir operator choice. We then show that the quadrupole operator in l=0 states provides a good measure of the deformation influence on the wave functions and on the Hilbert space spanned by them.
We approximately solve the deformed Dirac equation for a new improved Hulth en plus a class of Yukawa potential including a Coulomb-like tensor interaction (IHCYPCTI for short) in the context of extended relativistic quantum mechanics ERQM symmetries including an improved Coulomb-like tensor interaction with an arbitrary spin-orbit coupling quantum number k.Within the framework of the spin and pseudospin (p-spin) symmetry, we obtain the global new energy eigenvalue which equals the energy eigenvalue in usual relativistic QM plus the corrected energy induced by three innitesimal additive parts of the Hamiltonian corresponding to the spin-orbit interaction, the new modied Zeeman and the rotational Fermi term by using the parametric of Bopp's shift method and standard perturbation theory with an approximation to the centrifugal term.The new values that we got appeared sensitive to the quantum numbers (j, k, l, l, s, s, m, m), the mixed potential depths (A, B), the range of the potential δ, and noncommutativity parameters (Θ, σ, χ).The mixed potential which in some particular cases gives solutions of dierent potentials: the improved Hulth en, the improved Yukawa potential and the improved Coulomb-like problem along with their bound state energies are obtained.
Abstract In this paper, the Dirac equation with the q-deformed Scarf potential for spin symmetry is solved for an arbitrary spin-orbit quantum number κ, in the presence of Coulomb-like potential tensor. Using the Feynman path integral formalism and the Pekeris approximation of the centrifugal term, we obtain the bound state energy eigenvalues and the associated spinor of the Dirac particle. Furthermore, this method is used to determine the spectrum of two diatomic molecules Li 2(61Π u ) and KRb(B −1Π). The obtained results are compared to the experimental and numerical ones.
In this work, we obtained an exact solution to Schrodinger equation using q-deformed Woods-Saxon plus modified Coulomb potential Using conventional Nikiforov-Uvarov method.We also obtained the energy eigen value and its associated total wave function .This potential with some suitable conditions reduces to two well known potentials namely: the Yukawa and coulomb potential.Finally, we obtained the numerical results for energy eigen value with different values of q as dimensionless parameter.The result shows that the values of the energies for different quantum number(n) is negative(bound state condition) and increases with an increase in the value of the dimensionless parameter(arbitrary constant).The graph in figure (1) shows the different energy levels for a particular quantum number.
In this work, we propose a new potential called the "q-deformed Woods-Saxon plus hyperbolic tangent potential." We derive the generalized Schrödinger equation for quantum mechanical systems with position-dependent masses under these potentials using the Nikiforov-Uvarov method, with the mass relationship defined as m(x)=m_1⁄((1+qe^(-2λx))). The solutions to this equation, expressed in terms of hypergeometric functions and Jacobi polynomials, offer insights into the quantum behavior of particles. The energy eigenvalues depend on system parameters such as the deformation parameter q, potential parameters, and quantum numbers. We analyzed the effect of the deformation parameter q numerically and visually using different values of these parameters.