An analytical solution of the Dirac equation with a Cornell potential, with identical scalar and vectorial parts, is presented. The solution is obtained by using the linear potential solution, related to Airy functions, multiplied by another function
to be determined. The energy levels are obtained and we notice that they obey a band structure.
This Letter is based on the $kappa$-Dirac equation, derived from the $kappa$-Poincar{e}-Hopf algebra. It is shown that the $kappa$-Dirac equation preserves parity while breaks charge conjugation and time reversal symmetries. Introducing the Dirac osc
illator prescription, $mathbf{p}tomathbf{p}-imomegabetamathbf{r}$, in the $kappa$-Dirac equation, one obtains the $kappa$-Dirac oscillator. Using a decomposition in terms of spin angular functions, one achieves the deformed radial equations, with the associated deformed energy eigenvalues and eigenfunctions. The deformation parameter breaks the infinite degeneracy of the Dirac oscillator. In the case where $varepsilon=0$, one recovers the energy eigenvalues and eigenfunctions of the Dirac oscillator.
The effects of a Lorentz symmetry violating background vector on the Aharonov-Casher scattering in the nonrelativistic limit is considered. By using the self-adjoint extension method we found that there is an additional scattering for any value of th
e self-adjoint extension parameter and non-zero energy bound states for negative values of this parameter. Expressions for the energy bound states, phase-shift and the scattering matrix are explicitly determined in terms of the self-adjoint extension parameter. The expression obtained for the scattering amplitude reveals that the helicity is not conserved in this scenario.
The Aharonov-Casher problem in the presence of a Lorentz-violating background nonminimally coupled to a spinor and a gauge field is examined. Using an approach based on the self-adjoint extension method, an expression for the bound state energies is
obtained in terms of the physics of the problem by determining the self-adjoint extension parameter.
In this letter we study the Aharonov-Bohm problem for a spin-1/2 particle in the quantum deformed framework generated by the $kappa$-Poincar{e}-Hopf algebra. We consider the nonrelativistic limit of the $kappa$-deformed Dirac equation and use the spi
n-dependent term to impose an upper bound on the magnitude of the deformation parameter $varepsilon$. By using the self-adjoint extension approach, we examine the scattering and bound state scenarios. After obtaining the scattering phase shift and the $S$-matrix, the bound states energies are obtained by analyzing the pole structure of the latter. Using a recently developed general regularization prescription [Phys. Rev. D. textbf{85}, 041701(R) (2012)], the self-adjoint extension parameter is determined in terms of the physics of the problem. For last, we analyze the problem of helicity conservation.
Quantum walks constitute important tools in different applications, especially in quantum algorithms. To a great extent their usefulness is due to unusual diffusive features, allowing much faster spreading than their classical counterparts. Such beha
vior, although frequently credited to intrinsic quantum interference, usually is not completely characterized. Using a recently developed Greens function approach [Phys. Rev. A {bf 84}, 042343 (2011)], here it is described -- in a rather general way -- the problem dynamics in terms of a true sum over paths history a la Feynman. It allows one to explicit identify interference effects and also to explain the emergence of superdiffusivity. The present analysis has the potential to help in designing quantum walks with distinct transport properties.
In this work bound states for the Aharonov-Casher problem are considered. According to Hagens work on the exact equivalence between spin-1/2 Aharonov-Bohm and Aharonov-Casher effects, is known that the $boldsymbol{ abla}cdotmathbf{E}$ term cannot be
neglected in the Hamiltonian if the spin of particle is considered. This term leads to the existence of a singular potential at the origin. By modeling the problem by boundary conditions at the origin which arises by the self-adjoint extension of the Hamiltonian, we derive for the first time an expression for the bound state energy of the Aharonov-Casher problem. As an application, we consider the Aharonov-Casher plus a two-dimensional harmonic oscillator. We derive the expression for the harmonic oscillator energies and compare it with the expression obtained in the case without singularity. At the end, an approach for determination of the self-adjoint extension parameter is given. In our approach, the parameter is obtained essentially in terms of physics of the problem.
