In this study, we have constructed a generalized momentum operator based on the notion of backward–forward coordinates characterized by a low dynamical nonlocality decaying exponentially with position. We have derived the associated Schrodinger equation and we have studied the dynamics of a particle characterized by an exponentially decreasing position-dependent mass following the arguments of von Roos. In the absence of magnetic fields, it was observed that the dynamics of the particle is similar to the harmonic oscillator with damping and its energy state is affected by nonlocality. We have also studied the dynamics of a charged particle in the presence of Morse–Coulomb potentials and external magnetic and Aharonov-Bohm flux fields. Both the energy states and the thermodynamical properties were obtained. It was observed that all these physical quantities are affected by nonlocality and that for small magnetic fields and high quantum magnetic numbers, the entropy of the system decreases with increasing temperature unless the nonlocal parameter is negative. For positive value of the nonlocal parameter, it was found that the entropy increases with temperature and tends toward an asymptotically stable value similar to an isolated system.

Nonlocal Thermodynamics Properties of Position-Dependent Mass Particle in Magnetic and Aharonov-Bohm Flux Fields

In this study, we have constructed a generalized Schrodinger equation in fractal dimensions characterized by an effective potential which is generated by a position-dependent mass. Our analysis is based on the fractal anisotropy and product-like fractal measure approach introduced by Li and Ostoja-Starzewski in their formulation of continuum media. Our analysis is based on the von Roos phenomenological approach and is characterized by a parametric ordering ambiguity $${{\varvec{\upalpha}}},{{\varvec{\upbeta}}},{{\varvec{\upgamma}}}$$
 which arises in the Schrodinger Hamiltonian as a result of the non-unique representation of the kinetic energy term. We have selected numerical values of von Roos parameters different from the trivial case, and we have solved the corresponding generalized Schrodinger equation for different forms of the position-dependent mass. This study was proved to be practical to analyze several effective physical properties of nanostructures including quantum dots and quantum box that are relevant to overall photonics and exciton spectroscopy in quantum-well structures.

Quantum dots and cuboid quantum wells in fractal dimensions with position-dependent masses

In the context of the Dunkl derivative, we present the Dunkl–Duffin–Kemmer–Petiau oscillator (DDKPO) model of spin 0 and 1. For the spin 0 case , the exact solutions are determined, the wave functions are obtained using Cartesian and polar coordinates in the general form for the different eigenvalues of the reflection operators, and the corresponding spectrum energies are deduced. For the spin 1 case, the problem is complicated by the direct method, we obtain a differential equation of order four and we limit ourselves only to the study of the non-relativistic case.

The Dunkl–Duffin–Kemmer–Petiau Oscillator

Recent studies show that deformations in quantum mechanics are inevitable. In this contribution, we consider a relativistic quantum mechanical differential equation in the presence of Dunkl operator-based deformation and we investigate solutions for two important problems in three-dimensional spatial space. To this end, after introducing the Dunkl quantum mechanics, we examine the Dunkl–Klein–Gordon oscillator solutions with the Cartesian and spherical coordinates. In both coordinate systems, we find that the differential equations are separable and their eigenfunctions can be given in terms of the associate Laguerre and Jacobi polynomials. We observe how the Dunkl formalism is affecting the eigenvalues as well as the eigenfunctions. As a second problem, we examine the Dunkl–Klein–Gordon equation with the Coulomb potential. We obtain the eigenvalue, their corresponding eigenfunctions, and the Dunkl-fine structure terms. 

/pdf/dunkl-klein-gordon-equation-in-three-dimensions-the-klein-34u1yx5y.pdf

Dunkl–Klein–Gordon Equation in Three-Dimensions: The Klein–Gordon Oscillator and Coulomb Potential

In the context of new type of the extended uncertainty principle using the displacement operator method, we present an exact solution of some problems such as: the Klein–Gordon particle confined in a one dimensional box, the scalar particle with linear vector and scalar potentials and the case of inversely linear vector and scalar potentials of Coulomb-type. The expressions of bound state energies and the associated wave functions are exactly determined for these three cases.

Scalar Particle in New Type of the Extended Uncertainty Principle

The Green function for Dirac oscillator in 
$$(1+1)$$

 dimension in the context of the extended uncertainty principle (EUP) is calculated exactly via the path integral formalism. The spectrum energy is determined, the corresponding wave functions suitably normalized are derived and they are expressed by the Gegenbauer’s polynomials. Special cases are considered.

The EUP Dirac Oscillator: A Path Integral Approach

In this work, we present the exact solution of the bosonic oscillator subjected to the interaction of an uniform electric field on the de Sitter and anti de Sitter spaces. In both cases, we extract the energy spectrum and the associated wave functions. For de Sitter case, the wave functions are expressed according to the Jacobi polynomial and for anti de Sitter case, the wave functions are expressed according to Romanovski’s polynomial. Finally the limiting cases are studied.

Bosonic Oscillator on the de Sitter and the Anti-de Sitter Spaces

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Papers

The Dunkl–Duffin–Kemmer–Petiau Oscillator

Scalar Particle in New Type of the Extended Uncertainty Principle

The EUP Dirac Oscillator: A Path Integral Approach

Bosonic Oscillator on the de Sitter and the Anti-de Sitter Spaces