Non-additive quantum mechanics for a position-dependent mass system: Dirac delta and quasi-periodic potentials

Abstract

PACS 03.65.Ca -Quantum mechanics, field theories, and special relativity: Formalism PACS 71.20.-b -Electron density of states and band structure of crystalline solids PACS 05.90.+m -Other topics in statistical physics, thermodynamics, and nonlinear dynamical systems.

κ-Deformed quantum and classical mechanics for a system with position-dependent effective mass

“…We consider that the techniques employed in this work could stimulate the seek of other generalizations of classical and quantum mechanical aspects, as has been reported in recent research studies by means of the q-algebra. 7,[48][49][50][51][52][53][54][55][56]…”

“…2 Further studies in noncommuting quantum spaces led to a Schrödinger equation with a position-dependent effective mass (PDM). 3 Along the last decades, the PDM systems have attracted attention because of their wide range of applicability in semiconductor theory, [4][5][6][7] nonlinear optics, 8 quantum liquids, 9,10 inversion potential for NH 3 in density functional theory, 11 particle physics, 12 many body theory, 13 molecular physics, 14 Wigner functions, 15 relativistic quantum mechanics, 16 superintegrable systems, 17 nuclear physics, 18 magnetic monopoles, 19,20 astrophysics, 21 nonlinear oscillations, [22][23][24][25][26][27][28][29][30][31] factorization methods and supersymmetry, [32][33][34][35][36] coherent states, [37][38][39] etc.…”

“…Complementarily, it has been found that the mathematical foundations of the PDM systems rely on the assumption of the noncommutativity between the mass operator m(x) and the linear momentum operatorp, thus giving place to the ordering problem for the kinetic energy operator. 4,[40][41][42][43][44][45][46][47] In addition, the development of generalized translation operators motivated the introduction of a positiondependent linear momentum for characterizing a particle with a PDM 7,[48][49][50][51][52][53][54][55][56] that can be related to a generalized algebraic structure (called q-algebra 57 ) inherited from the mathematical background of nonextensive statistics. 58 Concerning these formal structures, the κ-deformed statistics, originated from the κ-exponential and κ-logarithm functions, allows us to develop an algebraic structure, called κ-algebra, [59][60][61][62][63][64][65][66][67][68][69][70][71][72][73][74] with similar properties to those of the q-algebra.…”

Supersymmetric quantum mechanics and coherent states for a deformed oscillator with position-dependent effective mass

“…[51][52][53] One of these formulations is derived from a a) Electronic mail: bruno.costa@ifsertao-pe.edu.br b) Electronic mail: cardoso.genilson@outlook.com c) Electronic mail: ignacio.sebastian@ufba.br translation operator that causes non-additive displacements of the type Tγ (ε)|x = |x + (1 + γx)ε , being γ a deformation parameter with inverse length dimension. [54][55][56][57][58][59][60][61][62][63][64][65][66][67] This translation operator leads to a position-dependent linear momentum operator pγ that generates non-additive translations. Consequently, the particle mass is a function of the position controlled by the parameter γ.…”

“…In the displacement-operator formalism, the time-independent Schrödinger equation can be expressed using a deformed derivative operator D γ = (1 + γx)d/dx, which results physically equivalent to introduce a particle with a PDM. Typical problems of quantum mechanics have been solved within this approach: infinite and finite square potential wells, [54][55][56] quantum dots and wells, 57,58 quasiperiodic 59 and Coulomb-like potentials, 60 harmonic oscillator, [61][62][63][64][65] Dirac fermions in graphene 66 and two dimensional electron gas. 67 It can be shown that the energy spectrum of the deformed harmonic oscillator corresponds to the Morse oscillator, i.e., an anharmonic oscillator.…”