A position-dependent mass harmonic oscillator and deformed space

Abstract

We consider canonically conjugated generalized space and linear momentum operatorsx q andp q in quantum mechanics, associated to a generalized translation operator which produces infinitesimal deformed displacements controlled by a deformation parameter q. A canonical transformation (x,p) → (x q ,p q ) leads the Hamiltonian of a position-dependent mass particle in usual space to another Hamiltonian of a particle with constant mass in a conservative force field of the deformed space. The equation of motion for the classical phase space (x, p) may be expressed in terms of the deformed (dual) q-derivative. We revisit the problem of a q-deformed oscillator in both classical and quantum formalisms. Particularly, this canonical transformation leads a particle with positiondependent mass in a harmonic potential to a particle with constant mass in a Morse potential. The trajectories in phase spaces (x, p) and (x q , p q ) are analyzed for different values of the deformation parameter. Lastly, we compare the results of the problem in classical and quantum formalisms through the principle of correspondence and the WKB approximation.

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

“…In the classical limit, one has that an,κ → Aκ once En → E, so the expectation value (64) recovers its classical average value (55). In Fig.…”

“…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]…”

“…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.…”

On q-Uniformly Mocanu Functions

“…We recall here some basic concepts from q-calculus for which we refer to [5][6][7][8][9][10][11][12][13][14][15][16] and the references therein.…”