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.

We propose a modification of a recently introduced generalized translation operator, by including a q-exponential factor, which implies in the definition of a Hermitian deformed linear momentum operatorpq, and its canonically conjugate deformed position operatorxq. A canonical transformation leads the Hamiltonian of a position-dependent mass particle to another Hamiltonian of a particle with constant mass in a conservative force field of a deformed phase space. The equation of motion for the classical phase space may be expressed in terms of the generalized dual q-derivative. A position-dependent mass confined in an infinite square potential well is shown as an instance. Uncertainty and correspondence principles are analyzed.

We present the quantum and classical mechanics formalisms for a particle with a position-dependent mass in the context of a deformed algebraic structure (named κ-algebra), motivated by the Kappa-statistics. From this structure, we obtain deformed versions of the position and momentum operators, which allow us to define a point canonical transformation that maps a particle with a constant mass in a deformed space into a particle with a position-dependent mass in the standard space. We illustrate the formalism with a particle confined in an infinite potential well and the Mathews-Lakshmanan oscillator, exhibiting uncertainty relations depending on the deformation.