We consider canonically conjugated generalized space and linear momentum operators x^q and p^q in quantum mechanics, associated with 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 position-dependent mass in a harmonic potential to a particle with constant mass in a Morse potential. The trajectories in phase spaces (x, p) and (xq, pq) are analyzed for different values of the deformation parameter. Finally, 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 operator ${\widehat{p}}_q$ , and its canonically conjugate deformed position operator ${\widehat{x}}_q$ . 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.

We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space where the linear (nonlinear) character of the FPE is associated with the employed deformed linear (nonlinear) derivative. The FPE for an inhomogeneous medium with a position-dependent diffusion coefficient is equivalent to a deformed FPE within a deformed space, described by generalized derivatives, and constant diffusion coefficient. The deformed FPE is consistent with the diffusion equation for inhomogeneous media when the temperature and the mobility have the same position-dependent functional form as well as with the nonlinear Langevin approach. The deformed version of the $H$-theorem permits to express the Boltzmann-Gibbs entropic functional as a sum of two contributions, one from the particles and the other from the inhomogeneous medium. The formalism is illustrated with the infinite square well and the confining potential with linear drift coefficient. Connections between superstatistics and position-dependent Langevin equations are also discussed.

We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator associated with a position-dependent effective mass by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique, we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass and uniform field interactions and to obtain a generalized uncertainty relation (GUP) that behaves as a distinguishability measure for the coherent states, these latter satisfying a periodic evolution for the corrections of the GUP.