A position-dependent mass harmonic oscillator and deformed space

Abstract

We consider canonically conjugated generalized space and linear momentum operators ${\hat{x}}_q$ and ${\hat{p}}_q$ in quantum mechanics, associated to a generalized translation operator which produces infinitesimal deformed displacements controlled by a deformation parameter $q$. A canonical transformation $(\hat{x},\hat{p})\to ({\hat{x}}_q,{\hat{p}}_q)$ 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 $(x_q,p_q)$ 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.

References in corpus (20)

• Statistical mechanics in the context of special relativity
• Non Linear Kinetics underlying Generalized Statistics
• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Effective Mass of One 4he Atom in Liquid 3he
• Generalized algebra within a nonextensive statistics
• Nonlinear Relativistic and Quantum Equations with a Common Type of Solution
• Displacement operator for quantum systems with position-dependent mass
• Morse potential derived from first principles
• Position Dependent Mass Oscillators and Coherent States
• Multi Parametric Deformed Heisenberg Algebras: A Route to Complexity
• Revisiting the displacement operator for quantum systems with position-dependent mass
• Generalized space and linear momentum operators in quantum mechanics
• Basic-deformed quantum mechanics
• Nonlinear Schroedinger Equation in the Presence of Uniform Acceleration
• Deformed quantum mechanics and q-Hermitian operators
• Dynamical Equations, Invariants and Spectrum Generating Algebras of Mechanical Systems with Position-Dependent Mass
• Algebraic solutions of shape-invariant position-dependent effective mass systems
• Quantum Confinement in Nonadditive Space with a Spatially Dependent Effective Mass for Si and Ge Quantum Wells
• Role of Quantum Confinement in Luminescence Efficiency of Group IV Nanostructures
• Effective Mass Quantum Systems with Displacement Operator: Inverse Square plus Coulomb-like Potential

Cited by in corpus (17)

• $\kappa$-Deformed quantum and classical mechanics for a system with position-dependent effective mass
• Dual Conformable Derivative: Definition, Simple Properties and Perspectives for Applications
• Deformed Fokker-Planck equation: inhomogeneous medium with a position-dependent mass
• Information-theoretic measures for a position-dependent mass system in an infinite potential well
• Landau Quantization for an electric quadrupole moment of Position-Dependent Mass Quantum Particles interacting with Electromagnetic fields
• Exact solution and coherent states of an asymmetric oscillator with position-dependent mass
• Supersymmetric quantum mechanics and coherent states for a deformed oscillator with position-dependent effective mass
• Coherent and squeezed states: introductory review of basic notions, properties and generalizations
• PDM-charged particles in PD-magnetic plus Aharonov-Bohm flux fields: unconfined "almost-quasi-free" and confined in a Yukawa plus Kratzer exact solvability
• Position-dependent mass charged particles in magnetic and Aharonov-Bohm flux fields: separability, exact and conditionally exact solvability
• Position-dependent mass systems: Classical and quantum pictures
• The Wigner function of a semiconfined harmonic oscillator model with a position-dependent effective mass
• Position-dependent mass in strong quantum gravitational background fields
• Quantum Smooth Boundary Forces from Constrained Geometries
• Two dimensional electron gas in a non-Euclidean space
• Hilbert space representation of maximal length and minimal momentum uncertainties
• Algebraic structures and deformed Schrödinger equations from groups entropies