Quantization of Hamiltonian systems with a position dependent mass: Killing vector fields and Noether momenta approach

Abstract

The quantization of systems with a position dependent mass (PDM) is studied. We present a method that starts with the study of the existence of Killing vector fields for the PDM geodesic motion (Lagrangian with a PDM kinetic term but without any potential) and the construction of the associated Noether momenta. Then the method considers, as the appropriate Hilbert space, the space of functions that are square integrable with respect to a measure related with the PDM and, after that, it establishes the quantization, not of the canonical momenta $p$, but of the Noether momenta $P$ instead. The quantum Hamiltonian, that depends on the Noether momenta, is obtained as an Hermitian operator defined on the PDM Hilbert space. In the second part several systems with position-dependent mass, most of them related with nonlinear oscillators, are quantized by making use of the method proposed in the first part.

arXiv comment21 pages, to appear in J.Phys. A:Math. Theor

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Cited by in corpus (8)

• Isochronous $n$-dimensional nonlinear PDM-oscillators: linearizability, invariance and exact solvability
• The Wigner function of a semiconfined harmonic oscillator model with a position-dependent effective mass
• On the classical and quantum dynamics of a class of nonpolynomial oscillators
• $n$-dimensional PDM-damped harmonic oscillators: Linearizability, and exact solvability
• On the singular position-dependent mass
• Dynamical symmetry of a semiconfined harmonic oscillator model with a position-dependent effective mass
• Position-Dependent Mass Quantum systems and ADM formalism
• Superintegrability on the 3-dimensional spaces with curvature. Oscillator-related and Kepler-related systems on the Sphere $S^3$ and on the Hyperbolic space $H^3$