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

TL;DR

This paper develops a novel quantization method for Hamiltonian systems with position-dependent mass using Killing vector fields and Noether momenta, and applies it to various nonlinear oscillator models.

Contribution

It introduces a new quantization approach based on Noether momenta and Killing vectors for PDM systems, extending the quantization framework beyond canonical momenta.

Findings

Quantization of PDM systems using Noether momenta.

Application to nonlinear oscillator models with PDM.

Construction of Hermitian quantum Hamiltonians for these systems.

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.