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Journal of Physics Conference Series

Bounded motion for classical systems with position-dependent mass

Sara Cruz y Cruz, C Santiago‐Cruz

Published Oct 10, 2014• 3 citations
Publisher

Abstract

In this work the dynamical equations for a system with position-dependent mass are considered. The phase space trajectories are constructed by means of the factorization method for classical systems. To illustrate how this formalism works the phase space trajectories for position-dependent mass oscillator, Scarf and Pöschl-Teller potentials with Gaussian and singular masses are presented.

DOI
10.1088/1742-6596/538/1/012006

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In this work the dynamical equations for a system with position-dependent mass are considered. The phase space trajectories are constructed by means of the factorization method for classical systems. To illustrate how this formalism works the phase space trajectories for position-dependent mass oscillator, Scarf and Pöschl-Teller potentials with Gaussian and singular masses are presented.
We revisit the problem of the deformed oscillator with position-dependent mass [da Costa et al., J. Math. Phys. 62, 092101 (2021)] in the classical and quantum formalisms by introducing the effect of the mass function in both kinetic and potential energies. The resulting Hamiltonian is mapped into a Morse oscillator by means of a point canonical transformation from the usual phase space (x, p) to a deformed one (xγ, Πγ). Similar to the Morse potential, the deformed oscillator presents bound trajectories in phase space corresponding to an anharmonic oscillatory motion in classical formalism and, therefore, bound states with a discrete spectrum in quantum formalism. On the other hand, open trajectories in phase space are associated with scattering states and continuous energy spectrum. Employing the factorization method, we investigate the properties of the coherent states, such as the time evolution and their uncertainties. A fast localization, classical and quantum, is reported for the coherent states due to the asymmetrical position-dependent mass. An oscillation of the time evolution of the uncertainty relationship is also observed, whose amplitude increases as the deformation increases.
Considering the position-dependent effective mass in the study of quantum mechanical systems, a wide range of solvable potentials has been obtained. These potentials are obtained by applying canonical transformations to the Schrödinger equation. In this method, the internal functions introduced by Levai for solvable potentials with constant mass have been used, and the eigenfunctions and eigenvalues have been fully obtained. The eigenfunction of these solvable potentials can be obtained based on specific known functions (Jacobian, generalized Laguerre, and Hermit polynomials). Some of these potentials are Scarf-II, Pöschl-Teller, Rosen-Mörse-II and Eckart, whose applications have been described in physical systems. Finally, all the results are placed in a table and the figures of the desired functions is drawn for specific values
Position-dependent mass (PDM) systems have diverse applications in physics, attracting significant scientific interest in the last decades. Classical PDM systems pose analytical challenges with their nonlinear equations of motion, while the quantum case is complicated because the kinetic operator is non-Hermitian. To address these challenges, this study explores Hamiltonian factorization and canonical transformations to investigate classical and quantum PDM systems, applying it to a Mathews-Lakshmanan-type oscillator with m⁢(x)=1/[1+(λ⁢x)2]. Classically, the phase space is examined, revealing increasingly pronounced deformities in the trajectories as energy and λ values increase. In the quantum realm, the solution to the ambiguous ordering problem for the PDM oscillator is presented, accompanied by an analysis of wave functions and probability densities. Further, the tunneling probabilities are analyzed. As λ increases, findings indicate that the tunneling probabilities of the PDM system decrease fast for higher excited states, offering novel insights into its behavior.
A quantum particle with potential energy V ( , t ) is considered in the frame of a phase-space picture of the quantum theory, and the interconnection between quantum mechanics and a -dependent extended classical dynamics is analysed. The initial position-space wavefunction determines the initial conditions for a set of Hamilton-like equations that leads up to an ensemble of complex-valued phase-space trajectories. The one-dimensional driven harmonic oscillator is used for illustrating the method, and for generating a complete set of phase-space functions.
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.