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arXiv:2302.02172 · Journal of Mathematical Physics

Exact solution and coherent states of an asymmetric oscillator with position-dependent mass

Bruno G. da Costa, Ignacio S. Gomez, Biswanath Rath

Published Jan 1, 2023• 14 citations

Abstract

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.

DOI: 10.1063/5.0094564

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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.

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.

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.

Abstract This paper presents the exact ground state solution for a diatomic particle system with position-dependent complex mass under action of a complex Morse potential in the quantum domain. By solving the position-dependent Schrödinger equation in extended complex phase space without assuming a specific mass profile, we derive both the eigenfunctions and corresponding eigenenergies using the analyticity conditions of the eigenfunctions. A key focus is placed on addressing the challenge of normalization inherent in non-Hermitian Hamiltonians. To overcome the limitations of conventional normalization methods in systems with complex potentials and spatially varying mass, we propose a modified normalization approach based on a two-dimensional integral over phase space. The results reveal that, under certain parameter constraints, real energy spectra can arise in non-Hermitian settings, supported by normalized and physically meaningful eigenfunctions. Probability density plots validate the existence of stable, localized bound states, maintaining essential characteristics of the traditional Morse potential. Moreover, the model offers potential applications in high-energy and cosmological physics, particularly in the quantum description of exotic systems like dark matter.

We construct a deformed oscillator whose energy spectra is similar to that of a Morse potential. We obtain a convenient algebraic representation of the displacement and the momentum of a Morse oscillator by expanding them in terms of deformed creation and annihilation operators and we compute their average values between approximate coherent states of the deformed oscillator, and we compare them to the results obtained using the exact Morse coordinate and momenta. Finally we evaluate the temporal evolution of the dispersion (Δx)(Δp) and show that these states are not minimum uncertainty states.