A position-dependent mass harmonic oscillator and deformed space

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.

PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative PDM-Hamiltonian

The binding energy of shallow hydrogenic impurities in parabolic GaAs/GaAlAs quantum dots is calculated as a function of dot radius in the influence of magnetic field. The binding energy has been calculated following a variational procedure within the effective-mass approximation. Calculations are presented with constant effective-mass and position dependent effective masses. A finite confining potential well with depth is determined by the discontinuity of the band gap in the quantum dot and the cladding. The results show that the impurity binding energy (i) increases as the dot radius decreases for the infinite case, (ii) reaches a peak value around 1R* as the dot radius decreases and then diminishes to a limiting value corresponding to the radius for which there are no bound states in the well for the infinite case, and (iii) increases with the magnetic field. Also it is found that (i) the use of constant effective mass (0.067 m0) is justified for dot sizes ≥ a* where a* is the effective Bohr radius which is about 100 Å for GaAs , in the estimation of ionization energy and (ii) the binding energy shows complicated behavior when the position dependent mass is included for the dot size ≤ a*. These results are compared with the available existing literatures.

Canonical transformations to action and angle variables and their representation in quantum mechanics IV. Periodic potentials

We study the non-relativistic charge-monopole system when the charged particle has a position-dependent mass written as M(r) = m0rw. The angular wave functions are the well-known monopole harmonics, and the radial ones are ordinary Bessel functions which depend on the magnetic and electric charge product as well as on the w parameter. We investigate mappings—approximate and exact—between the charge-monopole system with constant mass and the charge with a position-dependent mass solving the position-dependent mass Schrödinger equation for the mass distribution.

The effects of position-dependent effective mass on the absorption coefficients (ACs) of spherical quantum dot (QD) and quantum antidot (QAD) have been investigated. We show that the radial position-dependent effective mass Schrodinger equation can be reduced into ordinary one-dimensional Schrodinger equation with some suitable transformations. Our numerical calculations are performed using both a constant effective mass and the position-dependent effective mass. The comparative approach is used for presenting the results of both QD and QAD models. It is found that not only for the effective constant mass but also for the position-dependent effective mass, the ACs curves corresponding to QD and QAD are considerably different in shapes and behaviors. Our obtained results show that the spatial variation of the electron effective mass can not be neglected in the optical ACs of both QD and QAD nanostructures.

One of the most important scientific contributions of Professor Marcos Moshinsky has been his study on the harmonic oscillator in quantum theory vis à vis the standard Schrödinger equation with constant mass [1]. However, a simple description of the motion of a particle interacting with an external environment such as happen in compositionally graded alloys consist of replacing the mass by the so‐called effective mass that is in general variable and dependent on position. Therefore, honoring in memoriam Marcos Moshinsky, in this work we consider the position‐dependent mass Schrodinger equations (PDMSE) for the harmonic oscillator potential model as former potential as well as with equi‐spaced spectrum solutions, i.e. harmonic oscillator isospectral partners. To that purpose, the point canonical transformation method to convert a general second order differential equation (DE), of Sturm‐Liouville type, into a Schrödinger‐like standard equation is applied to the PDMSE. In that case, the former potential associated to the PDMSE and the potential involved in the Schrödinger‐like standard equation are related through a Riccati‐type relationship that includes the equivalent of the Witten superpotential to determine the exactly solvable positions‐dependent mass distribution (PDMD) m(x). Even though the proposed approach is exemplified with the harmonic oscillator potential, the procedure is general and can be straightforwardly applied to other DEs.

In the domain of light emissions, quantum mechanics has been an immensely successful guiding tool for us. In the propagation of light and optical instrument design, Huygens-Fresnel diffraction integral (HFDI) (or its advanced versions) and Maxwell’s wave equation are continuing to be the essential guiding tools for optical scientists and engineers. In fact, most branches of optical science and engineering, like optical instrument design, image processing, Fourier optics, Holography, etc., cannot exist without using the foundational postulates behind the Huygens-Fresnel diffraction integral. Further, the field of structured light is also growing where phases and the state of polarizations are manipulated usually with suitable classical macro-devices to create wave fronts that restructured through light-matter interactions through these devices. Mathematical modeling of generating such complex wave fronts generally follows classical concepts and classical macro tools of physical optics. Some of these complex light beams can impart mechanical angular momentum and spin-like properties to material particles inserted inside these structured beams because of their electromagnetic dipolar properties and/or structural anisotropy. Does that mean these newly structured beams have acquired new quantum properties without being generated through quantum devices and quantum transitions? In this chapter, we bridge the classical and quantum formalism by defining a hybrid photon (HP). HP is a quantum of energy, hν, at the initial moment of emission. It then immediately evolves into a classical time-finite wave packet, still transporting the original energy, hν, with a classical carrier frequency ν (oscillation of the E-vector). This chapter will raise enquiring questions whether all these observed “quantum-like” behaviors are manifestations of the joint properties of interacting material particles with classical EM waves or are causal implications of the existence of propagation of “indivisible light quanta” with exotic properties like spin, angular momentum, etc.

We analyze the dynamical equations obeyed by a classical system with position-dependent mass. It is shown that there is a non-conservative force quadratic in the velocity associated to the variable mass. We construct the Lagrangian and the Hamiltonian for this system and find the modifications required in the Euler-Lagrange and Hamilton's equations to reproduce the appropriate Newton's dynamical law. Since the Hamiltonian is not time invariant, we get a constant of motion suited to write the dynamical equations in the form of the Hamilton's ones. The time-dependent first integrals of motion are then obtained from the factorization of such a constant. A canonical transformation is found to map the variable mass equations to those of a constant mass. As particular cases, we recover some recent results for which the dependence of the mass on the position was already unnoticed, and find new solvable potentials of the P\"oschl-Teller form which seem to be new. The latter are associated to either the su(1,1) or the su(2) Lie algebras depending on the sign of the Hamiltonian.

Klein-Gordon (KG) particles in cosmic string rainbow gravity spacetime and a uniform magnetic field are studied in the context of the so called, effectively speaking, position-dependent mass (PDM) settings. We show that the corresponding KG-equation collapses into a two-dimensional radial Schrödinger-Coulomb like model. The exact textbook solution of which is used to find the energies and wave functions of KG-Coulomb particles (both constant mass and PDM ones). In so doing, we consider, with y=E/EP, four pairs of rainbow functions: (a) g0(y)=1, g1(y)=1−ϵy2, (b) g0(y)=1, g1(y)=1−ϵy, (c) g0(y)=g1(y)=(1−ϵy)−1, and (d) g0(y)=(eϵy−1)/ϵy, g1(y)=1. Interestingly, we observe that the first pair in (a) introduces the Planck energy Ep as a maximum possible KG-particle/antiparticle energy value.

An always extremum variational principle for classical mechanics via a canonical transformation∗

We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schrödinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering, we show that the class of systems can also be exactly solvable and is also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Liénard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

Effect of position-dependent effective mass on linear and nonlinear optical properties of a cubic quantum dot

We construct arbitrary-order Darboux transformations for Schrödinger equations with energy-dependent potential and position-dependent mass within the Dunkl formalism. Our construction is based on a point transformation that interrelates our equations with the standard Schrödinger case. We apply our method to generate several solvable Dunkl–Schrödinger equations.

In this paper, we show how to construct a general class of non-relativistic quantum systems with position-dependent masses, which are isospectral to a given system with constant mass. Then, we use this approach in order to construct the Wigner functions in three examples which are isospectral to the harmonic oscillator. Finally, we discuss possible general behavior of these systems.