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Ordering ambiguity revisited via position dependent mass pseudo-momentum operators

Mustafa, O, Mazharimousavi, S H

Published Jul 23, 2006• 49 citations
OpenAlex

Abstract

Ordering ambiguity associated with the von Roos position dependent mass (PDM) Hamiltonian is considered. A PDM-pseudo-momentum operator is introduced and generalized to d-dimensions. Intertwining our Hamiltonian with the von Roos d-dimensional PDM-Hamiltonian we observed that the so-called von Roos ambiguity parameters are strictly determined. The ordering ambiguity seems to be removed through our intertwining process, therefore.

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Using azimuthally symmetrized cylindrical coordinates, we report the consequences of zero-energy quantal states on the von Roos Hamiltonian. A position-dependent mass (PDM) M(ρ, φ, z) = bzjρ2υ+1/2 is used. We show that the zero-energy setting not only offers an additional degree of freedom toward feasible separability for the von Roos Hamiltonian, but also manifestly yields auxiliary quantized ambiguity parametric constraints (i.e. the ambiguity parameters are given in terms of quantum numbers).

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Comment on the "Classical and quantum position-dependent mass harmonic oscillators" and ordering-ambiguity resolution

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We recycle Cruz et al.'s (Phys. Lett. A 369 (2007) 400) work on the classical and quantum position-dependent mass (PDM) oscillators. To elaborate on the ordering ambiguity, we properly amend some of the results reported in their work and discuss the classical and quantum mechanical correspondence for the PDM harmonic oscillators. We use a point canonical transformation and show that one unique quantum PDM oscillator Hamiltonian (consequently, one unique ordering-ambiguity parametric set j=l=-1/4 and k=-1/2) is obtained. To show that such a parametric set is not just a manifestation of the quantum PDM oscillator Hamiltonian, we consider the classical and quantum mechanical correspondence for quasi-free PDM particles moving under the influence of their own PDM force fields.

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Classification Scheme for Kinetic Energy Operators with Position-Dependent Mass

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In this paper we present a complete classification scheme for kinetic energy operators (KEO's) describing a particle endowed with position-dependent mass (PDM). We first present a generalized formulation of KEO's with PDM and show that it is equivalent to three-parameter linear formulation. This reduces further to two independent parameters under Hermiticity condition. Based on this linear formulation we prove that, contrary to what was widely believed, von Roos family is not the most general ordering. We found an entire new family of Hermitian KEO's that does not fit into von Roos ordering. We were able to construct all the Hermitian KEO's and classify them into two-parameter classes. As one application, we solve the puzzling case of Yang and Yee KEO. We also find, under certain conditions, some kind of duality between von Roos ordering and one of the new classes.

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Abstract Revisiting the issue associated with Position-Dependent Mass (PDM), we reaffirm that the appropriate framework for addressing a generic PDM is the symmetrization proposed by BenDaniel and Duke. To accomplish this result adopts the effective mass Hamiltonian proposed by von Roos, corrected by a symmetrized kinematic term. After verifying the appropriate ordering to approach the PDM issue, one investigates a crystalline lattice with a defect described by a singular PDM. The singular mass profile proves intriguing as it yields an atom’s cluster in the neighborhood of the singularity. Considering that a restoring force acts on the atoms, one notes that the confluent Heun function describes the quantum states. Furthermore, one highlights that when the effective mass distribution tends to a constant profile, we recover a system similar to the harmonic oscillator.

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Algebraic Approach to the Position-Dependent Mass Quantum Systems

G Ovando Zuniga, José Juan Peña Gil, Jesús Morales Rivas

The position-dependent mass Schrodinger equation (PDMSE) comes from the O von Roos quantum Hamiltonian which models to position-dependent effective mass quantum systems. The algebraic study of its factorization is compared in this work with the factorization of the traditional constant mass Schrödinger equation so that both equations are related by similarity transformations. The approach allows building solvable cases of the PDMSE for any value of the ambiguity parameters in the general Hamiltonian of O von Roos, so it can be considered as a unified treatment of the PDMSE that contains as particular cases those Hamiltonians of various authors such as BenDaniel-Duke, Gora-Williams, Zhu-Kroemer and Li- Kuhn, among others. We explicitly show the PDMSE solutions coming from the harmonic, Morse and multiparameter exponential-type potentials. The method is general and can be easily extended to other potential models and position dependent mass distributions useful in the mathematical modeling of quantum systems.

Book Publisher International (a part of SCIENCEDOMAIN International)•Apr 20, 2023•0 citations
Ordering ambiguity associated with the von Roos position dependent mass (PDM) Hamiltonian is considered. A PDM-pseudo-momentum operator is introduced and generalized to d-dimensions. Intertwining our Hamiltonian with the von Roos d-dimensional PDM-Hamiltonian we observed that the so-called von Roos ambiguity parameters are strictly determined. The ordering ambiguity seems to be removed through our intertwining process, therefore.
Using azimuthally symmetrized cylindrical coordinates, we report the consequences of zero-energy quantal states on the von Roos Hamiltonian. A position-dependent mass (PDM) M(ρ, φ, z) = bzjρ2υ+1/2 is used. We show that the zero-energy setting not only offers an additional degree of freedom toward feasible separability for the von Roos Hamiltonian, but also manifestly yields auxiliary quantized ambiguity parametric constraints (i.e. the ambiguity parameters are given in terms of quantum numbers).
We recycle Cruz et al.'s (Phys. Lett. A 369 (2007) 400) work on the classical and quantum position-dependent mass (PDM) oscillators. To elaborate on the ordering ambiguity, we properly amend some of the results reported in their work and discuss the classical and quantum mechanical correspondence for the PDM harmonic oscillators. We use a point canonical transformation and show that one unique quantum PDM oscillator Hamiltonian (consequently, one unique ordering-ambiguity parametric set j=l=-1/4 and k=-1/2) is obtained. To show that such a parametric set is not just a manifestation of the quantum PDM oscillator Hamiltonian, we consider the classical and quantum mechanical correspondence for quasi-free PDM particles moving under the influence of their own PDM force fields.
In this paper we present a complete classification scheme for kinetic energy operators (KEO's) describing a particle endowed with position-dependent mass (PDM). We first present a generalized formulation of KEO's with PDM and show that it is equivalent to three-parameter linear formulation. This reduces further to two independent parameters under Hermiticity condition. Based on this linear formulation we prove that, contrary to what was widely believed, von Roos family is not the most general ordering. We found an entire new family of Hermitian KEO's that does not fit into von Roos ordering. We were able to construct all the Hermitian KEO's and classify them into two-parameter classes. As one application, we solve the puzzling case of Yang and Yee KEO. We also find, under certain conditions, some kind of duality between von Roos ordering and one of the new classes.
Abstract Revisiting the issue associated with Position-Dependent Mass (PDM), we reaffirm that the appropriate framework for addressing a generic PDM is the symmetrization proposed by BenDaniel and Duke. To accomplish this result adopts the effective mass Hamiltonian proposed by von Roos, corrected by a symmetrized kinematic term. After verifying the appropriate ordering to approach the PDM issue, one investigates a crystalline lattice with a defect described by a singular PDM. The singular mass profile proves intriguing as it yields an atom’s cluster in the neighborhood of the singularity. Considering that a restoring force acts on the atoms, one notes that the confluent Heun function describes the quantum states. Furthermore, one highlights that when the effective mass distribution tends to a constant profile, we recover a system similar to the harmonic oscillator.
The position-dependent mass Schrodinger equation (PDMSE) comes from the O von Roos quantum Hamiltonian which models to position-dependent effective mass quantum systems. The algebraic study of its factorization is compared in this work with the factorization of the traditional constant mass Schrödinger equation so that both equations are related by similarity transformations. The approach allows building solvable cases of the PDMSE for any value of the ambiguity parameters in the general Hamiltonian of O von Roos, so it can be considered as a unified treatment of the PDMSE that contains as particular cases those Hamiltonians of various authors such as BenDaniel-Duke, Gora-Williams, Zhu-Kroemer and Li- Kuhn, among others. We explicitly show the PDMSE solutions coming from the harmonic, Morse and multiparameter exponential-type potentials. The method is general and can be easily extended to other potential models and position dependent mass distributions useful in the mathematical modeling of quantum systems.