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Revista Mexicana de Física

Management of the von Roos Operator in a Confined System

Martín E. Molinar-Tabares, Lamberto Castro-Arce, Carlos Figueroa-Navarro, J. Campos-García

Published Jan 1, 2016• 2 citations

Abstract

In this article, an analysis of the kinetic energy operator introduced by O. von Roos is performed. Such analysis allows for determining the energy of a particle when its mass is position-dependent within a specific medium. The work, focused on the discussion of this concept in quantum mechanics courses, reveals how to apply the operator to numerically solve the one-dimensional Schrödinger equation for an electron rigidly confined within a semiconductor structure having a specific width (considering that the particle is subject to the action of a position-dependent quadratic potential). The Schrödinger equation is solved considering the mass of the electron constant, allowing a comparison of the obtained energies for this case as well as the ones that correspond to the case involving a position-dependent mass and taking into consideration several values of the structure width where the particle is confined. Observations during the analysis reveal how the ambiguity of the parameters that appear in the von Roos operator leads us to conclude that several dynamic systems can be associated to a given form of the potential energy.

DOI: 10.31349/revmexfis.62.5.409

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In this article, an analysis of the kinetic energy operator introduced by O. von Roos is performed. Such analysis allows for determining the energy of a particle when its mass is position-dependent within a specific medium. The work, focused on the discussion of this concept in quantum mechanics courses, reveals how to apply the operator to numerically solve the one-dimensional Schrödinger equation for an electron rigidly confined within a semiconductor structure having a specific width (considering that the particle is subject to the action of a position-dependent quadratic potential). The Schrödinger equation is solved considering the mass of the electron constant, allowing a comparison of the obtained energies for this case as well as the ones that correspond to the case involving a position-dependent mass and taking into consideration several values of the structure width where the particle is confined. Observations during the analysis reveal how the ambiguity of the parameters that appear in the von Roos operator leads us to conclude that several dynamic systems can be associated to a given form of the potential energy.

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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.

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A systematic procedure to study one-dimensional Schrödinger equation with a position-dependent effective mass (PDEM) in the kinetic energy operator is explored. The conventional free-particle problem reveals a new and interesting situation in that, in the presence of a mass background, formation of bound states is signalled. We also discuss coordinate-transformed, constant-mass Schrödinger equation, its matching with the PDEM form and the consequent decoupling of the ambiguity parameters. This provides a unified approach to many exact results known in the literature, as well as to a lot of new ones.

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