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AIP conference proceedings

The harmonic oscillator and the position dependent mass Schrodinger equation: isospectral partners and factorization operators

J. Morales, Gerardo Ovando, José Juan Peña, Kurt Bernardo Wolf, Luis Benet, Juan Mauricio Torres, Peter Oto Hess

Published Jan 1, 2010• 0 citations
Publisher

Abstract

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.

DOI
10.1063/1.3537852

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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.
Abstract Quantum chemical systems with a position‐dependent mass have attracted the attention due to their relevance in describing the features of many microstructures of current interest. In this work, the point canonical transformation method applied to Schrödinger equations with a position‐dependent mass (SEPDM) is presented. Essentially, the proposal is aimed to transform the Schrödinger equation with a position‐dependent mass into a standard Schrödinger‐like equation for constant mass in such a way that the position‐dependent mass distribution (PDMD) becomes incorporated into the effective potential. As an useful application of the proposal, it is considered as effective potential the one‐dimensional harmonic oscillator potential model, which leads to those isospectral potentials related to different forms of PDMD. For example, the exactly solvable isospectral potentials involved in the SEPDM for some PDMD such as , , , , , xα, and , are worked out explicitly including their raising and lowering operators that factorize the SEPDM for each PDMD allowing HO eigenvalues. However, the proposal is general and can be straightforwardly applied to other effective potential models as well as other PDMD that could be useful in quantum chemical applications. © 2008 Wiley Periodicals, Inc. Int J Quantum Chem, 2008
In this work, the quantum operator approach is applied to both, the position-dependent mass Schrödinger equation (PDMSE) and the Schrodinger equation with constant mass (CMSE). This fact enable us to find the factorization operators that relates both Hamiltonians by means of a kinetic energy operator that comes from the proposal of Morrow and Brownstein. With this approach is possible to find the exactly-solvable PDMSE, for any value of the parameters α and γ in the von Roos's Hamiltonian. For that, our proposal can be considered as a unified treatment of the PDMSE because it contains as particular cases, the kinetic energy operators of various authors such as BenDaniel-Duke, Gora-Williams, Zhu-Kroemer and Li-Kuhn among others. To show the usefulness of our result, we show the solvable PDMSE that comes from the harmonic oscillator potential model for the CMSE. The proposal is general and can easily be extended to other potential models and mass distributions which will be given in the extended paper.
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.
The position dependent mass Schrödinger equation (PDMSE) has a wide range of quantum applications such as the study of semiconductors, quantum wells, quantum dots and impurities in crystals, among many others. On the other hand, the Morse potential is one of the most important potential models used to study the electronic properties of diatomic molecules. In this work, the solution of the effective mass one-dimensional Schrödinger equation for the Morse potential is presented. This is done by means of the canonical transformation method in algebraic form. The PDMSE is solved for any model of the proposed kinetic energy operators as for example the BenDaniel-Duke, Gora-Williams, Zhu-Kroemer or Li-Kuhn. Also, in order to solve the PDMSE with Morse potential, we consider a superpotential leading to a special form of the exactly solvable Schrödinger equation of constant mass for a class of multiparameter exponential-type potential along with a proper mass distribution. The proposed approach is general and can be applied in the search of new potentials suitable on science of materials by looking into the viable choices of the mass function.
The von Roos’ position dependent mass Schrödinger Equation (PDMSE) is given in terms of an effective potential, which is constructed by the potential under study and additional terms involving the position dependent mass and the ambiguity ordering parameters. At this regard, some of the most important approaches to the ambiguity parameters are those of BenDaniel-Duke, Gora-Williams, Zhu-Kroemer, Li-Kuhn and so on. In this work we show that for the cases of Zhu-Kroemer and BenDaniel-Duke it is possible to express any solvable potential through a couple of quantum superpotentials. For this, we stablish the factorization of the PDMSE Hamiltonian and the use of a point canonical transformation; these allow to identify a first superpotential W(x). Next, the term in the effective potential with ambiguity parameters is expressed through a function Z(x) and later the cases when Z(x) is a quantum superpotential are discussed. This way, the proposed superpotentials determine the solvable potentials in the PDMSE, the energy spectra and mass functions. We give some examples to explain the approach which can be straightforwardly applied to the solution of the PDMSE for other different potential models and mass distributions.