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Journal of Physics Conference Series

Effective mass Schrödinger equation with Thomas-Fermi potential

Gerardo Ovando, José Juan Peña, J. Morales

Published Jan 21, 2015• 0 citations
Publisher

Abstract

The exactly-solvable position-dependent mass Schrodinger equation (PDMSE) for the Thomas-Fermi potential is presented. The PDMSE is transformed into a standard Schrodinger equation (SSE) with constant mass by means of a point canonical transformation scheme. By proposing an exponential type potential of the SSE it is possible to determine a PDMSE with the Thomas-Fermi potential. The resulting PDMSE is carried to the Sturm- Liouville form and the corresponding theory is developed for the particular resulting problem in order to obtain closed solutions in terms of Bessel functions of the first kind. Beyond the case considered in this work, the approach is general and can be useful in the study of electronic properties of non-uniform materials in which the carrier effective mass depends on the position as well as in the search of new solvable potentials suitable for quantum systems.

DOI
10.1088/1742-6596/574/1/012108

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The exactly-solvable position-dependent mass Schrodinger equation (PDMSE) for the Thomas-Fermi potential is presented. The PDMSE is transformed into a standard Schrodinger equation (SSE) with constant mass by means of a point canonical transformation scheme. By proposing an exponential type potential of the SSE it is possible to determine a PDMSE with the Thomas-Fermi potential. The resulting PDMSE is carried to the Sturm- Liouville form and the corresponding theory is developed for the particular resulting problem in order to obtain closed solutions in terms of Bessel functions of the first kind. Beyond the case considered in this work, the approach is general and can be useful in the study of electronic properties of non-uniform materials in which the carrier effective mass depends on the position as well as in the search of new solvable potentials suitable for 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.
Abstract The exactly solvable Position Dependent Mass Schrödinger Equation (PDMSE) for Mie-type potentials is presented. To that, by means of a point canonical transformation the exactly solvable constant mass Schrödinger equation is transformed into a PDMSE. The mapping between both Schrödinger equations lets obtain the energy spectra and wave functions for the potential under study. This happens for any selection of the O von Roos ambiguity parameters involved in the kinetic energy operator. The exactly solvable multiparameter exponential-type potential for the constant mass Schrödinger equation constitutes the reference problem allowing to solve the PDMSE for Mie potentials and mass functions of the form given by m(x) = skx s-1/(xs + 1))2. Thereby, as a useful application of our proposal, the particular Lennard-Jones potential is presented as an example of Mie potential by considering the mass distribution m(x) = 6kx 5/(x 6 + 1))2. The proposed method is general and can be straightforwardly applied to the solution of the PDMSE for other potential models and/or with different position-dependent mass distributions.
Abstract Exactly solvable Schrödinger equation (SE) with a position‐dependent mass distribution allowing Morse‐like eigenvalues is presented. For this, the position‐dependent mass Schrödinger equation is transformed into a standard SE, with constant mass, by means of the point canonical transformation scheme. In that method, the choice of potential for the position‐dependent mass Schrödinger equation allows us to obtain the transformation that should be used to find the exactly solvable SE. As a useful application of the proposal, the equivalent of the Witten superpotential is chosen to be constant to find the position‐dependent mass distribution and the exactly solvable potential V(m(x)) allowing Morse‐type energy spectra. This V(m(x)) is shown to have a Coulomb potential structure and can be useful in the study of the electronic properties of materials in which the carrier effective mass depends on the position. Moreover, the worked example, the approach is general and can be applied in the search of new potentials suitable on the study of quantum chemical systems. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2010
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 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.