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International Journal of Quantum Chemistry

Position‐dependent mass Schrödinger equations allowing harmonic oscillator (HO) eigenvalues

José Juan Peña, Gerardo Ovando, J. Morales, Jesús García‐Ravelo, C. Pacheco‐García

Published Jan 1, 2008• 5 citations
Publisher

Abstract

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

DOI
10.1002/qua.21879

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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
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
Abstract In this work, an alternative approach for finding exact solutions of Schrödinger equations with a position‐dependent mass is presented. Essentially, the method consists in transforming the position‐dependent mass Schrödinger equation into a standard constant mass Schrödinger equation. The constant mass equation is formulated in terms of a new variable and includes, as effective potential, a term related to the position‐dependent mass and another term connected with the original potential. For exactly solvable potentials, our proposal leads to a Riccati equation for an equivalent Witten superpotential, which means that this kind of problems can be studied within the quantum supersymmetric framework with the aim to find the explicit solutions and the energy spectra. To show the usefulness of the proposed approach, we consider the particular case of exactly solvable Schrödinger equations for different position‐dependent mass situations in a null potential. However, the method can directly be applied to other effective potentials as well as to find isospectral potentials for specific position‐dependent mass distributions. © 2007 Wiley Periodicals, Inc. Int J Quantum Chem, 2007
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 The aim of the present study is to obtain exact solutions for the one‐dimensional position‐dependent effective mass Schrödinger equation. After the change of wave function on position‐dependent effective mass Schrödinger equation and comparison between the resulted equation with the PT‐symmetric invariant anharmonic potential, three distinct classes of exact solutions are obtained for the Schrödinger equation. As an illustrative example for a Gaussian form of m(x), energy spectrum, potential energy function and wave function are obtained explicitly. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004
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.
We study the generalized harmonic oscillator that has both the position-dependent mass and the potential depending on the form of mass function in a more general framework. The explicit expressions of the eigenvalue and eigenfunction for such a system are given, they have the same forms as those for the usual harmonic oscillator with constant mass. The coherent state and its properties for the system with PDM are also discussed. We give the corresponding effective potentials for several mass functions, the systems with such potentials are isospectral to the usual harmonic oscillator.