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International Journal of Quantum Chemistry

Exactly solvable effective mass Schrödinger equation with coulomb‐like potential

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

Published Sep 8, 2010• 4 citations
Publisher

Abstract

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

DOI
10.1002/qua.22898

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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
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
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 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.