We construct a singular oscillator Hamiltonian with a position-dependent effective mass. We find that an su(1, 1) algebra is the hidden symmetry of this quantum system and the isospectral potentials V(x) depend on the different choices of the m(x). The complete solutions are also presented by using this Lie algebra.
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.
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 2m(x) = e −α 2 x 2 , 1/(α 2 x 2 + 1), exp(2αx)/ cosh 2 (αx), 1/ cos 2 (αx), exp(−α|x|), x α , 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.
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.