Algebraic solutions of shapeinvariant positiondependent effective mass systems
Abstract
Keeping in view the ordering ambiguity that arises due to the presence of positiondependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with positiondependent effective mass is discussed. We quantize the Hamiltonian of the pertaining system by using symmetric ordering of the operators concerning momentum and the spatially varying mass, initially proposed by von Roos and LévyLeblond. The algebraic method, used to obtain the solutions, is based on the concepts of supersymmetric quantum mechanics and shape invariance. In order to exemplify the general formalism a class of nonlinear oscillators has been considered. This class includes the particular example of a onedimensional oscillator with different positiondependent effective mass profiles. Explicit expressions for the eigenenergies and eigenfunctions in terms of generalized Hermite polynomials are presented. Moreover, properties of these modified Hermite polynomials, like existence of generating function and recurrence relations among the polynomials have also been studied. Furthermore, it has been shown that in the harmonic limit, all the results for the linear harmonic oscillator are recovered.
 Authors:
 School of Electrical Engineering and Computer Sciences, National University of Sciences and Technology, Islamabad (Pakistan)
 School of Natural Sciences, National University of Sciences and Technology, Islamabad (Pakistan)
 Publication Date:
 OSTI Identifier:
 22596685
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; EFFECTIVE MASS; EIGENFUNCTIONS; HAMILTONIANS; HARMONIC OSCILLATORS; HERMITE POLYNOMIALS; MATHEMATICAL SOLUTIONS; NONLINEAR PROBLEMS; ONEDIMENSIONAL CALCULATIONS; OSCILLATORS; QUANTUM MECHANICS; RECURSION RELATIONS; SUPERSYMMETRY
Citation Formats
Amir, Naila, Email: naila.amir@live.com, Email: naila.amir@seecs.edu.pk, and Iqbal, Shahid, Email: sic80@hotmail.com, Email: siqbal@sns.nust.edu.pk. Algebraic solutions of shapeinvariant positiondependent effective mass systems. United States: N. p., 2016.
Web. doi:10.1063/1.4954283.
Amir, Naila, Email: naila.amir@live.com, Email: naila.amir@seecs.edu.pk, & Iqbal, Shahid, Email: sic80@hotmail.com, Email: siqbal@sns.nust.edu.pk. Algebraic solutions of shapeinvariant positiondependent effective mass systems. United States. doi:10.1063/1.4954283.
Amir, Naila, Email: naila.amir@live.com, Email: naila.amir@seecs.edu.pk, and Iqbal, Shahid, Email: sic80@hotmail.com, Email: siqbal@sns.nust.edu.pk. 2016.
"Algebraic solutions of shapeinvariant positiondependent effective mass systems". United States.
doi:10.1063/1.4954283.
@article{osti_22596685,
title = {Algebraic solutions of shapeinvariant positiondependent effective mass systems},
author = {Amir, Naila, Email: naila.amir@live.com, Email: naila.amir@seecs.edu.pk and Iqbal, Shahid, Email: sic80@hotmail.com, Email: siqbal@sns.nust.edu.pk},
abstractNote = {Keeping in view the ordering ambiguity that arises due to the presence of positiondependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with positiondependent effective mass is discussed. We quantize the Hamiltonian of the pertaining system by using symmetric ordering of the operators concerning momentum and the spatially varying mass, initially proposed by von Roos and LévyLeblond. The algebraic method, used to obtain the solutions, is based on the concepts of supersymmetric quantum mechanics and shape invariance. In order to exemplify the general formalism a class of nonlinear oscillators has been considered. This class includes the particular example of a onedimensional oscillator with different positiondependent effective mass profiles. Explicit expressions for the eigenenergies and eigenfunctions in terms of generalized Hermite polynomials are presented. Moreover, properties of these modified Hermite polynomials, like existence of generating function and recurrence relations among the polynomials have also been studied. Furthermore, it has been shown that in the harmonic limit, all the results for the linear harmonic oscillator are recovered.},
doi = {10.1063/1.4954283},
journal = {Journal of Mathematical Physics},
number = 6,
volume = 57,
place = {United States},
year = 2016,
month = 6
}

We consider onedimensional stationary positiondependent effective mass quantum model and derive a generalized Kortewegde Vries (KdV) equation in (1+1) dimension through Lax pair formulation, one being the effective mass Schrödinger operator and the other being the timeevolution of wave functions. We obtain an infinite number of conserved quantities for the generated nonlinear equation and explicitly show that the new generalized KdV equation is an integrable system. Inverse scattering transform method is applied to obtain general solution of the nonlinear equation, and then Nsoliton solution is derived for reflectionless potentials. Finally, a special choice has been made for the variable massmore »

On the construction of coherent states of position dependent mass Schroedinger equation endowed with effective potential
In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schroedinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schroedinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new. 
An exactly solvable Schroedinger equation with finite positive positiondependent effective mass
The solution of the onedimensional Schroedinger equation is discussed in the case of positiondependent mass. The general formalism is specified for potentials that are solvable in terms of generalized Laguerre polynomials and mass functions that are positive and bounded on the whole real x axis. The resulting fourparameter potential is shown to belong to the class of 'implicit' potentials. Closed expressions are obtained for the boundstate energies and the corresponding wave functions, including their normalization constants. The constant mass case is obtained by a specific choice of the parameters. It is shown that this potential contains both the harmonic oscillatormore » 
Wave packet dynamics for a system with position and timedependent effective mass in an infinite square well
The problem of a particle with position and timedependent effective mass in a onedimensional infinite square well is treated by means of a quantum canonical formalism. The dynamics of a launched wave packet of the system reveals a peculiar revival pattern that is discussed. . 
Explicit solutions for Ndimensional Schroedinger equations with positiondependent mass
With the consideration of spherical symmetry for the potential and mass function, onedimensional solutions of nonrelativistic Schroedinger equations with spatially varying effective mass are successfully extended to arbitrary dimensions within the frame of recently developed elegant nonperturbative technique, where the BenDanielDuke effective Hamiltonian in one dimension is assumed like the unperturbed piece, leading to wellknown solutions, whereas the modification term due to possible use of other effective Hamiltonians in one dimension and, together with the corrections coming from the treatments in higher dimensions, are considered as an additional term like the perturbation. Application of the model and its generalization formore »