# Algebraic solutions of shape-invariant position-dependent effective mass systems

## Abstract

Keeping in view the ordering ambiguity that arises due to the presence of position-dependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with position-dependent 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 Lévy-Leblond. 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 non-linear oscillators has been considered. This class includes the particular example of a one-dimensional oscillator with different position-dependent 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; ONE-DIMENSIONAL CALCULATIONS; OSCILLATORS; QUANTUM MECHANICS; RECURSION RELATIONS; SUPERSYMMETRY

### Citation Formats

```
Amir, Naila, E-mail: naila.amir@live.com, E-mail: naila.amir@seecs.edu.pk, and Iqbal, Shahid, E-mail: sic80@hotmail.com, E-mail: siqbal@sns.nust.edu.pk.
```*Algebraic solutions of shape-invariant position-dependent effective mass systems*. United States: N. p., 2016.
Web. doi:10.1063/1.4954283.

```
Amir, Naila, E-mail: naila.amir@live.com, E-mail: naila.amir@seecs.edu.pk, & Iqbal, Shahid, E-mail: sic80@hotmail.com, E-mail: siqbal@sns.nust.edu.pk.
```*Algebraic solutions of shape-invariant position-dependent effective mass systems*. United States. doi:10.1063/1.4954283.

```
Amir, Naila, E-mail: naila.amir@live.com, E-mail: naila.amir@seecs.edu.pk, and Iqbal, Shahid, E-mail: sic80@hotmail.com, E-mail: siqbal@sns.nust.edu.pk. Wed .
"Algebraic solutions of shape-invariant position-dependent effective mass systems". United States.
doi:10.1063/1.4954283.
```

```
@article{osti_22596685,
```

title = {Algebraic solutions of shape-invariant position-dependent effective mass systems},

author = {Amir, Naila, E-mail: naila.amir@live.com, E-mail: naila.amir@seecs.edu.pk and Iqbal, Shahid, E-mail: sic80@hotmail.com, E-mail: siqbal@sns.nust.edu.pk},

abstractNote = {Keeping in view the ordering ambiguity that arises due to the presence of position-dependent effective mass in the kinetic energy term of the Hamiltonian, a general scheme for obtaining algebraic solutions of quantum mechanical systems with position-dependent 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 Lévy-Leblond. 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 non-linear oscillators has been considered. This class includes the particular example of a one-dimensional oscillator with different position-dependent 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 = {Wed Jun 15 00:00:00 EDT 2016},

month = {Wed Jun 15 00:00:00 EDT 2016}

}