Rational extensions of an oscillator-shaped quantum well potential in a position-dependent mass background

Abstract: We show that a recently proposed oscillator-shaped quantum well model associated with a position-dependent mass can be solved by applying a point canonical transformation to the constant-mass Schr\"odinger equation for the Scarf I potential. On using the known rational extension of the latter connected with $X_1$-Jacobi exceptional orthogonal polynomials, we build a rationally-extended position-dependent mass model with the same spectrum as the starting one. Some more involved position-dependent mass models associated with $X_2$-Jacobi exceptional orthogonal polynomials are also considered.

• The paper presents a systematic method to extend oscillator-shaped quantum well potentials by applying point canonical transformation in position-dependent mass scenarios.
• It leverages exceptional orthogonal polynomials, notably Jacobi EOPs, to develop rational extensions that preserve the original energy spectrum.
• The study provides explicit analytical solutions for wavefunctions and potential models, offering practical insights for applications in semiconductor physics and quantum dots.

Rational Extensions of an Oscillator-Shaped Quantum Well Potential in a Position-Dependent Mass Background

Introduction

The paper focuses on extending a quantum mechanical model involving position-dependent mass (PDM) systems, specifically addressing solutions to the Schrödinger equation with PDM within oscillator-shaped quantum wells. The significance of PDM arises in several physical contexts, such as semiconductor physics and quantum dots, where the mass can vary due to compositional or structural changes. The study integrates concepts from supersymmetric quantum mechanics and exceptional orthogonal polynomials (EOPs) to create more generalized models.

Potential Model and Point Canonical Transformation

The model considers a PDM Schrödinger equation characterized by a mass function $M(x)$ and an effective potential $V_{\text{eff}}(x)$. The mass is defined between two walls, creating a confined space for quantum oscillations. This PDM system is transformed via a point canonical transformation (PCT) from a constant-mass scenario using the Scarf I potential. This transformation preserves the spectral properties and provides a way to map known solutions to new PDM configurations.

Rational Extensions and Exceptional Orthogonal Polynomials

The research leverages the framework of exceptional orthogonal polynomials, specifically $X_1$- and $X_2$-Jacobi polynomials, to extend the Scarf I potential. The incorporation of these EOPs allows the construction of rational extensions of the quantum well potential which maintain the spectral properties of the original system. Such extensions introduce additional terms into the potential that modify its shape but preserve the overall eigenvalue structure, enabling a richer variety of quantum phenomena to be explored.

Implementation and Results

To implement the extended models, the study uses explicit formulas for both the potential and the wavefunctions. The energy spectrum remains unchanged, but the wavefunctions now involve $X_m$ Jacobi EOPs, providing a more complex nodal structure. Calculations demonstrate how these extensions affect the localization and distribution of wavefunctions within the potential well.

The PDM models are expressed analytically, highlighting their applicability in quantum systems with non-trivial mass configurations. The technique of using PCT ensures that these solutions are direct extensions of well-known forms, making them easier to apply in practical physics problems.

Conclusion

The research provides a systematic approach to deriving rationally-extended models of PDM systems, expanding the toolkit available for dealing with non-standard quantum mechanical problems. By integrating EOPs, the approach allows for potential extensions that offer deeper insights into the underlying physics of PDM scenarios. This method paves the way for future explorations into more complex systems, potentially impacting fields such as semiconductor technology and quantum computing.

This work underscores the importance of PCT and EOPs in building generalizable quantum mechanical models, facilitating advancements in theoretical and applied physics.