Impact of the non-canonical approach to the exact solution of the ideal one-dimensional electron gas confined with an anisotropic quantum wire of oscillator-shaped profile

TL;DR

This paper presents an exact analytical solution for a one-dimensional electron gas in an anisotropic quantum wire with position-dependent effective mass, using both canonical and non-canonical approaches, and derives explicit wavefunctions and energy spectra.

Contribution

It provides new exact solutions for the Schrödinger equation with position-dependent mass in an anisotropic quantum wire, including wavefunctions and energy levels expressed via special polynomials.

Findings

Analytical wavefunctions in terms of Laguerre and Gegenbauer polynomials.

Explicit discrete energy spectrum derived for the model.

Comparison of canonical and non-canonical solution approaches.

Abstract

We study an exactly solvable model that can be interpreted as an ideal one-dimensional electron gas confined with an anisotropic quantum wire potential of oscillator-shaped profile. The homogeneous nature of the quantum wire is broken by the introduction of the effective electron mass, which changes with radial distance. We solve the problem described both within the canonical and the non-canonical approach. Analytical expressions of the wavefunctions of the stationary states for both cases in terms of the Laguerre polynomials are obtained, as well as the discrete energy spectrum related to these wavefunctions. Additionally, an exact solution to the angular position part of the position-dependent mass Schr\"odinger equation within the non-canonical approach leads to the angular-part wavefunctions of the even and odd states expressed through the Gegenbauer polynomials. Possible limit relations and special cases are studied too.