Removal of ordering ambiguity for a class of position dependent mass quantum systems with an application to the quadratic Li\'enard type nonlinear oscillators

TL;DR

This paper develops a method to eliminate ordering ambiguity in position-dependent mass quantum systems, enabling exact solutions for both Hermitian and non-Hermitian cases, with applications to nonlinear oscillators.

Contribution

It introduces a point canonical transformation approach to remove ordering ambiguity in position-dependent mass Hamiltonians, including non-Hermitian systems, and applies it to quadratic Liénard oscillators.

Findings

Order ambiguity can be removed for certain mass functions and potentials.

Both Hermitian and non-Hermitian systems can be exactly solvable and iso-spectral.

Eigenfunctions can be normalized for both types of orderings.

Abstract

We consider the problem of removal of ordering ambiguity in position dependent mass quantum systems characterized by a generalized position dependent mass Hamiltonian which generalizes a number of Hermitian as well as non-Hermitian ordered forms of the Hamiltonian. We implement point canonical transformation method to map one-dimensional time-independent position dependent mass Schr$\"{o}$dinger equation endowed with potentials onto constant mass counterparts which are considered to be exactly solvable. We observe that a class of mass functions and the corresponding potentials give rise to solutions that do not depend on any particular ordering, leading to the removal of ambiguity in it. In this case, it is imperative that the ordering is Hermitian. For non-Hermitian ordering we show that the class of systems can also be exactly solvable and are also shown to be iso-spectral using suitable similarity transformations. We also discuss the normalization of the eigenfunctions obtained from both Hermitian and non-Hermitian orderings. We illustrate the technique with the quadratic Li$\'{e}$nard type nonlinear oscillators, which admit position dependent mass Hamiltonians.