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

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$\ddot{o}$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$\acute{e}$nard type nonlinear oscillators, which admit position dependent mass Hamiltonians.

Submitted for publication in J. Math. Phys. (2014)

References in corpus (3)

• Exact quantization of a PT-symmetric (reversible) Liénard-type nonlinear oscillator
• Point Canonical Transformation versus Deformed Shape Invariance for Position-Dependent Mass Schrödinger Equations
• Classification Scheme for Kinetic Energy Operators with Position-Dependent Mass

Cited by in corpus (5)

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• Quantum solvability of a general ordered position dependent mass system: Mathews-Lakshmanan oscillator
• Exact solution and coherent states of an asymmetric oscillator with position-dependent mass
• The Wigner function of a semiconfined harmonic oscillator model with a position-dependent effective mass
• Liénard Type Nonlinear Oscillators and Quantum Solvability