PDM creation and annihilation operators of the harmonic oscillators and the emergence of an alternative PDM-Hamiltonian

Abstract

The exact solvability and impressive pedagogical implementation of the harmonic oscillator creation and annihilation operators make it a problem of great physical relevance and the most fundamental one in quantum mechanics. So would be the position dependent mass oscillator for the position dependent mass quantum mechanics. We, hereby, construct the position dependent mass creation and annihilation operators for the position dependent mass oscillator via two different approaches. First, via von Roos position dependent mass Hamiltonian and show that the commutation relation between the position dependent mass creation and annihilation operators not only offers a unique position dependent mass Hamiltonian but also suggests a position dependent mass deformation in the coordinate system. Next, we use a position dependent mass point canonical transformation of the textbook constant mass harmonic oscillator analog and obtain yet another set of position dependent mass creation and annihilation operators, hence an apparently new position dependent mass Hamiltonian is obtained. The new position dependent mass Hamiltonian turned out to be not only correlated but also represents an alternative and most simplistic user-friendly position dependent mass Hamiltonian that has never been reported before.