Abstract
The exact solvability and impressive pedagogical implementation of the harmonic oscillator's 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 (PDM) oscillator for the PDM quantum mechanics. We, hereby, construct the PDM creation and annihilation operators for the PDM oscillator via two different approaches. First, via von Roos PDM Hamiltonian and show that the commutation relation between the PDM creation Aˆ+ and annihilation Aˆ operators, [Aˆ,Aˆ+]=1⇔AˆAˆ+−1/2=Aˆ+Aˆ+1/2, not only offers a unique PDM-Hamiltonian Hˆ1 but also suggests a PDM-deformation in the coordinate system. Next, we use a PDM point canonical transformation of the textbook constant mass harmonic oscillator analog and obtain yet another set of PDM creation Bˆ+ and annihilation Bˆ operators, hence an “apparently new” PDM-Hamiltonian Hˆ2 is obtained. The “new” PDM-Hamiltonian Hˆ2 turned out to be not only correlated with Hˆ1 but also represents an alternative and most simplistic user-friendly PDM-Hamiltonian, Hˆ=(pˆ/2m(x))2+V(x); pˆ=−iħ∂x, that has never been reported before.
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