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'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 (A) over cap (+) and annihilation A operators, [(A) over cap, (A) over cap (+)] = 1 double left right arrow (A) over cap(A) over cap (+) - 1/2 = (A) over cap (+) (A) over cap + 1/2, not only offers a unique PDM-Hamiltonian hi 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 (B) over cap (+) and annihilation (B) over cap operators, hence an apparently new PDM-Hamiltonian (H) over cap (2) is obtained. The new PDM-Hamiltonian (H) over cap (2) turned out to be not only correlated with (H) over cap (1) but also represents an alternative and most simplistic user-friendly PDM-Hamiltonian, (H) over cap = ((p) over cap/root 2m(x))(2) + V (x); (p) over cap = -i (h) over bar partial derivative(x), that has never been reported before. (C) 2020 Elsevier B.V. All rights reserved.