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 + and annihilation operators, [Â, + ] = 1 ⇔ + − 1/2 = + + 1/2, not only offers a unique PDM-Hamiltonian H1 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 creationB + and annihilationB operators, hence an "apparently new" PDM-HamiltonianĤ2 is obtained. The "new" PDM-HamiltonianĤ2 turned out to be not only correlated withĤ1 but also represents an alternative and most simplistic user-friendly PDM-Hamiltonian, H = p/ 2m (x) 2 + V (x) ;p = −i ∂x, that has never been reported before.

κ-Deformed quantum and classical mechanics for a system with position-dependent effective mass

“…2 Further studies in noncommuting quantum spaces led to a Schrödinger equation with a position-dependent effective mass (PDM). 3 Along the last decades, the PDM systems have attracted attention because of their wide range of applicability in semiconductor theory, [4][5][6][7] nonlinear optics, 8 quantum liquids, 9,10 inversion potential for NH 3 in density functional theory, 11 particle physics, 12 many body theory, 13 molecular physics, 14 Wigner functions, 15 relativistic quantum mechanics, 16 superintegrable systems, 17 nuclear physics, 18 magnetic monopoles, 19,20 astrophysics, 21 nonlinear oscillations, [22][23][24][25][26][27][28][29][30][31] factorization methods and supersymmetry, [32][33][34][35][36] coherent states, [37][38][39] etc.…”

Investigation of the Infinite Discrete Levels from Finite Discrete Levels in Position Dependent Mass Quantum Systems

“…In this paper we have shown that DDL quantum systems can reflect infinite distinct discrete energy levels under suitable form of position dependent mass. The model mass [15] and used by others [1,3,7,9]. It is worth mentioning that only this typical form of T can be derived [8].…”

“…In fact, for N=1 model Hamiltonian was also previously proposed by Mathews and Lakshnman for the study of nonlinear analysis in view of its potential applications in quantum field theory [4]. As such the Hamiltonian is not self-adjoint(H P DM ̸ = H † P DM ) [1,3,7,9,[13][14][15]. In its self adjoint form, it is expressed as [15]:…”

Vibrational resonances in driven oscillators with position-dependent mass

“…As mentioned in the Introduction, §1, the nature of the problem or potential function considered determines the type of mass variation functions to be employed [14]. For instance, m ( x ) can be a quadratic or exponential function of position x [20,102,103]. The former has been classified on the basis of its singularity property: as either regular mass-functions without singularity or as singular mass-functions with single or dual singularities [20].…”