Supersymmetric quantum mechanics and coherent states for a deformed oscillator with position-dependent effective mass

Abstract

We study the classical and quantum oscillator in the context of a non-additive (deformed) displacement operator associated with a position-dependent effective mass by means of the supersymmetric formalism. From the supersymmetric partner Hamiltonians and the shape invariance technique, we obtain the eigenstates and the eigenvalues along with the ladders operators, thus showing a preservation of the supersymmetric structure in terms of the deformed counterpartners. The deformed space in supersymmetry allows to characterize position-dependent effective mass and uniform field interactions and to obtain a generalized uncertainty relation (GUP) that behaves as a distinguishability measure for the coherent states, these latter satisfying a periodic evolution for the corrections of the GUP.

Exact solution and coherent states of an asymmetric oscillator with position-dependent mass

“…61. This work is a continuation of Ref. 61. However, we address here the problem of the deformed oscillator subject to an asymmetric potential (which also includes the effect of PDM), as well as properties of its coherent states.…”

“…Some theoretical studies have been developed with the aim of introducing the effect of a PDM by means of deformed algebraic structures. [56][57][58][59][60][61][62][63] More specifically, the effect of a PDM can be described by a Schrödinger equation where the usual derivative is replaced by a deformed derivative operator, and the Hamiltonian operator is expressed in terms of a deformed linear momentum operator. 56,60,62 For instance, Costa Filho et al 56 introduced a displacement operator that leads to non-additive spatial translations Tγ (ε)|x = |x + (1 + γx)ε , in which γ is a deformation parameter with inverse length dimension.…”

“…Within this approach, a harmonic oscillator with PDM subjected to a quadratic potential has been proposed. 57,58,60,61 The deformed harmonic oscillator presents an anharmonic spectrum, which it can describe diatomic molecules. From factorization methods, the coherent states of the deformed oscillator have been obtained recently in Ref.…”

Darboux transformations for Dunkl-Schroedinger equations with energy dependent potential and position dependent mass

“…In order to find cases of our Dunkl-Schrödinger equation that admits closed-form solutions, we adapt point transformations and Darboux transformations to the present scenario. Both of these techniques have been widely applied to nonrelativistic quantum systems with position-dependent mass, for example in the study of finite gap systems [4], the problem of operator ordering in Hamiltonians [16], and the construction of coherent states [8]. In contrast to these and related applications, to the best of our knowledge point transformations and Darboux transformations have not been applied yet to Dunkl-Schrödinger equations equipped with both a position-dependent mass and an energy-dependent potential.…”

Dynamics of free time-dependent effective mass

“…With the help of the uncertainty relation in x and p, we have shown that the ground state is a squeezed coherent state (CS). Being the minimum uncertainty state, a CS mostly resembles the classical states [64][65][66][67]. On the other hand, the Wigner quasiprobability distributions (WQD) provide a phase-space representation of QM.…”