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arXiv:2106.08467 · Journal of Mathematical Physics

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

Bruno G. da Costa, Genilson A. C. da Silva, Ignacio S. Gomez

Published Sep 1, 2021• 16 citations
arXivHTMLPDF

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.

DOI
10.1063/5.0058920

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We consider the application of the supersymmetric quantum-mechanical formalism to the Schr\"odinger equation describing a particle characterized by a position-dependent effective mass m(x).m(x).m(x). We show that any one-dimensional quantum system with effective mass has a supersymmetric partner system characterized by the same position dependence of the mass, but with a new potential function. The form of this supersymmetric partner potential V2(x){V}_{2}(x)V2​(x) depends on both the form of the original potential V1(x){V}_{1}(x)V1​(x) and the form of the mass x dependence. We also generalize the concept of shape invariance to the nonconstant mass scenario. As illustrative examples we provide, for a given form m(x)m(x)m(x) of the effective mass, shape-invariant potentials exhibiting (a) harmonic-oscillator-like spectra and (b) Morse-like spectra. In both cases the energy eigenvalues and eigenfunctions can be obtained in algebraic fashion.
In the context of a two-parameter (α, β) deformation of the canonical commutation relation leading to nonzero minimal uncertainties in both position and momentum, the harmonic oscillator spectrum and eigenvectors are determined by using an extension of the techniques of conventional supersymmetric quantum mechanics (SUSYQM) combined with shape invariance under parameter scaling. The resulting supersymmetric partner Hamiltonians correspond to different masses and frequencies. The exponential spectrum is proved to reduce to a previously found quadratic spectrum whenever one of the parameters α, β vanishes, in which case shape invariance under parameter translation occurs. In the special case where α = β ≠ 0, the oscillator Hamiltonian is shown to coincide with that of the q -deformed oscillator with q > 1 and its eigenvectors are therefore n - q -boson states. In the general case where 0 ≠ α ≠ β ≠ 0, the eigenvectors are constructed as linear combinations of n - q -boson states by resorting to a Bargmann representation of the latter and to q -differential calculus. They are finally expressed in terms of a q -exponential and little q -Jacobi polynomials.
A nonlinear oscillator with variable mass is studied in the approach of Supersymmetric quantum mechanics. Ladder operators in association with the shape invariance condition allowed us to find the coherent states of the system in the sense of Barut and Girardello. The statistical properties of these particular states are studied using the probability of distribution. In addition, the vibrational partition function is calculated. We see that the thermodynamic functions of the system such as mean energy and entropy depend on the nonlinearity parameter, except for the specific heat.
In this paper, we construct and analyze a class of squeezed coherent states within the framework of supersymmetric quantum mechanics (SUSYQM) involving a position-dependent mass (PDM). Using a deformed algebraic structure, we generalize the creation and annihilation operators to accommodate spatially varying mass profiles. The resulting states exhibit non-classical features, such as squeezing, coherence, and modified uncertainty relations, strongly influenced by both the deformation parameters and the mass function. We explore their physical properties through expectation values, variances, and probability densities. This work provides a pathway toward extending coherent state theory to more complex quantum systems with geometrical and algebraic richness.
A new method for generating exactly solvable Schrödinger equations with a position-dependent mass is proposed. It is based on a relation with some deformed Schrödinger equations, which can be dealt with by using a supersymmetric quantum mechanical approach combined with a deformed shape-invariance condition. The solvability of the latter is shown to impose the form of both the deformed superpotential and the position-dependent mass. The conditions for the existence of bound states are determined. A lot of examples are provided and the corresponding bound-state spectrum and wavefunctions are reviewed.