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arXiv:1908.02785

Algebraic structures and deformed Schrödinger equations from groups entropies

Ignacio S. Gomez, Ernesto P. Borges

Published Aug 7, 2019• 0 citations
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Abstract

Motivated by the group entropy theory, in this work we generalize the algebra of real numbers (that we called G-algebra), from which we develop an associated G-differential calculus. Thus, the algebraic structures corresponding to the Tsallis and Kappa statistics are obtained as special cases when the Tsallis and Kappa group classes are chosen. We employ the G-algebra to formulate a generalized G-deformed Schrödinger equation and we illustrate it with the infinite potential well, where the effective mass is related with the G-algebra structure and the $q$-deformed (standard) Schrödinger equation results an special case for the Tsallis (Boltzmann-Gibbs) group class. The non-uniform zeros spacing of the G-deformed eigenfunctions is expressed in terms of the generalized sum of the G-algebra.

DOI
10.48550/arxiv.1908.02785

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Motivated by the group entropy theory, in this work we generalize the algebra of real numbers (that we called G-algebra), from which we develop an associated G-differential calculus. Thus, the algebraic structures corresponding to the Tsallis and Kappa statistics are obtained as special cases when the Tsallis and Kappa group classes are chosen. We employ the G-algebra to formulate a generalized G-deformed Schrödinger equation and we illustrate it with the infinite potential well, where the effective mass is related with the G-algebra structure and the qqq-deformed (standard) Schrödinger equation results an special case for the Tsallis (Boltzmann-Gibbs) group class. The non-uniform zeros spacing of the G-deformed eigenfunctions is expressed in terms of the generalized sum of the G-algebra.
The Heisenberg algebra is first deformed with the set of parameters q,l,λ{q, l, λ}q,l,λ to generate a new family of generalized coherent states. In this framework, the matrix elements of relevant operators are exactly computed. A proof on sub-Poissonian character of the statistics of the main deformed states is provided. This property is used to determine a generalized metric. A unified method of calculating structure functions from commutation relations of deformed single-mode oscillator algebras is then presented. A natural approach to building coherent states associated to deformed algebras is deduced. Known deformed algebras are given as illustration. Futhermore, we generalize a class of two-parameter deformed Heisenberg algebras related to meromorphic functions, called R(p,q){\cal R}(p,q)R(p,q)-deformed algebra. Relevant families of coherent states maps are probed and their corresponding hypergeometric series are computed. The latter generalizes known hypergeometric series and gives to a generalization of the binomial theorem. The involved notions of differentiation and integration generalize the usual qqq- and (p,q)(p,q)(p,q)-differentiation and integration. A Hopf algebra structure compatible with the R(p,q){\cal R}(p,q)R(p,q)-algebra is deduced. We succeed in giving a new characterization of Rogers- Szegö polynomials, called R(p,q){\cal R}(p,q)R(p,q)-deformed Rogers-Szegö polynomials, by their three-term recursion relations and the associated quantum algebra built with corresponding creation and annihilation operators. Continuous R(p,q){\cal R}(p,q)R(p,q)-deformed Hermite polynomials and their recursion relation are also deduced. Novel algebraic relations are provided and discussed. The whole formalism is performed in a unified way, generalizing known relevant results which are straightforwardly derived as particular cases.
In this paper, we use the q-derivative emerging in the non-extensive statistical physics to formulate the q-deformed quantum mechanics. We find the algebraic structure related to this deformed theory and investigate some properties of the q-deformed elementary functions. Using this mathematical background, we formulate the q-deformed Heisenberg algebra and q-deformed time-dependent Schrödinger equation. As an example, we deal with the infinite potential well and compute the Fermi energy in the q-deformed theory.
An algebraic approach is used to obtain the canonical form of the position-dependent mass Schrödinger equation from where a couple of canonical quantum variables, the q-deformed operators for the position xq, and the hermitian linear momentum operator pq are derived. In this q-deformed coordinate space, the commutator remains invariant namely [xq, pq] = iħ. By taking advantage of these q-deformed variables, one gets to a q-deformed exponential function expq(x) as well as its corresponding q-deformed logarithm function lnq(x). From these q-deformed mathematical relations and from the fact that thermodynamic properties such as the internal energy U, entropy S, free energy F, heat capacity C, and others are related to the partition function Z and ln(Z), it is proposed their generalizations in terms of the q-deformed exponential and q-deformed logarithmic functions. As a result, the structure of Legendre transformations between these statistical properties remains invariant. The usefulness of the proposal is exemplified by considering two specific position-dependent mass distributions. In the same way, other possibilities could be used to generalize the statistical properties straightforwardly.
Bohmian mechanics and Fisher information for qqq-deformed Schrödinger equation
We discuss the Bohmian mechanics by means of the deformed Schrödinger equation for position dependent mass, in the context of a qqq-algebra inspired by nonextensive statistics. A deduction of the Bohmian quantum formalism is performed by means of a deformed Fisher information functional, from which a deformed Cramér-Rao bound is derived. Lagrangian and Hamiltonian formulations, inherited by the qqq-algebra, are also developed. Then, we illustrate the results with a particle confined in an infinite square potential well. The preservation of the deformed Cramér-Rao bound for the stationary states shows the role played by the qqq-algebraic structure.