q-deformed quantum mechanical evolution equations via a q-deformed Schrödinger algebra

Won Sang Chung, Shi‐Hai Dong, HASSAN HASSANABADI

Reidun Twarock

In an earlier paper a q -Schrödinger equation was obtained based on a particular quantization procedure, called Borel quantization, and a related q -deformation of the Witt algebra. This q -deformation is a deformation in the category of Lie algebras and hence the corresponding q -Witt algebra has a trivial Hopf algebra structure. In this paper, we extend the above algebra by the addition of a set of shift-type generators, which appear in the expression for the quantum mechanical position operator and hence lead to a new type of quantum kinematics. The latter gives rise to a new kind of evolution equation and it is shown that in the limit q 1 a specific class of Schrödinger equations is obtained from it. This specification of a particular class is a new phenomenon, because in earlier references, where a different q -deformation has been implemented or no deformation has been used at all, such a class could not be determined uniquely. The extended algebra used here has a nontrivial Hopf structure. The appearance of the shift-type generator in the q -deformed picture hence leads to a selection of a particular type of dynamics and delivers in the limit q 1 new information for the characterization of the corresponding dynamics in the undeformed situation.

Won Sang Chung, HASSAN HASSANABADI

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.

Bruno G. da Costa, Ernesto P. Borges

M. Micu

Bruno G. da Costa, Ignacio S. Gomez

We discuss the Bohmian mechanics by means of the deformed Schrödinger equation for position dependent mass, in the context of a $q$-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 $q$-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 $q$-algebraic structure.