One-Dimensional Potentials in q Space

Abstract

abstract. We study the one dimensional potentials in q space and the new features that arise. In particular we show that the probability of tunneling of a particle through a barrier or potential step is less than the one of the same particle with the same energy in ordinary space which is somehow unexpected. We also show that the tunneling time for a particle in q space is less than the one of the same particle in ordinary space. Key words. Q spaces, Perturbation theory.Pacs number. 03.65.-w, 02.40.Gh 1 Introduction. Quantum groups are a generalization of the concept of symmetries [1][2][3][4][5]. The mathematical theory of quantum groups arose as an abstraction from constructions developed in the frame of the inverse scattering method of solution of quantum integrable models, but because of its rich and powerful structure, it has been applied to different problems far beyond the original area. quantum groups act on noncommutative spaces. If the space structure at short distances (much smaller than 10 −18 cm, according to the present test of quantum electrodynamics, the usual Heisenberg's commutation relations are correct at least down to 10 −18 cm) shows a noncommutative property and thus governed by quantum group symmetry, then quantum mechanics based on q-deformed Heisenberg algebra is a possible candidate for quantum theory to study the phenomena at short distances . Different frameworks of q−deformed Heisenberg algebra have been established [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22], but physically, the one presented in Refs. [13,17] is more clear : its relation to the corresponding q−deformed boson commutation relations and the limiting process of the q−deformed harmonic oscillator to the undeformed one are clear. By the results of string theory arguments, some applications of quantum mechanics on a non-commutative plane has been studied in [27]. Perturbative aspects of the schroedinger equation in q space has been studied in [23]. There are two perturbative Hamiltonians corresponding to two ways of realizing the q−deformed Heisenberg algebra by the undeformed variables(which are related by the cononical transformation): One includes it in the kinetic energy term, and the other includes it in the potential.