The Generalized Uncertainty Principle and Harmonic Interaction in Three Spatial Dimensions

Few-Body Syst (2015) 56:1927

DOI 10.1007/s00601-014-0910-7

Received: 15 March 2014 / Accepted: 29 September 2014 / Published online: 8 November 2014 Springer-Verlag Wien 2014

Abstract In three spatial dimensions, the generalized uncertainty principle is considered under an isotropic harmonic oscillator interaction in both non-relativistic and relativistic regions. By using novel transformations and separations of variables, the exact analytical solution of energy eigenvalues as well as the wave functions is obtained. Time evolution of the non-relativistic region is also reported.

1 Introduction

The old wave equations of quantum mechanics such as Schrdinger and KleinGordon equations have become the subject of quite recent researches because of the implications of the quantum gravity and string theory [14] which imply modications to the equations. In more precise words, these theories imply the modication of our common Heisenberg commutation relation into a generalized form. Quantum gravity, quantum geometry, string theory, black hole physics and doubly special relativity imply the existence of a minimal length of order of Planck length lpl = hG

c3 1035m [5], where G is the Newton constant, i. e. there is no length below the

latter. In the jargon, the minimal length uncertainty relation is alternatively called the generalized uncertainty principle (GUP). Modication of the uncertainty relation is equivalent to the modication of the wave equation under consideration and as a result, the spectrum and the wave function of the systems are changed [68]. Experimental results of string theory and black holes physics lead authors to re-examine usual uncertainty principle of Heisenberg [915].

Due to the energy scale engaged with the problem, the theoretical studies become more appealing than many other elds. Till now, various novel approaches have been proposed to deal with the problem incorporated with a desired interaction term such as harmonic, Coulomb, Woods-Saxon, etc. [1633]. These papers imply both exact and perturbative schemes to solve the equation. Likewise various novels in the framework of gravity effect have been published. The mentioned papers are limited to one or two spatial dimensions [34,35]. There are also some papers that the authors considered the general form of d-dimensional harmonic oscillator with the framework of minimal length quantum mechanics [28,3640]. In this study, we consider the three-dimensional KleinGordon and Schrdinger equations in the presence of an isotropic...