Classical and quantum mechanics in the generalized non-commutative plane

¹  Key Lab for Nanomaterials, Ministry of Education, Beijing University of Chemical Technology Beijing 100029, PRC
²  Department of Physics and Electronic, School of Science, Beijing University of Chemical Technology Beijing 100029, PRC

Corresponding author: jingjian@mail.buct.edu.cn

Received: 25 August 2008
Accepted: 12 November 2008

Abstract

A particle confined by a quadratic potential in the generalized non-commutative plane is studied from both the classical and the quantum aspects. We show that there are two parameters in the full theory which may lead to two different reduced theories when these two parameters take zero value, respectively. The classical aspect has the continuous limit when both of the two parameters take zero value in the full theory. However, the quantum aspect of the full theory behaves interestingly. The spectra of the full theory have the continuous limit when one of the parameters takes zero value while they do not when the other parameter, say, the mass of the particle, takes the same value. The spectra of the full theory will be divergent when the mass tends to zero. In order to get a finite result, we must regularize the spectra properly. Finally, we resort to the constrained theories to verify that the regularization we made is reasonable.