Perturbation foundation of q-deformed dynamicsJian-zu Zhang1, 2
1 Department of Physics, University of Kaiserslautern, P.O. Box 3049, 67653 Kaiserslautern, Germany
2 Institute for Theoretical Physics, Box 316, East China University of Science and Technology, Shanghai 200237, P.R. China
(Received: 6 September 2002 / Revised version: 21 October 2002 / Published online: 14 April 2003 )
In the q-deformed theory the perturbation approach can be expressed in terms of two pairs of undeformed position and momentum operators. There are two configuration spaces. Correspondingly there are two q-perturbation Hamiltonians; one originates from the perturbation expansion of the potential in one configuration space, the other one originates from the perturbation expansion of the kinetic energy in another configuration space. In order to establish a general foundation of the q-perturbation theory, two perturbation equivalence theorems are proved. The first is Equivalence Theorem I: Perturbation expressions of the q-deformed uncertainty relations calculated by two pairs of undeformed operators are the same, and the two q-deformed uncertainty relations undercut Heisenberg's minimal one in the same style. The general Equivalence Theorem II is: for any potential (regular or singular) the expectation values of two q-perturbation Hamiltonians in the eigenstates of the undeformed Hamiltonian are equivalent to all orders of the perturbation expansion. As an example of singular potentials the perturbation energy spectra of the q-deformed Coulomb potential are studied.
© Società Italiana di Fisica, Springer-Verlag 2003