DOI: 10.1140/epjc/s2003-01086-1
Perturbation foundation of q-deformed dynamics
Jian-zu Zhang1, 21 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 )
Abstract
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
