Perturbation Foundation of q-Deformed Dynamics

TL;DR

This paper develops a foundational perturbation theory for q-deformed quantum systems, establishing equivalence theorems for uncertainty relations and expectation values across different configuration spaces, with applications to Coulomb potentials.

Contribution

It introduces two perturbation equivalence theorems for q-deformed dynamics, unifying different perturbation approaches and extending to singular potentials.

Findings

Perturbation expressions of q-deformed uncertainty relations are equivalent.

Expectation values of q-perturbation Hamiltonians are equivalent for all potentials.

Application to Coulomb potential demonstrates the theory's effectiveness.

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: (I) Equivalence theorem {\it 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. (II) The general equivalence theorem {\it II}: for {\it 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.