Perturbative Equivalent Theorem in q-Deformed Dynamics

TL;DR

This paper establishes a general theorem showing the expectation values of two different q-perturbative Hamiltonians are equivalent for regular, singularity-free potentials in q-deformed dynamics, ensuring reliable perturbative calculations.

Contribution

The paper proves a general q-perturbative equivalent theorem for regular potentials, clarifying the relationship between two q-perturbative Hamiltonians in q-deformed quantum mechanics.

Findings

Expectation values of two q-perturbative Hamiltonians are equivalent for regular potentials.

The perturbative Hamiltonian from the kinetic energy does not cause energy shifts for the free particle.

The theorem applies to a broad class of singularity-free potentials in q-deformed systems.

Abstract

Corresponding to two ways of realizing the q-deformed Heisenberg algebra by the undeformed variables there are two q-perturbative Hamiltonians with the additional momentum-dependent interactions, one originates from the perturbative expansion of the potential, the other originates from that of the kinetic energy term. At the level of operators, these two q-perturbative Hamiltonians are different. In order to establish a reliable foundation of the perturbative calculations in q-deformed dynamics, except examples of the harmonic-oscillator and the Morse potential demonstrated before, the general q-perturbative equivalent theorem is demonstrated, which states that for any regular potential which is singularity free the expectation values of two q-perturbative Hamiltonians in the eigenstates of the undeformed Hamiltonian are equivalent. For the q-deformed ``free'' particle case, the perturbative Hamiltonian originated from the kinetic energy term still keeps its general expression, but it does not lead to energy shift.