The q-Deformed Harmonic Oscillator, Coherent States, and the Uncertainty Relation

TL;DR

This paper explores the properties of a q-deformed harmonic oscillator, deriving explicit representations and analyzing how q-deformation affects the uncertainty relation, revealing violations of the standard quantum limit in coherent states.

Contribution

It provides explicit coordinate representations of operators and states for the q-deformed harmonic oscillator, and demonstrates how q-deformation alters the uncertainty relation.

Findings

Ground state uncertainty product equals 1/2, as in standard quantum mechanics.

Coherent states exhibit a violation of the standard uncertainty relation, with the product less than 1/2.

Minimum uncertainty tends to zero near the convergence radius of the q-exponential.

Abstract

For a q-deformed harmonic oscillator, we find explicit coordinate representations of the creation and annihilation operators, eigenfunctions, and coherent states (the last being defined as eigenstates of the annihilation operator). We calculate the product of the coordinate momentum uncertainties in qoscillator eigenstates and in coherent states. For the oscillator, this product is minimum in the ground state and equals 1/2, as in the standard quantum mechanics. For coherent states, the $q$-deformation results in a violation of the standard uncertainty relation; the product of the coordinate- and momentumoperator uncertainties is always less than 1/2. States with the minimum uncertainty, which tends to zero, correspond to the values of $\lambda$ near the convergence radius of the $q$-exponential.