Generalized Uncertainty Relations and Quantum Speed Limits

TL;DR

This paper introduces a unified mathematical framework for hybrid quantum mechanics, deriving generalized uncertainty relations and quantum speed limits that encompass standard, q-deformed, and fractional quantum mechanics, with implications for experimental platforms.

Contribution

It develops a rigorous operator formalism combining algebraic deformation and non-locality, deriving new bounds and speed limits that interpolate between different quantum theories.

Findings

Derived exact generalized uncertainty relations for hybrid quantum systems.

Established a quantum speed limit theorem sensitive to deformation and fractional parameters.

Identified experimental signatures distinguishing hybrid quantum effects in various platforms.

Abstract

We propose a mathematically rigorous unified framework for hybrid quantum mechanics that systematically combines algebraic deformation and spatial non-locality within a single operator formalism. By constructing a self-adjoint hybrid kinetic operator through spectral calculus, we derive exact generalized uncertainty relations that interpolate between $q$-deformed and fractional quantum mechanical bounds. Furthermore, we establish a rigorous quantum speed limit theorem for the hybrid Hamiltonian, revealing how deformation parameters, fractional orders, and external potentials tune the fundamental evolution rate of quantum states. We prove that algebraic deformation accelerates coherent dynamics through discrete momentum quantization, while fractional non-locality induces spectral broadening that suppresses evolution speed. The framework recovers standard quantum mechanics, $q$-quantum mechanics, and fractional quantum mechanics as limiting cases, and provides explicit phenomenological signatures for experimental discrimination in trapped-ion, superconducting, and cold-atom platforms.