# Energy-dependent noncommutative quantum mechanics

## TL;DR

This paper introduces an energy-dependent noncommutative quantum mechanics model where noncommutative parameters vary with energy, affecting quantum systems like free particles and oscillators, and derives solutions for these cases.

## Contribution

It presents a novel energy-dependent noncommutative framework and derives the corresponding Schrödinger equation, including solutions and energy spectra for specific systems.

## Key findings

- Energy dependence modifies quantum behavior at high energies.
- Solutions include fractional differential equations in certain models.
- Explicit energy levels are obtained for free particles and harmonic oscillators.

## Abstract

We propose a model of dynamical noncommutative quantum mechanics in which the noncommutative strengths, describing the properties of the commutation relations of the coordinate and momenta, respectively, are arbitrary energy dependent functions. The Schr\"{o}dinger equation in the energy dependent noncommutative algebra is derived for a two dimensional system for an arbitrary potential. The resulting equation reduces in the small energy limit to the standard quantum mechanical one, while for large energies the effects of the noncommutativity become important. We investigate in detail three cases, in which the noncommutative strengths are determined by an independent energy scale, related to the vacuum quantum fluctuations, by the particle energy, and by a quantum operator representation, respectively. Specifically, in our study we assume an arbitrary power law energy-dependence of the noncommutative strength parameters, and of their algebra. In this case, in the quantum operator representation, the Schr\"{o}dinger equation can be formulated mathematically as a fractional differential equation. For all our three models we analyze the quantum evolution of the free particle, and of the harmonic oscillator, respectively. The general solutions of the noncommutative Schr\"{o}dinger equation as well as the expressions of the energy levels are explicitly obtained.