The Noncommutative Harmonic Oscillator based in Simplectic Representation of Galilei Group

TL;DR

This paper develops a phase space formulation of quantum mechanics using symplectic representations of the Galilei group, deriving the Schrödinger equation and analyzing noncommutative harmonic oscillators with Wigner functions.

Contribution

It introduces a symplectic unitary representation framework for the Galilei group, leading to a novel phase space approach for quantum systems with noncommutative structures.

Findings

Derived the Schrödinger equation in phase space.

Analyzed the 3D harmonic oscillator in noncommutative phase space.

Determined Wigner functions for both standard and noncommutative oscillators.

Abstract

In this work we study symplectic unitary representations for the Galilei group. As a consequence the Schr\"odinger equation is derived in phase space. The formalism is based on the non-commutative structure of the star-product, and using the group theory approach as a guide a physical consistent theory in phase space is constructed. The state is described by a quasi-probability amplitude that is in association with the Wigner function. The 3D harmonic oscillator and the noncommutative oscillator are studied in phase space as an application, and the Wigner function associated to both cases are determined.