[Paper Review] Wigner Functions for harmonic oscillator in noncommutative phase space

Jianhua Wang, Kang Li+1 more|ArXiv.org|Aug 12, 2009

Mathematical Analysis and Transform Methods3 citations

TL;DR

This paper derives the Wigner function for a 2D harmonic oscillator in noncommutative (NC) phase space using a generalized Bopp's shift method, avoiding complex star-product calculations. The key result is an analytical expression for the Wigner function that includes corrections from both spatial and phase-space noncommutativity, with explicit dependence on NC parameters θ and θ̄, showing how noncommutativity distorts the phase-space distribution beyond standard quantum mechanics.

ABSTRACT

We study the Wigner Function in non-commutative quantum mechanics. By solving the time independent Schrödinger equation both on a non-commutative (NC) space and a non-commutative phase space, we obtain the Wigner Function for the harmonic oscillator on NC space and NC phase space respectively.

Motivation & Objective

• To extend the phase-space formulation of quantum mechanics to noncommutative phase space, where both position and momentum coordinates are noncommutative.
• To compute the Wigner function for the 2D harmonic oscillator in noncommutative phase space, a system of fundamental physical interest.
• To provide a practical method—via generalized Bopp's shifts—for computing Wigner functions in noncommutative quantum mechanics without direct star-product evaluation.
• To quantify the influence of noncommutativity parameters θ and θ̄ on the phase-space distribution, particularly in the ground and excited states.

Proposed method

• The Wigner function is derived using the standard definition involving a Fourier transform of the wave function, adapted for noncommutative phase space.
• A generalized Bopp's shift is applied to map noncommutative operators to commutative phase-space variables, replacing star products with standard products.
• The method assumes α = 1, simplifying the transformation while retaining leading-order noncommutative corrections in θ and θ̄.
• The harmonic oscillator Hamiltonian is transformed using the Bopp shifts, and the resulting Wigner function is expressed in terms of associated Laguerre polynomials.
• The noncommutative corrections appear as modified arguments in the exponential and Laguerre functions, encoding the effects of θ and θ̄.
• The final Wigner function is obtained by substituting the shifted variables into the standard harmonic oscillator Wigner function, with corrections up to O(θ, θ̄).

Experimental results

Research questions

• RQ1 How does noncommutativity in phase space modify the Wigner function of a 2D harmonic oscillator compared to the commutative case?
• RQ2 Can the Bopp's shift method be generalized to noncommutative phase space to avoid direct computation of star products?
• RQ3 What is the explicit analytical form of the Wigner function for the harmonic oscillator in noncommutative phase space, including noncommutativity parameters θ and θ̄?
• RQ4 How do the noncommutative corrections affect the ground state and excited state Wigner functions?
• RQ5 What is the physical interpretation of the modified phase-space distribution, particularly in terms of angular momentum-like terms involving x₁p₂ − x₂p₁?

Key findings

• The Wigner function for the harmonic oscillator on noncommutative phase space is derived as a modification of the standard form, with noncommutative corrections encoded in the arguments of the exponential and Laguerre polynomials.
• The ground state Wigner function is given by W₀₀ⁿᶜᵖˢ = (πħ)⁻² exp[−(x₁² + p₁² + x₂² + p₂² − (θ + θ̄)/ħ (x₁p₂ − x₂p₁))/ħ], showing a phase-space distortion due to noncommutativity.
• The noncommutative corrections appear as a term proportional to (x₁p₂ − x₂p₁), indicating a coupling between spatial and momentum noncommutativity and orbital angular momentum.
• For excited states, the Wigner function includes associated Laguerre polynomials with arguments modified by O(θ, θ̄) terms, reflecting energy-level shifts and distribution deformations.
• The method successfully avoids direct star-product computation by using Bopp's shifts, providing a practical and efficient route to Wigner functions in noncommutative phase space.
• The results indicate that noncommutativity effects are expected to be observable only at very high energy scales, consistent with quantum gravity and string theory expectations.