Extended Wigner function for the harmonic oscillator in the phase space

Abstract

New time dependent extended Wigner functions for the quantum harmonic oscillator have been obtained in this work. The Moyal equation for the harmonic oscillator has been associated with the wave equation of a 2D membrane in the phase plane. The values of the extended Wigner function are equal to the deviation values of the points on the surface of the membrane from the equilibrium state. The positive and negative values of the extended Wigner function correspond to the direction of the deviation from the equilibrium state. As an example, a time dependent extended Wigner function corresponding to the standing wave of quasi-probability density arising in the phase plane is considered.

Keywords

• Wigner function
• Harmonic oscillator
• Wave equation
• Exact solution
• Moyal equation
• Rigorous result