Time-Dependent Harmonic Oscillator and the Wigner Function

Mar Sánchez–Córdova, Jasel Berra–Montiel, Alberto Molgado +1 more

Seoktae Koh

Yuta Nasuda

Jean Michel Sellier, Ivan Dimov

E. E. Perepelkin, Б. И. Садовников, Наталья Германовна Иноземцева +1 more

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.

G. Schrade, Vladimir I. Man’ko, Wolfgang P. Schleich +1 more

The authors review the theory of the harmonic oscillator with time-dependent frequency by means of an approach based on an operator constant of the motion. With the help of this operator constant we define the ground state, the excited states and a coherent state of the oscillator and discuss the time dependence of these states through their Wigner functions. The authors derive the Wigner function of an arbitrary state at time t evolving in the time-dependent harmonic potential. Moreover, they calculate the correlation coefficient between position and momentum, which appears in the Schrodinger uncertainty relation. The authors illustrate their results for the example of a charged particle in the Paul trap.