Etats cohérents de l'oscillateur harmonique inversé et Angle de Hannay pour le champ de Yang-Mills

Abstract

This work is made of two parts; In the first one we have determine the linear and Hermitian invariant of the inverted oscillator then solving the eigenvalues equation and calculating the phase as well as the solution of the Schrödinger equation of the system. This solution in the form of a Gaussian wave packet allows us to define the coherent states of the inverted oscillator. These coherent states reproduce the classical evolution, they are eigenstates of the annihilation operator and they are obtained by applying the displacement operator to the ground state.In the second part, we used the fact that the evolution equation of the Yang-Mills field is that of a particle confined in a quartic potential. The adiabatic evolution allows us to study the classical evolution of the quartic oscillator and introduce the notion geometric angle of this system.