Three examples of quantum dynamics on the half-line with smooth bouncing

Abstract: This article is an introductory presentation of the quantization of the half-plane based on affine coherent states (ACS). The half-plane is viewed as the phase space for the dynamics of a positive physical quantity evolving with time, and its affine symmetry is preserved due to the covariance of this type of quantization. We promote the interest of such a procedure for transforming a classical model into a quantum one, since the singularity at the origin is systematically removed, and the arbitrariness of boundary conditions can be easily overcome. We explain some important mathematical aspects of the method. Three elementary examples of applications are presented, the quantum breathing of a massive sphere, the quantum smooth bouncing of a charged sphere, and a smooth bouncing "dust" sphere as a simple model of quantum Newtonian cosmology.